Qualitative analysis of set-valued differential equations [1st ed] 9783030076436, 9783030076443, 303007644X [PDF]

Zitiervorschau

Anatoly A. Martynyuk

Qualitative Analysis of Set-Valued Differential Equations

Anatoly A. Martynyuk

Qualitative Analysis of Set-Valued Differential Equations

Anatoly A. Martynyuk Institute of Mechanics National Academy of Sciences of Ukraine Kiev, Ukraine

ISBN 978-3-030-07643-6 ISBN 978-3-030-07644-3 (eBook) https://doi.org/10.1007/978-3-030-07644-3 Library of Congress Control Number: 2018968441 Mathematics Subject Classification (2010): 34A34, 34A37, 34A60, 49J24, 49K24, 34D20, 34D40, 34G99, 93B12, 93B27, 93C41, 93D09, 93D30 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Simplicity is the only soil on which we can erect the building of our generalizations. A. Poincaré

Preface

The construction of the theory of dynamic system trajectories is rooted in the classical works by Poincare and Lyapunov. The notion of phase space introduced by Gibbs allows one to consider the motion of a mechanical system as the motion of an image point in some n-dimensional space of configurations or, what is the same, in the phase space with given metric. Darboux treated a dynamical system as a point moving in n-dimensional space. This idea was successively applied in the papers by Hertz, where the trajectories were considered as the geodesic lines. In the papers by Painleve, the trajectories of mechanical systems were studied in the context of the ideas of multidimensional geometry with the use of Euclidean metric. The ideas of multidimensional geometry were extensively applied in the investigation of the trajectories of mechanical systems in the works by Ricci and Levi-Civita, Sing, and Belenkiy and many others. The recent studies of complex mechanical and electrophysical systems have required some development of the “tool” for modeling the processes in such systems. As is known, the main requirement to the mathematical model of a process is an adequate description of the system functioning. One of the approaches developed for such systems is based on the consideration of systems of ordinary differential equations with multivalued right-hand part. One of the factors generating multi-valuedness of the right-hand part is an uncertainty in determination of system parameters. In view of these facts, there arises a necessity to study sets (bundles) of trajectories of system of perturbed motion equations. General requirements to the set of systems of equations are its closedness and self-consistency as well as correctness with respect to the entering parameters. The construction of the theory is at the initial stage, and there are many open problems to be investigated in this area. The solution of some of them is brought to the attention of the readers. The monograph consists of eight chapters, a list of references, and subject index. In the first chapter, we discuss the general properties of equations with a set of trajectories. Here, we propose a procedure for regularizing the set of inaccurate equations and establish sufficient conditions for the existence and uniqueness of the set solutions. In addition, we present estimates of the solutions of the perturbed vii

viii

Preface

motion systems in which the change of the state vector is subject to a generalized derivative. In the second chapter, for the set of equations with generalized derivative, sufficient conditions for various types of boundedness of trajectories and for the stability of a stationary set of trajectories are established. To this end, the scalar and vector Lyapunov functions, constructed on the basis of an auxiliary matrix-valued function, are used. In the third chapter, for the set of discrete-time equations, a comparison principle with matrix Lyapunov function is established and sufficient conditions for the stability of a certain type of stationary solution are obtained. The analysis is carried out on the basis of the matrix Lyapunov function of a special structure. For essentially nonlinear multiply connected difference system with switching, we establish the conditions that guarantee the asymptotic stability of its zero solution under any switching law. In the fourth chapter, for the family of impulse equations, a heterogeneous matrix Lyapunov function is considered, a theorem of the principle of comparison is obtained, and stability conditions for a set of stationary solutions are established. In the fifth chapter, we consider sets of equations with aftereffect and uncertain values of parameters. As a result of regularization of the family of equations according to the scheme adopted in the book, a set of equations with aftereffect is obtained, for which the conditions for the existence of solutions are established, the estimate of the distance between the extreme sets of solutions is given, and the stability conditions for the set of stationary solutions on a finite-time interval are found as well as the damping conditions for the set of trajectories. In the sixth chapter, the families of equations with aftereffect and uncertain values of parameters are considered. As a result of regularization of the family of equations, a new family of equations with aftereffect is obtained, for which conditions for the existence of solutions are established, an estimate of the distance between the extreme sets of solutions is given, and the stability conditions for the set of stationary solutions on a finite-time interval are found as well as the damping conditions for the set of trajectories. The seventh chapter presents the results of dynamic analysis of the family equations with a causal robust operator. The conditions of local and global existence of solutions of the regularized equation are found, an estimate of the funnel for the set of trajectories is given, and the stability conditions for the set of stationary solutions are found. The generalized direct Lyapunov method and the comparison principle with the Lyapunov matrix function are applied. In Chap. 8, for standard form nonlinear equations with generalized derivative, estimates of deviation of the set of exact solutions from the averaged ones are established and the deviation of the set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integration and stability over finite interval are considered as applications.

Preface

ix

In this book, for the first time, we present: 1. A procedure for regularizing a family of equations with respect to an uncertain parameter 2. A technique for estimating the “funnel” solutions of S.A. Chaplygin for the set of trajectories of the families of equations 3. A generalization of the direct Lyapunov method on the basis of the matrix-valued function for a dynamical analysis of families of equations 4. The idea of a pseudo-linear representation of nonlinear integral inequalities with respect to the problems of deviation of the set of trajectories from the equilibrium state 5. The stability conditions for the set of trajectories of the family of difference equations for any switching law This book is designed for the experts working in the field of qualitative analysis of differential and other types of equations. The application of general results given in the book is associated with the need to construct suitable Lyapunov functions on compact convex spaces or the use of classical results of the theory of differential and integral inequalities.

Acknowledgments Acquainted with some chapters of this book are Professor N.A. Izobov, an academician of the National Academy of Sciences of Belarus, and Professors A.Yu. Aleksandrov (Russia), T.A. Burton (USA), I.M. Stamova (USA), and A.S. Vatsala (USA). Their comments and discussion of specific issues have contributed to improving the presentation of the results. Employees of the Processes Stability Department of the S.P. Timoshenko Institute of Mechanics NAS of Ukraine, L.N. Chernetskaya and S.N. Rasshyvalova, have done a great work on the preparation of this manuscript for printing. My heartfelt gratitude to all the scientists and specialists mentioned for the work done. Finally, we are grateful to the editors and production staff of Birkhäuser for their assistance, good ideas, and patience in the publication of this book. Kiev, Ukraine 2018

Anatoly A. Martynyuk

Contents

1

General Properties of Set-Valued Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Elements of Multivalued Analysis . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Regularization of a Family of Uncertain Equations .. . . . . . . . . . . . . . . . . 1.4 Estimates for the Funnel of Family Equations . . .. . . . . . . . . . . . . . . . . . . . 1.5 Existence Conditions for a Set of Trajectories . . .. . . . . . . . . . . . . . . . . . . . 1.6 Monotone Iterative Technique . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Iterative Technique for a Family of Equations . . .. . . . . . . . . . . . . . . . . . . . 1.8 Conditions for Global Existence of Solutions .. . .. . . . . . . . . . . . . . . . . . . . 1.9 Approximate Solution of the Family Equations ... . . . . . . . . . . . . . . . . . . . 1.10 Euler Solutions for the Family of Equations (1.4) . . . . . . . . . . . . . . . . . . . 1.11 Invariance of the Set of Euler Solutions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.12 Deviation of Trajectories from the Equilibrium State. . . . . . . . . . . . . . . . 1.13 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 2 6 8 13 17 20 26 28 29 32 36 44

2 Analysis of Continuous Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Ideas with Many Applications . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Lyapunov-Like Functions and Their Applications .. . . . . . . . . . . . . . . . . . 2.4 Stability of the Set of Stationary Solutions .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Theorems on Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Application of Strengthened Lyapunov Function . . . . . . . . . . . . . . . . . . . . 2.7 Theorems on Boundedness .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8 Differential Equations on Product of Convex Spaces . . . . . . . . . . . . . . . . 2.9 Hyers–Ulam–Rassias Stability of the Set of Equations .. . . . . . . . . . . . . 2.10 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 47 48 53 56 58 65 70 73 79 83

3 Discrete-Time Systems with Switching . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Statement of the Problem.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Structure of Auxiliary Matrix Function.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85 85 86 87 88 xi

xii

Contents

3.5 3.6 3.7 3.8 3.9 3.10

Stability of a Stationary Solution . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 89 Multi-Connected Switched Difference System . .. . . . . . . . . . . . . . . . . . . . 91 Construction of a Comparison System . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 92 Construction of a Common Lyapunov Function .. . . . . . . . . . . . . . . . . . . . 94 Example .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100

4 Qualitative Analysis of Impulsive Equations . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Stability Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Monotone Iterative Technique . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 104 106 108 113 121

5 Stability of Systems with Aftereffect . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Statement of the Problem and Designations .. . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Existence of the Set of Solutions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Funnel for the Set of Trajectories . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Finite-Time Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Damping Time for the Set Trajectories .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

123 123 124 125 126 128 132 134

6 Impulsive Systems with Aftereffect . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Stability of the Set of Stationary Solutions .. . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Funnel for the Set of Uncertain Equations . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Regularization of the Set of Equations with Aftereffect . . . . . . . . . . . . . 6.7 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

135 135 136 137 141 143 145 149

7 Dynamics of Systems with Causal Operator .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Regularization of the Set of Causal Equations . . .. . . . . . . . . . . . . . . . . . . . 7.4 Funnel for the Set of Solutions of Causal Equations .. . . . . . . . . . . . . . . 7.5 Global Existence of Solutions.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.7 Stability Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.8 Hyers–Ulam–Rassias Stability of the Set of Causal Equations.. . . . . 7.9 Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

151 151 152 152 153 156 161 162 167 170

8 Finite-Time Stability of Standard Systems Sets . . . . . .. . . . . . . . . . . . . . . . . . . . 173 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173 8.2 Statement of the Problem.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 174

Contents

8.3 8.4 8.5 8.6

xiii

On Distance Between the Sets of Solutions . . . . . .. . . . . . . . . . . . . . . . . . . . Finite-Time Stability for the Set of Averaged Equations . . . . . . . . . . . . Boundedness of the Set of Solutions of Standard Affine Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Notes and References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

174 181 183 186

Postface .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 191 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197

Chapter 1

General Properties of Set-Valued Equations

The chapter addresses general properties of equations with set trajectories. A regularization procedure is proposed for the set of uncertain equations and sufficient conditions for the existence and uniqueness of solutions are established. Moreover, estimates are given for solutions of perturbed motion of systems, in which the change of the state vector is subject to the generalized derivative.

1.1 Introduction The chapter discusses the general properties of equations with a set of trajectories. Here a regularization procedure for the set of uncertain equations is proposed. Sufficient conditions for the existence and uniqueness of solutions are obtained. Also, we present estimates for the solutions of perturbed motion of systems in which the change of the state vector is subject to the generalized derivative. The families of equations modeling perturbed motion of a physical or other nature system whose parameters are given uncertainly are considered. In the chapter we show the application of the principle of comparison in integral or differential form for the purpose of analysis of the properties of trajectories of both original and intermediate families of equations. The chapter is arranged as follows. Section 1.2 recalls the information necessary for the presentation of data from the multivalued analysis. In Sect. 1.3 we introduce a procedure of regularizing a family of equations with uncertainty parameter. In Sect. 1.4 estimates are given for the funnel of set trajectories of the family of equations. In Sect. 1.5 we establish conditions for the existence of a set of trajectories of regularized equations. Section 1.6 contains the main estimates of the monotone iterative technique. In Sect. 1.7 we present an iterative technique for the family of regularized equations. © Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_1

1

2

1 General Properties of Set-Valued Equations

In Sect. 1.8 we investigate the conditions for the global existence of set trajectories of the family of regularized equations. In Sect. 1.9 we present a theorem on estimation of approximate solution to the family of equations. In Sects. 1.10–1.11 we discuss the Euler sets of trajectories and conditions for their invariance. Section 1.12 provides estimates for the deviation of the set of trajectories from the equilibrium state. The concluding Sect. 1.13 contains bibliographical details and comments to this chapter.

1.2 Elements of Multivalued Analysis Let Rn be an n-dimensional vector space with the norm  · . We designate by Kc (Rn ) a nonempty subset in Rn , containing all nonempty compact convex subsets of Rn ; K(Rn ) contains all nonempty compact subsets in Rn and C(Rn ) is a subset of all nonempty closed subsets in Rn . For λ ∈ R and for the nonempty subsets A and B in the space Rn , we define the operations of summation and multiplication by a scalar value: A + B = {a + b : a ∈ A, b ∈ B}; λA = {λa : a ∈ A}. For the subsets A, B, C ∈ Kc (Rn ) the following correlations hold true: 1. 2. 3. 4. 5. 6. 7.

A +  =  + A = A, where  ∈ Rn is a zero element in Rn ; (A + B) + C = A + (B + C); A + B = B + A; A + C = B + C implies that A = B; λ(A + B) = λA + λB; (λ + μ)A ⊂ λA + μA, where λ, μ ∈ R; 1 · A = A.

Because of nonlinearity of the space Kc (Rn ) there is no notion of difference of two sets, i.e., A +(−A) = . A solution to this problem was proposed by Hukuhara [34]. Let A and B be fixed subsets in Kc (Rn ). If there exists an element C ∈ Kc (Rn ) such that A = B + C, then we will claim that there exists a Hukuhara difference A − B. If in Kc (Rn ) a Hukuhara difference exists, it is unique. The following existence condition for the Hukuhara difference of two subsets A and B from Kc (Rn ) is known. In order that the difference A − B exists, it is necessary and sufficient that the following condition is satisfied. If a ∈ ∂A, then there exists at least one point c such

1.2 Elements of Multivalued Analysis

3

that a ∈ B + c ⊂ A. In general case, note that the existence of the difference A − B for any A, B ∈ Kc (Rn ) does not yield the existence of the difference B − A. For any nonempty subset A of the space Rn we designate its convex shell by co A. If A is convex, then A ⊆ co A, and co A is closed if A is compact. Let x be a point in the space Rn and A be a nonempty subset in Rn . The distance from the point x to the set A is specified by the formula d(x, A) = inf{x − a : a ∈ A}, and ε-neighborhood of the set A is determined as Sε (A) = {x ∈ Rn : d(x, A) < ε}. The closure of ε-neighborhood Sε (A) is the set S ε (A) = {x ∈ Rn : d(x, A) ≤ ε}. Note that the product of a scalar β and a set A is specified by the formula C = βA = {c = βa : a ∈ A}. For nonempty subsets A and B of the space Rn we define the Hausdorff separation of these sets by the formula dH (B, A) = sup{d(b, A) : b ∈ B} or in the equivalent form n

dH (B, A) = inf{ε > 0 : B ⊆ A + εS 1 }, n

where S 1 = S 1 (θ ), θ ∈ Rn is a zero element in Rn . Note that in general case dH (A, B) = dH (B, A). The distance between the nonempty closed subsets A and B of the space Rn is defined by the formula D[A, B] = max{dH (A, B), dH (B, A)} and is called the Hausdorff metric. It is known that (a) D[A, B] ≥ 0 D[A, B] = 0, iff A = B; (b) D[A, B] = D[B, A]; (c) D[A, B] ≤ D[A, C] + D[C, B]

4

1 General Properties of Set-Valued Equations

for any nonempty subsets A, B, and C of the space Rn . We recall that the set Z ∈ Kc (Rn ) that satisfies the relation A = B + Z is called the difference in the sense of Hukuhara of the sets A and B in Kc (Rn ) and is denoted by A−B. The pair (C(Rn ), D) is a complete separable metric space, where K(Rn ) and Kc (Rn ) are closed subsets. Further we assume that for the subsets A, B, C, D ∈ Kc (Rn ) there exist Hukuhara differences A − C; B − D; C − D; A − B. Moreover, the following correlations hold: 1. 2. 3. 4. 5. 6.

(A + B) − (C + D) = (A − C) + (B − D); (A − B) − (C − D) = (A − C) − (B − D); (A − B) − (C − D) = (A − C) + (D − B); (A − D) = (A − B) + (D − D); λ(A − B) = λA − λB, where λ ∈ R; (αA + βB) − (λD + μC) = (α − λ)A + λ(A − D) + (β − μ)B + μ(B − D), if λ ≥ 0, μ ≥ 0, α − λ ≥ 0, β − μ ≥ 0.

For the proof of these correlations, see the paper by de Blasi and Iervolino [22]. Let F be a mapping of the domain Q of the space Rk into the metric space (Kc (Rn ), D) , i.e., F : Q → Kc (Rn ), which is equivalent to the inclusion F (t) ∈ Kc (Rn ) for all t ∈ Q. Such mappings are called multivalued mappings of Q into Rn . If there exists a constant L > 0 such that D[F (t ∗ ), F (t)] ≤ Lt ∗ − t for all (t ∗ , t) ∈ Q, the multivalued mapping F is Lipschitz. Note that the distance d(x, F (t)) of the mapping F (t) from the point x ∈ Rn satisfies the estimate |d(x, F (t)) − d(y, F (t ∗ ))| ≤ x − y + D[F (t), F (t ∗ )] for all x, y ∈ Rn and (t ∗ , t) ∈ Q and is continuous if the mapping F (t) is continuous, or it is Lipschitz continuous if the mapping F (t) is continuous and satisfies Lipschitz condition. The support function of mapping s(·, F (t)) possesses the same properties, the fact follows from the inequality |s(x, F (t ∗ )) − s(y, F (t))| ≤ F (t) x − y + D[F (t ∗ ), F (t)] n

for all (t ∗ , t) ∈ Q and x, y ∈ S 1 = {x ∈ Rn : x ≤ 1}. The selector of the multivalued mapping F (t) from Q into Rn is a one-valued mapping f : Q → Rn such that f (t) ∈ F (t) for all t ∈ Q. If the mapping F : Q → Kc (Rn ) is measurable, it has a measurable selector f : Q → Rn .

1.2 Elements of Multivalued Analysis

5

Let a multivalued function I ⊂ R be given on the interval X : I → Kc (Rn ). The function X is differentiable at the point t ∈ I in the Hukuhara sense (see Hukuhara [34]) if there exists a value DH X(t) ∈ Kc (Rn ) such that the limits   lim [X(t + τ ) − X(t)]τ −1 : τ → 0+ ,   lim [X(t) − X(t − τ )]τ −1 : τ → 0+ exist and both limits are equal to DH X(t). The set DH X(t) is called the Hukuhara derivative of the multivalued mapping X : I → Kc (Rn ) at the point t. Moreover, it is assumed that for the sufficiently small τ > 0 the Hukuhara differences X(t + τ ) − X(t) and X(t) − X(t − τ ) exist. If the multivalued mappings X, Y : I → Kc (Rn ) are differentiable at the point t ∈ I , then 1. the mapping X + Y : I → Kc (Rn ) is differentiable at the point t and DH (X(t) + Y (t)) = DH X(t) + DH Y (t); 2. the mapping λX : I → Kc (Rn ) is differentiable at the point t and DH (λX(t)) = λDH X(t). If for the multivalued mappings X, Y : I → Kc (Rn ) in the neighborhood I ∗ of the point t0 the Hukuhara difference X − Y exists, then the mapping X − Y : I ∗ → Kc (Rn ) is differentiable at the point t and DH (X(t) − Y (t)) = DH X(t) − DH Y (t). For the multivalued function F : [a, b] → Kc (Rn ) the integral is defined as t F (s) ds = F (t).

DH a

Let diam(X(t)) be a diameter of the set X(t) for all t ∈ I . The following result is known. If the multivalued function X : I → Kc (Rn ) is differentiable on I in the Hukuhara sense, the real-valued function t → diam(X(t)), t ∈ I , is nondecreasing on I . This result should be taken into account in the statement of stability problems for a set of trajectories of nonlinear dynamics. Note also that the set of values of the mapping X(t) is constant iff DH X(t) = 0 on I .

6

1 General Properties of Set-Valued Equations

Let the mapping F : [0, 1] → Kc (Rn ) be measurable and integrally bounded. Then the mapping A : [0, 1] → Kc (Rn ), specified by the expression t A(t) =

F (s) ds, 0

for all t ∈ [0, 1], is differentiable in the Hukuhara sense for almost all t ∈ (0, 1) with the Hukuhara derivative DH A(t) = F (t). In the problems of dynamical analysis of the families of perturbed motion equations the Hukuhara and Aumann integrals are used. It is known that if the multivalued mapping F : I → Kc (Rn ) is Hukuhara integrable, it is also Aumann integrable, though in general case the converse is not true.

1.3 Regularization of a Family of Uncertain Equations Let Kc (Rn ) be a family of all nonempty compact and convex subsets of the space Rn , I ⊂ R+ be a finite interval of changes of t and a set of system states be defined as DH X = F (t, X, α), X(t0 ) = X0 , X0 ∈ Kc (Rn ), α ∈ I. Here X(t) ∈ Kc (Rn ) for all t ∈ I , F ∈ C(I × Kc (Rn ) × I, Kc (Rn )) is a multivalued mapping, DH X is a generalized derivative of the set of system states X(t) at time t ∈ I , α ∈ I is an uncertainty parameter of the mapping F , and I is a compact set in the space Rd . Consider a family of perturbed motion equations DH X = F (t, X, α),

X(t0 ) = X0 ∈ Kc (Rn ),

(1.1)

and compute the limiting mappings Fm (t, ·) = co



F (t, ·, α),

α∈I

FM (t, ·) = co



F (t, ·, α),

α∈I

where the symbol co means the closure of convex shell of the corresponding set. We assume that Fm (t, ·) and FM (t, ·) belong to the space Kc (Rn ).

1.3 Regularization of a Family of Uncertain Equations

7

Along with system (1.1) we shall consider the following families of equations: DH Y = Fm (t, Y ),

Y (t0 ) = Y0 ∈ Kc (Rn ),

(1.2)

DH V = FM (t, V ),

V (t0 ) = V0 ∈ Kc (Rn ).

(1.3)

W (t0 ) = W0 ∈ Kc (Rn ),

(1.4)

and

The family of equations DH W = Fβ (t, W ), where Fβ (t, ·) = Fm (t, ·)β + FM (t, ·)(1 − β),

β ∈ [0, 1],

is called a regularized family of equations of uncertain family of equations (1.1). The regularization procedure for the family of equations (1.1) is called correct if for any ε > 0 there exists a 0 < δ(ε) < ε such that the condition D[X0 , W0 ] < δ implies the estimate D[X(t), W (t)] ≤ ε for all t ≥ t0 , for which there exist solutions X(t) and W (t) to the families of equations (1.1) and (1.4). We shall indicate a simple condition for correct regularization of the family of equations (1.1). Assume that for all α ∈ I and β ∈ [0, 1] for the families of equations (1.1) and (1.4) there exists a positive function g(t), integrable on any finite interval, such that (a) D[F (t, X, α), Fβ (t, W )] ≤ g(t)D[X, W ]; t  (b) g(s) ds ≤ ln εδ t0

for all t ≥ t0 , then the regularization of the family of equations with respect to the uncertainty parameter α ∈ I is correct. Remark 1.1 Depending on the problem of investigation of the family of equations (1.1) the condition (a) can be changed, and then one will have other conditions for correct regularization. Further we shall obtain estimates of the distance between the solutions Y (t) and V (t) of the families of equations (1.2) and (1.3) that put upper and lower limit of the set of trajectories X(t) of the family of equations (1.1).

8

1 General Properties of Set-Valued Equations

1.4 Estimates for the Funnel of Family Equations In order to estimate the funnel span, the integral and differential inequalities are employed that form the basis of the comparison principle in the qualitative theory of equations. Definition 1.1 The multivalued mapping Y (t) : J → Kc (Rn ), J ⊂ I , (V (t) : J → Kc (Rn )) is a solution to problem (1.2) ((1.3)) if it is continuously differentiable on J and satisfies Eq. (1.2) ((1.3)) for all t ∈ J . Since the mappings Y (t) and V (t) are continuously differentiable, we have t Y (t) = Y0 +

DH Y (s) ds,

t ∈ J,

(1.5)

DH V (s) ds,

t ∈ J.

(1.6)

t0

t V (t) = V0 + t0

Therefore, in view of (1.2) and (1.3), we get t Y (t) = Y0 +

Fm (s, Y (s)) ds,

t ∈ J,

(1.7)

FM (s, V (s)) ds,

t ∈ J.

(1.8)

t0

t V (t) = V0 + t0

In correlations (1.5)–(1.8) the integral is understood in the sense of Hukuhara (see Hukuhara [34]). Theorem 1.1 For Eqs. (1.2) and (1.3) assume that (1) Fm ∈ C(I × Kc (Rn ), Kc (Rn )), FM ∈ C(I × Kc (Rn ), Kc (Rn )) and there exists a monotone function g(t, w), g ∈ C(I × R+ , R) such that D[Fm (t, y), F (t, X)] + D[F (t, X), FM (t, V )] ≤ g(t, D[Y, X] + D[X, V ]) for all Y, X, V ∈ Kc (Rn ) and for all t ∈ I ; (2) on I there exists a maximal solution r(t, t0 , w0 ) of the scalar equation dw = g(t, w), dt

w(t0 ) = w0 ≥ 0;

1.4 Estimates for the Funnel of Family Equations

9

(3) for the set of solutions Y(t), V(t) of Eqs. (1.2) and (1.3) the initial conditions (t0 , Y0 ) and (t0 , V0 ) are such that D[Y0 , X0 ] + D[X0 , V0 ] ≤ w0 . Then for all t ∈ J ⊂ I the following estimate is valid: D[Y (t), V (t)] ≤ r(t, t0 , w0 ).

(1.9)

Proof Designate m(t) = D[Y (t), X(t)]+D[X(t), V (t)] and choose (X0 , Y0 , V0 ) ∈ Kc (Rn ) so that m(t0 ) = D[Y0 , X0 ] + D[X0 , V0 ] ≤ w0 . In view of the Hausdorff distance properties we arrive at the following sequence of estimates:

t

m(t) = D Y0 +

Fm (t, Y (t))dt, X0 + t0

F (t, X(t))dt t0



t

+ D X0 +

F (t, X(t))dt, V0 + t0

FM (t, V (t))dt t0

t Fm (t, Y (t))dt,

t0



t

t ≤D



t

 F (t, X(t))dt + D[Y0 , X0 ]

t0

t +D

t F (t, X(t))dt,

t0

 FM (t, V (t))dt + D[X0 , V0 ]

t0

t ≤

D [Fm (t, Y (t)), F (t, X(t))] dt + D[Y0 , X0 ] t0

t D [F (t, X(t)), FM (t, V (t))] dt + D[X0 , V0 ].

+ t0

Hence we find that t m(t) ≤ m(t0 ) + t0

 D [Fm (t, Y (t)), F (t, X(t))]

 + D [F (t, X(t)), FM (t, V (t))] dt

10

1 General Properties of Set-Valued Equations

t ≤ m(t0 ) +

g(s, D [Y (s), X(s)] + D [X(s), V (s)]) ds t0

t = m(t0 ) +

g(s, m(s)) ds,

t ∈ J.

(1.10)

t0

Estimate (1.10) and Theorem 1.6.1 from Lakshmikantham et al. [51] imply m(t) ≤ r(t, t0 , w0 )

for all t ∈ J.

Further, taking into account the fact that D [Y (t), V (t)] ≤ D [Y (t), X(t)] + D [X(t), V (t)] , we get estimate (1.9). Estimate (1.9) forms a funnel for the set of solutions X(t) of Eq. (1.1) in terms of Eqs. (1.2) and (1.3), because Fm (t, X) ≤ F (t, X, α) ≤ FM (t, X) for all (t, X) ∈ I × Kc (Rn ) and α ∈ I. Further we shall investigate the problem on estimating the distance between the sets of solutions to the family of equations (1.1) and (1.4). Recall that for Eq. (1.4) the correlation t Wβ (t) = W0 +

Fβ (t, W (s))ds,

t ∈ J,

t0

is valid for all β ∈ [0, 1]. Theorem 1.2 Assume that for Eqs. (1.1) and (1.4) the following conditions hold: (1) Fβ ∈ C(I × Kc (Rn ), Kc (Rn )), F ∈ C(I × Kc (Rn ) × I, Kc (Rn )) and there exists a monotone nondecreasing function g1 (t, w), g1 ∈ C(I × R+ , R) such that D[F (t, X, α), Fβ (t, W )] ≤ g1 (t, D[X, W ]) for all (t, Y, W ) ∈ I × Kc (Rn ) × Kc (Rn ) and for any α ∈ I and β ∈ [0, 1]; (2) there exists a maximal solution r(t, t0 , w0 ) of the scalar equation dw = g1 (t, w), dt

w(t0 ) = w0 ≥ 0;

1.4 Estimates for the Funnel of Family Equations

11

(3) the initial conditions (t0 , X0 ), (t0 , W0 ) for the set of solutions X(t) and Wβ (t) of Eqs. (1.1) and (1.4) are such that D[X0 , W0 ] ≤ w0 . Then for all t ∈ J and β ∈ [0, 1] the following estimate takes place: D[X(t), Wβ (t)] ≤ r(t, t0 , w0 ).

(1.11)

Proof Letting m(t) = D[X(t), Wβ (t)] and performing calculations similar to those in the proof of Theorem 1.1 we arrive at estimate (1.11). Remark 1.2 The monotonicity condition for functions g(t, w) and g1 (t, w) in Theorems 1.1 and 1.2 can be weakened, provided that comparison theorems for differential inequalities are used. Theorem 1.3 Suppose that for Eqs. (1.1) and (1.4) the following conditions are satisfied: (1) in condition (1) of Theorem 1.2 the function g1 (t, w) ∈ C(I × R+ , R) is such that  lim sup (D[X + hF (t, X, α), W + hFβ (t, W )] − D[X, W ])h−1 :  h → 0+ ≤ g1 (t, D[X, W ]) for all t ∈ I , α ∈ I and β ∈ [0, 1]; (2) conditions (2)–(3) of Theorem 1.2 are satisfied for all t ∈ J and β ∈ [0, 1]. Then estimate (1.11) is valid for all t ∈ J and β ∈ [0, 1]. Proof Let m(t) = D[X(t), W (t)] and D[X0 , W0 ] ≤ w0 . For arbitrary small h > 0 there exist the Hukuhara differences X(t + h) − X(t) and W (t + h) − W (t) for all t ∈ J and β ∈ [0, 1]. Therefore, we can compute m(t + h) − m(t) = D[X(t + h), W (t + h)] − D[X(t), W (t)] ≤ D[X(t + h), X(t) + hF (t, X(t), α)] + D[W (t) + hFβ (t, W (t)), W (t + h)] + D[X(t) + hF (t, X, α), W (t) + hFβ (t, W (t))] − D[X(t), W (t)] for any value of β ∈ [0, 1]. Hence we find 1 D + m(t) = lim sup [m(t + h) − m(t)] h→0+ h  ≤ lim sup D[X(t) + hF (t, X, α), W (t) + hFβ (t, W (t))] h→0+

 X(t + h) − X(t) 1 , F (t, X, α) − D[X(t), W (t)] + lim sup D h h→0+ h 

12

1 General Properties of Set-Valued Equations

 1 W (t + h) − W (t) + lim sup D Fβ (t, W (t)), h h→0+ h ≤ g1 (t, D[X, W ]) = g1 (t, m(t))

(1.12)

for all t ∈ J and β ∈ [0, 1]. Applying Theorem 1.5.2 by Lakshmikantham et al. [51] to estimate (1.12) we arrive at estimate (1.11). Further we shall consider Eq. (1.4) and the set 0 ∈ Kc (Rn ) of stationary solutions of (1.4), for which for all t ≥ t0 the relation F (t, 0 , α) = 0 is true for all α ∈ I. The following result holds. Theorem 1.4 For Eq. (1.4) let the following conditions be satisfied: (1) Fβ (t, W ) ∈ C(R+ × Kc (Rn ), Kc (Rn )) for all β ∈ [0, 1]; (2) there exists a function g(t, u) ∈ C(R+ × R+ , R), monotone increasing with respect to u and such that (a) D[Fβ (t, W ), 0 ] ≤ g(t, D[W, 0 ]) for all β ∈ [0, 1] or   (b) lim sup h1 D[W + hFβ (t, W ), 0 ] − D[X, 0 ] ≤ h→0+ g1 (t, D[W, 0 ]),

where the function g1 (t, ·) satisfies conditions of Theo-

rem 1.3; (3) the maximal solution ri (t, t0 , w0 ), i = 1, 2, of the problems dw = g(t, w), dt dv = g1 (t, v), dt

w(t0 ) = w0 ≥ 0, (1.13) v(t0 ) = v0 ≥ 0,

exists for t ≥ t0 . Then under the initial conditions D[W0 , 0 ] ≤ min{w0 , v0 } = u0 the estimate D[Wβ (t), 0 ] ≤ r(t, t0 , u0 ) holds true for all t ≥ t0 and β ∈ [0, 1], where r(t, t0 , u0 ) = max{r1 (t, t0 , w0 ), r2 (t, t0 , v0 )}. The proof of Theorem 1.4 is carried out in the same way as that of Theorem 1.3. Corollary 1.1 In Theorem 1.4 let the function g(t, D[W, 0 ]) = λ(t)D[W, 0 ],

λ(t) > 0.

1.5 Existence Conditions for a Set of Trajectories

13

Then t D[Wβ (t), 0 ] ≤ D[W (t0 ), 0 ] exp

λ(s)ds

(1.14)

t0

for all t ≥ t0 and β ∈ [0, 1]. Estimate (1.14) follows from the fact that for the solution r(t, t0 , w0 ) of Eq. (1.13) the following inequality holds: t r(t, t0 , w0 ) ≤ r(t0 ) exp

λ(s) ds ,

t ≥ t0 .

t0

If in estimate (1.14) λ(t) = −λ = const < 0, then D[Wβ (t), 0 ] ≤ D[W (t0 ), 0 ]e−λ(t −t0) for all t ≥ t0 and β ∈ [0, 1]. If I = [0, ∞), one can easily see that lim D[Wβ (t), 0 ] = 0 and therefore t →∞

the set of solutions of the family of equations (1.4) tends to the stationary solution 0 ∈ Kc (Rn ).

1.5 Existence Conditions for a Set of Trajectories Consider a problem on the existence and uniqueness of solution to the family of equations (1.4) under conditions weaker than the Lipschitz condition. Introduce the designations T = I × B(W0 , a), Tc = I × [0, 2a] and B(W0 , a) = {W ∈ Kc (Rn ) : D[W, W0 ] ≤ a}. The following result is valid. Theorem 1.5 Let the following conditions be satisfied: (1) for all β ∈ [0, 1] Fβ (t, W ) ∈ C(T, Kc (Rn )) and D[Fβ (t, W ), 0 ] ≤ M0 , where M0 = M0 (β) > 0; (2) there exists a function g(t, w) ∈ C(Tc , R+ ), g(t, w) ≤ M1 on Tc and g(t, 0) = 0, which is nondecreasing with respect to w, and such that the equation dw = g(t, w), dt possesses zero solution on I ;

w(t0 ) = 0,

(1.15)

14

1 General Properties of Set-Valued Equations

(3) for any (t, W ) ∈ T the estimate D[Fβ (t, W ), Fβ (t, V )] ≤ g(t, D[W, V ]) holds true for all β ∈ [0, 1]. Then the successive approximations t Wn+1 (t) = W0 +

n = 0, 1, 2, . . . ,

Fβ (s, Wn (s))ds,

(1.16)

t0

exist on I0 = I ∩ [t0 , t0 + η], where η = min{I, a/M},

M = max



 min M0 (β), M1 ,

β∈[0,1]

as continuous functions and converge uniformly to the solution W (t) of problem (1.4) for all β ∈ [0, 1]. Proof For any n = 0, 1, 2, . . . for the elements of sequence (1.16) we have the estimate



t

D[Wn+1 (t), W0 ] = D W0 +

Fβ (s, Wn (s))ds, W0 t0



t =D

Fβ (s, Wn (s))ds, 0 t0

t ≤

  D Fβ (s, Wn (s)), 0 ds

t0

≤ M0 (β)(t − t0 ) ≤ M0 (β)I ≤ a for all β ∈ [0, 1]. Hence it follows that sequence (1.16) is correctly defined on I0 ⊆ I . For the initial problem (1.15) the successive approximations {wn (t)} are defined as w0 (t) = M(t − t0 ), t wn+1 (t) =

g(s, wn (s)) ds t0

1.5 Existence Conditions for a Set of Trajectories

15

for all n = 0, 1, 2, . . . for the values t ∈ I0 . Condition (2) of the theorem implies that 0 ≤ wn+1 (t) ≤ wn (t)

for all t ∈ I0 .

(1.17)

Condition (1.17) and the fact that    dwn     dt  ≤ g(t, wn−1 (t)) ≤ M1 , by virtue of the Ascoli–Arzela theorem, yield the limiting correlation lim wn (t) = n→∞

w(t) uniformly with respect to t ∈ I0 . The function w(t) is the solution of problem (1.15) and w(t) ≡ 0 for all t ∈ I0 . It can be easily seen that t D[W1 (t), W0 ] ≤

D[Fβ (s, W0 ), 0 ]ds ≤ M(t − t0 ) = w0 (t). t0

Let for some k > 1 the estimate D[Wk (t), Wk−1 (t)] ≤ wk−1 (t) hold for all t ∈ I0 . Since t D[Wk+1 (t), Wk (t)] ≤

D[Fβ (s, Wk (s)), Fβ (s, Wk−1 (s))]ds, t0

we get for all β ∈ [0, 1] that t D[Wk+1 (t), Wk (t)] ≤

g(s, D[Wk (s), Wk−1 (s)])ds t0

t ≤

g(s, wk−1 (s))ds = wk (t). t0

From here it follows by induction that D[Wn+1 (t), Wn (t)] ≤ wn (t), for all n = 0, 1, 2, . . ..

t ∈ I0 ,

16

1 General Properties of Set-Valued Equations

Let u(t) = D[Wn+1 (t), Wn (t)], t ∈ I0 . Then we find that D + u(t) ≤ g(t, D[Wn (t), Wn−1 (t)]) ≤ g(t, wn−1 (t))

for all

t ∈ I0 .

Then let n ≤ m and v(t) = D[Wn (t), Wm (t)]. In view of sequence (1.16), we get D + v(t) ≤ D[DH Wn (t), DH Wm (t)] = D[Fβ (t, Wn−1 (t)), Fβ (t, Wm−1 (t))] ≤ D[Fβ (t, Wn (t)), Fβ (t, Wn−1 (t))] + D[Fβ (t, Wn (t)), Fβ (t, Wm (t))] + D[Fβ (t, Wm (t)), Fβ (t, Wm−1 (t))] ≤ g(t, wn−1 (t)) + g(t, wm−1 (t)) + g(t, D[Wn (t), Wm (t)]) ≤ g(t, v(t)) + 2g(t, wn−1 (t)),

t ∈ I0 ,

for any values of β ∈ [0, 1]. Here we take into account the fact that g(t, w) is a nondecreasing function, wm−1 ≤ wn−1 , n ≤ m, and the sequence wn (t) is decreasing. The principle of comparison provides the estimate v(t) ≤ Rn (t)

for all

t ∈ I0 ,

where Rn (t) is a maximal solution to the scalar equation drn = g(t, rn ) + 2g(t, wn−1 (t)), dt

rn (t0 ) = 0,

for all n = 0, 1, 2, . . .. Since 2g(t, wn−1 (t)) → 0 uniformly with respect to t, we have that Rn (t) → 0 with respect to t on I0 as n → ∞. Hence it follows the uniform convergence of the sequence Wn (t) to W (t) for any β ∈ [0, 1]. Further on we shall need the following result. Lemma 1.1 Let Fβ ∈ C(I0 × Kc (Rn ), Kc (Rn )) and Gβ (t, r) =



(D[Fβ (t, W ), 0 ]).

D[W,W0 ]≤r

Assume that the maximal solution R ∗ (t, t0 , 0) = max (rβ∗ (t, t0 , 0)) of the family of β∈[0,1]

equations dw = Gβ (t, w), dt exists for all t ∈ I0 .

w(t0 ) = 0,

(1.18)

1.6 Monotone Iterative Technique

17

Then for the solution W (t) of the family of equations (1.4) the following estimate is true: D[W (t), W0 ] ≤ R ∗ (t, t0 , 0)

for all

t ∈ I0 .

Proof Designate m(t) = D[W (t), W0 ] and compute D + m(t): D + m(t) = D[DH W (t), 0 ] = D[Fβ (t, W ), 0 ]  (D[Fβ (t, W ), 0 ]) = Gβ (t, r). ≤ D[W,W0 ]≤r

By the principle of comparison it follows for Eq. (1.18) that D[W (t), W0 ] ≤ max rβ∗ (t, t0 , 0) = R ∗ (t, t0 , 0). β∈[0,1]

This proves the assertion of Lemma 1.1. Remark 1.3 If conditions (1)–(3) of Theorem 1.5 are satisfied and the solution w(t, t0 , w0 ) of the comparison equation (1.13) exists uniformly with respect to (t0 , w0 ), the solution W (t) of the family of systems (1.4) is continuous with respect to (t0 , w0 ). For the proof of this assertion, Lemma 1.1 and the principle of comparison for scalar equations are applied.

1.6 Monotone Iterative Technique We introduce a partial ordering in the metric space (Kc (Rn ), D). Let K(K 0 ) be a subset of Kc (Rn ) such that for any u ∈ X, X ∈ Kc (Rn ), the condition ui ≥ 0 (ui > 0) is satisfied for i = 1, 2, . . . , n. Besides, K is a cone in Kc (Rn ), and K 0 is its interior. Definition 1.2 For any U and V ∈ Kc (Rn ) we shall write U ≥ V (U > V ) if there exists Z ∈ Kc (Rn ) such that U = V + Z. The ordering U ≤ V (U < V ) is determined in the same way. Definition 1.3 Let the mapping R(t) = max Rβ (t) be a solution of the set of β∈[0,1]

equations (1.4). We shall claim that R(t) is a maximal solution for the set W (t) of solutions to equations (1.4) if for any t ∈ I0 the estimate W (t) ≤ R(t) is true. The minimal solution of the set of equations (1.4) is defined in the same way.

18

1 General Properties of Set-Valued Equations

Theorem 1.6 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the mapping Fβ (t, W ) ∈ C(R+ × Kc (Rn ), Kc (Rn )) and for (W, V ) ∈ Kc (Rn ) Fβ (t, W ) ≤ Fβ (t, V ) for all t ∈ R+ ; (2) for any W, V ∈ C 1 (R+ , Kc (Rn )) the estimates DH W < Fβ (t, W )

and DH V ≥ Fβ (t, V )

are true for all t ∈ R+ and β ∈ [0, 1]; (3) W (t0 ) < V (t0 ). Then for all t ≥ t0 min Wβ (t) < max Vβ (t).

β∈[0,1]

β∈[0,1]

Proof Assume that under conditions (1)–(3) of Theorem 1.6 there exists a pair (t ∗ , β ∗ ) ∈ I0 × [0, 1] such that under condition (3) the equality Wβ ∗ (t ∗ ) = Vβ ∗ (t ∗ ) is true. Then for t0 < t < t ∗ and [0, β ∗ ) ⊂ [0, 1] the estimate Wβ (t) ≤ Vβ (t) is valid. By virtue of condition (1) of Theorem 1.6 we have DH W (t ∗ ) < Fβ ∗ (t, W ) ≤ Fβ ∗ (t, V ) ≤ DH V (t ∗ ). Hence we get DH (Vβ ∗ (t) − Wβ ∗ (t)) > 0

for all t ∈ [t0 , t ∗ ].

Since Vβ ∗ (t) − Wβ ∗ (t) is a nondecreasing function on [t0 , t ∗ ], (Vβ ∗ (t) − Wβ ∗ (t)) > (Vβ ∗ (t0 ) − Wβ ∗ (t0 )) > 0 and therefore Wβ ∗ (t ∗ ) < Vβ ∗ (t ∗ ). This contradicts the assumption made on the existence of the pair (t ∗ , β ∗ ), for which the equality Wβ ∗ (t ∗ ) = Vβ ∗ (t ∗ ) is true. This completes the proof of Theorem 1.6. We shall cite one more result in which condition (2) of Theorem 1.6 is nonstrict. Theorem 1.7 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the mapping Fβ (t, W ) ∈ C(R+ × Kc (Rn ), Kc (Rn )) and for (W, V ) ∈ Kc (Rn ) the inequality Fβ (t, W ) ≤ Fβ (t, V ) is true for all t ∈ R+ and DH W ≤ Fβ (t, W ),

DH V ≥ Fβ (t, V );

(2) for any X, Y ∈ Kc (Rn ) such that X ≥ Y , the estimate Fβ (t, X) ≤ Fβ (t, Y ) + L(X − Y ) is valid for any β ∈ [0, 1] and t ∈ R+ , where L > 0;

1.6 Monotone Iterative Technique

19

(3) W (t0 ) ≤ V (t0 ). Then for all t ≥ t0 min Wβ (t) < max Vβ (t).

β∈[0,1]

β∈[0,1]

 = V + βe2Lt , where β ∈ [0, 1], L = const > 0. Since Proof Consider the set V (t0 ), it is sufficient to show that W (t0 ) ≤ V (t0 ) < V β (t) Wβ (t) < V

for all t > t0

and any β ∈ [0, 1]. Let there exist a pair (t ∗ , β ∗ ) ∈ R+ ×[0, 1] such that under the condition W (t0 ) < (t0 ) the equality Wβ ∗ (t ∗ ) = V β ∗ (t ∗ ) is true. Then for t0 < t < t ∗ and [0, β ∗ ) ⊂ V [0, 1] the estimate Wβ (t) < Vβ (t) is true. Condition (1) of Theorem 1.7 implies (t ∗ )) DH W (t ∗ ) ≤ Fβ ∗ (t ∗ , W (t ∗ )) ≤ Fβ ∗ (t ∗ , V  − V ) ≤ DH V (t ∗ ) + Lβe2Lt ≤ Fβ ∗ (t ∗ , V (t ∗ )) + L(V

∗

∗

(t ∗ ). ≤ DH V (t ∗ ) + 2Lβe2Lt = DH V Hence we have β ∗ (t)) > 0 for all DH (Vβ ∗ (t) − V

t ∈ [t0 , t ∗ ].

β ∗ (t) is nondecreasing on [t0 , t ∗ ], Since the function Wβ ∗ (t) − V β ∗ (t)) > (Wβ ∗ (t0 ) − V β ∗ (t0 )) > 0, (Wβ ∗ (t) − V and hence it follows that β ∗ (t ∗ ) < Vβ ∗ (t ∗ ). W This inequality contradicts the assumption on the existence of a pair (t ∗ , β ∗ ), for β ∗ (t ∗ ). Theorem 1.7 is proved. which Wβ ∗ (t ∗ ) = V Corollary 1.2 For any β ∈ [0, 1] let Fβ (t, W ) = σ , σ ∈ C(R+ , Kc (Rn )), and DH W ≤ σ

and

DH V ≥ σ

for all

t ≥ t0 .

Then if condition (3) of Theorem 1.7 is satisfied, the estimate W (t) ≤ V (t) holds true for all t ≥ t0 .

20

1 General Properties of Set-Valued Equations

1.7 Iterative Technique for a Family of Equations The set of equations (1.1) contains the uncertainty parameter α ∈ I and that is why it is impossible to directly apply the monotone iterative technique proposed in the monograph by Lakshmikantham et al. [49] for the estimation of solutions. We transform the set of equations (1.1) to the form DH X = Fβ (t, X) + G(t, X, α), X(0) = X0 ,

X0 ∈ Kc (Rn ).

(1.19)

Here Fβ (t, X) = Fm (t, X)β + FM (t, X)(1 − β), and we assume that Hukuhara difference G(t, X, α) = F (t, X, α) − Fβ (t, X) for all α ∈ I exists. We shall further assume that Fβ ∈ C(I × Kc (Rn ), Kc (Rn )) for all β ∈ [0, 1] and G ∈ C(I × Kc (Rn ) × I, Kc (Rn )), where I ⊂ [0, T ]. For the set of equations (1.19) it is of interest to investigate the pairs of upper and lower solutions for β ∈ [0, 1] and any α ∈ I. Definition 1.4 Let V , W ∈ C 1 (I, Kc (Rn )). The sets V and W are called: (a) natural lower and upper solutions of the family of equations (1.19) if DH V ≤ Fβ (t, V ) + G(t, V , α), DH W ≥ Fβ (t, W ) + G(t, W, α)

(1.20)

for all β ∈ [0, 1] and any α ∈ I and t ∈ I ; (b) coupled pair of lower and upper solutions of the family of equations (1.19) of type I if DH V ≤ Fβ (t, V ) + G(t, W, α), DH W ≥ Fβ (t, W ) + G(t, V , α)

(1.21)

for all β ∈ [0, 1] and any α ∈ I and t ∈ I ; (c) coupled pair of lower and upper solutions of the family of equations (1.19) of type II if DH V ≤ Fβ (t, W ) + G(t, V , α), DH W ≥ Fβ (t, V ) + G(t, W, α) for all β ∈ [0, 1], and any α ∈ I t ∈ I ;

(1.22)

1.7 Iterative Technique for a Family of Equations

21

(d) coupled pair of lower and upper solutions of the family of equations (1.19) of type III if DH V ≤ Fβ (t, W ) + G(t, W, α), DH W ≥ Fβ (t, V ) + G(t, V , α) for all β ∈ [0, 1] and any α ∈ I and t ∈ I . Note that if V (t) ≤ W (t) for all t ∈ I , and if the mapping Fβ (t, X) for all β ∈ [0, 1] is nondecreasing with respect to X for any t ∈ I and the mapping G1 (t, Y ) is nonincreasing with respect to Y for any t ∈ I , then the pairs of lower and upper solutions defined in cases (a) and (d) are reduced to definition (b). Hence it is sufficient to consider the cases specified by inequalities (1.21) and (1.22). Theorem 1.8 Assume that: (1) there exists a pair of lower and upper solutions satisfying inequalities (1.21) and, moreover, Vβ (t) < Wβ (t) for all t ∈ I ; (2) for any β ∈ [0, 1] the mapping Fβ ∈ C(I × Kc (Rn ), Kc (Rn )) and Fβ is nondecreasing with respect to X for every t ∈ I ; (3) for any α ∈ I the mapping G ∈ C(I × Kc (Rn ) × I, Kc (Rn )) and G(t, Y, α) is nonincreasing with respect to Y for any t ∈ I ; (4) the mappings Fβ and G map the bounded sets into the bounded sets in Kc (Rn ) for any β ∈ [0, 1] and α ∈ I. Then there exist monotone sequences {Vn (t)} and {Wn (t)} in Kc (Rn ) such that Vn (t) → P (t) and Wn (t) → Q(t) as n → ∞. The pair (P (t), Q(t)) is a pair of minimal and maximal solutions for the family of equations (1.19) and, besides, DH P (t) = Fβ (t, P ) + G(t, Q, α), DH Q(t) = Fβ (t, Q) + G(t, P , α) for any β ∈ [0, 1] and all α ∈ I. Proof Let β ∈ [0, 1] and α ∈ I. Consider a pair of mappings (Vn+1 (t), Wn+1 (t)), solving for n ≥ 0 the family of equations DH Vn+1 = Fβ (t, Vn ) + G(t, Wn , α), DH Wn+1 = Fβ (t, Wn ) + G(t, Vn , α) under the initial conditions Vn+1 (0) = U0 ∈ Kc (Rn ), Wn+1 (0) = U0 ∈ Kc (Rn ),

22

1 General Properties of Set-Valued Equations

where V0 ≤ U0 ≤ W0 . We shall prove the sequence of inequalities V0 ≤ V1 ≤ V2 ≤ . . . ≤ Vn ≤ Wn ≤ . . . ≤ W2 ≤ W1 ≤ W0 for all t ∈ I and β ∈ [0, 1], α ∈ I. For V0 ≤ W0 conditions (1) and (2) of Theorem 1.8 yield DH V0 ≤ Fβ (t, V0 ) + G(t, W0 , α),

(1.23)

and for n = 0 correlation (29) implies DH V1 = Fβ (t, V0 ) + G(t, W0 , α).

(1.24)

By Theorem 1.7 we get from inequality (1.23) that V0 ≤ V1 for all t ∈ I and β ∈ [0, 1]. By similar reasoning it is easy to show that W1 ≤ W0 for all t ∈ I and β ∈ [0, 1]. Further we shall prove that V1 ≤ W1 for all t ∈ I . Consider a family of equations DH V1 = Fβ (t, V0 ) + G(t, W0 ), DH W1 = Fβ (t, W0 ) + G(t, V0 ),

(1.25)

V1 (0) = W1 (0) = U0 . By virtue of conditions (2) and (3) of Theorem 1.8 we get from (1.25) the inequalities DH V1 ≤ Fβ (t, W0 ) + G(t, W0 , α), DH W1 ≥ Fβ (t, W0 ) + G(t, W0 , α), which imply by Theorem 1.7 that V1 ≤ W1 for all t ∈ I and β ∈ [0, 1]. Thus, we have proved that V0 ≤ V1 ≤ W1 ≤ W0 for all t ∈ I . By similar reasoning the inequality Vj ≤ Vj +1 ≤ Wj +1 ≤ Wj ,

j > 1,

(1.26)

is established for any t ∈ I , β ∈ [0, 1]. It follows from estimate (1.26) that the sequences {Vn } and {Wn } are uniformly bounded on I for any β ∈ [0, 1]. The fact that D[Vn (t), Vn (s)] ≤ M1 t − s

for any

s < t,

(t, s) ∈ I,

1.7 Iterative Technique for a Family of Equations

23

and D[Wn (t), Wn (s)] ≤ M2 t − s

for any

s < t,

(t, s) ∈ I,

where M1 = D[Fβ (τ, Vn−1 (τ ) + G(τ, Wn−1 (τ ), α)), 0 ], M2 = D[Fβ (τ, Wn−1 (τ ) + G(τ, Vn−1 (τ ), α)), 0 ], implies in view of the Ascoli–Arzela theorem that {Vn } → P (t)

and {Wn } → Q(t)

as n → ∞

and V0 ≤ P (t) ≤ Wβ (t) ≤ Q(t) ≤ W0

(1.27)

for all β ∈ [0, 1], i.e., the pair (P (t), Q(t)) is a minimal and maximal solution for the family of equations (1.19). This completes the proof of Theorem 1.8. Since the mapping G(t, X) is a majorant for the difference F (t, X, α)−Fβ (t, X) for any α ∈ I, it is of interest to consider some properties of G(t, X) in terms of Theorem 1.8. We shall focus our attention on some corollaries of Theorem 1.8. Corollary 1.3 If in the totality of equations (1.19) the mapping G(t, X, α) ≡ 0, then (1.19) becomes a totality of equations (1.4) and under additional condition on function Fβ (t, X) being nondecreasing with respect to X estimate (1.27) remains valid. Corollary 1.4 Let in the totality of equations (1.19) the mapping G(t, X, α) ≡ 0 and Fβ (t, X) be not a mapping nondecreasing with respect to X for all t ∈ I and β (t, X) = MX + Fβ (t, X) β ∈ [0, 1]. If there exists a constant M > 0 such that F is nondecreasing with respect to X for all t ∈ I and β ∈ [0, 1], then for the totality of the initial problems β (t, U ), DH U + MU = F

U (0) = U0 ,

the estimate of (1.27) type is valid. Corollary 1.5 In system (1.19) let G(t, Y, α) be nondecreasing with respect to Y and Fβ (t, X) be not monotone for all t ∈ I and β ∈ [0, 1]. If there exists a constant β (t, X) = MX + F (t, X) is a nondecreasing mapping for all M > 0 such that F β ∈ [0, 1], then for the totality of the initial problems β (t, U ) + G(t, U, α), DH U + MU = F the assertion of Theorem 1.8 is true.

U (0) = U0 ,

24

1 General Properties of Set-Valued Equations

Corollary 1.6 In the totality of equations (1.19) let Fβ(t, X) (nondecreasing with respect to X for all t) and G(t, X) be not monotone. If there exist a constant M > 0  X) such that and a mapping G(t,  X), G(t, X, α) = MX + G(t, and  X, ) = G(t, X, α) − MX G(t, is nonincreasing with respect to X, then for the totality of the initial problems ) + G(t,  U ),  = Fβ0 (t, U DH U

(0) = U0 , U

) = Fβ (t, U eMt )e−Mt , G(t,  U ) = G(t, U eMt )e−Mt , the assertion where Fβ0 (t, U of Theorem 1.8 holds true. We shall further consider the iterative technique for the family of equations (1.19) in the case of a pair of solutions of type II satisfying equations (1.22). The following result holds. Theorem 1.9 Let conditions (2)–(4) of Theorem 1.8 be fulfilled. Then for any solution Xβ (t) of the family of equations (1.19) such that V0 ≤ Xβ (t) ≤ W0 for all t ∈ I and β ∈ [0, 1], there exist the sequences {Vn } and {Wn } such that V0 ≤ V2 ≤ . . . ≤ V2n ≤ Xβ (t) ≤ V2n+1 ≤ . . . ≤ V3 ≤ V1 W1 ≤ W3 ≤ . . . ≤ V2n+1 ≤ Xβ (t) ≤ W2n ≤ . . . ≤ W2 ≤ W0 for all t ∈ I and β ∈ [0, 1] provided that V0 ≤ V2 and W2 ≤ W0 on I . Besides, the totalities of iteration schemes DH Vn+1 = Fβ (t, Wn ) + G(t, Vn , α),

Vn+1 (0) = X0 ,

DH Wn+1 = Fβ (t, Vn ) + G(t, Wn , α),

Wn+1 (0) = X0 ,

generate the monotone sequences {V2n }, {V2n+1 }, {W2n }, {W2n+1 } ∈ Kc (Rn ), converging to the sets P (t), Q(t), P ∗ (t), Q∗ (t) ∈ Kc (Rn ), respectively, and satisfying the totality of equations DH Q = Fβ (t, Q∗ ) + G(t, P (t), α), ∗

DH P = Fβ (t, P ) + G(t, Q(t), α),

Q(0) = X0 , P (0) = X0 ,

DH Q∗ = Fβ (t, Q) + G(t, P ∗ , α),

Q∗ (0) = X0 ,

DH P ∗ = Fβ (t, P ) + G(t, Q∗ , α),

P ∗ (0) = X0 ,

for all t ∈ I and β ∈ [0, 1].

1.7 Iterative Technique for a Family of Equations

25

Proof Define V0 and W0 by the relations R0 + V0 = Z

and W0 = Z + R0 ,

where R0 = (R01 , . . . , R0n ) ∈ Kc (Rn ) and Z(t) is a solution to the family of equations DH Z = Fβ (t, ) + G(t, , α),

Z(0) = X0 ,

where  ∈ Kc (Rn ) and V0 ≤  ≤ W0 , β ∈ [0, 1]. Since the mappings Fβ and G are monotone, we get the estimate DH V0 = DH Z = Fβ (t, ) + G(t, , α) ≤ Fβ (t, W0 ) + G(t, V0 , α), V0 (0) = Z(0) − R0 < Z(0) = X0 . The inequality for W0 is obtained in the same way: DH W0 ≥ Fβ (t, V0 ) + G(t, W0 , α),

W0 (0) ≥ X0 .

Therefore the pair (V0 , W0 ) ∈ Kc (Rn ) is a pair of lower and upper solutions for problem (1.19) on I for all β ∈ [0, 1] and α ∈ I. Let Xβ (t) be a solution of the totality of systems (1.19), such that V0 ≤ Xβ (t) ≤ W0 for all t ∈ I and β ∈ [0, 1]. Show that V0 ≤ V2 ≤ Xβ (t) ≤ V3 ≤ V1 , W1 ≤ W3 ≤ Xβ (t) ≤ W2 ≤ W0 for all t ∈ I and β ∈ [0, 1]. In view of monotonicity of the mappings Fβ and G for all t ∈ I and β ∈ [0, 1], and the fact that V0 ≤ Xβ (t) ≤ W0 , we get DH W = Fβ (t, W ) + G(t, W, α) ≤ Fβ (t, W0 ) + G(t, V0 , α), W (0) = X0 , DH V1 = Fβ (t, W0 ) + G(t, V0 , α),

V1 (0) = X0 ,

for all t ∈ I and β ∈ [0, 1]. Hence it follows that Xβ (t) ≤ V1 on I for β ∈ [0, 1]. Similarly, W1 ≤ Wβ (t) on I for β ∈ [0, 1]. In order to show that V2 ≤ Xβ (t) on I for β ∈ [0, 1], we consider a totality of the initial problems DH V2 = Fβ (t, W1 ) + G(t, V1 , α),

V2 (0) = X0 ,

and the inequality DH X = Fβ (t, X) + G(t, X, α) ≥ Fβ (t, W1 ) + G(t, V1 , α), X(0) = X0 ,

26

1 General Properties of Set-Valued Equations

which leads to the estimate V2 ≤ Xβ (t) on I for all β ∈ [0, 1]. Repeating arguments similar to the above, it is easy to show that V2 ≤ Xβ (t) and Xβ (t) ≤ W2

for all

t∈I

and β ∈ [0, 1].

Proceeding further for n > 2, we get V2n−4 ≤ V2n−2 ≤ Xβ (t) ≤ V2n−1 ≤ V2n−2 , W2n−3 ≤ W2n−1 ≤ Xβ (t) ≤ W2n−2 ≤ W2n−4 for all t ∈ I and β ∈ [0, 1]. Since Vn , Wn ∈ Kc (Rn ) for all n = 0, 1, 2, . . ., we find, as in Theorem 1.8, that lim V2n = P (t),

n→∞

lim W2n+1 = P ∗ (t),

n→∞

lim V2n+1 = Q(t),

n→∞

lim W2n = Q∗ (t)

n→∞

exist in Kc (Rn ) uniformly with respect to t ∈ I for all β ∈ [0, 1] and the relations for the sets P (t), Q(t), P ∗ (t) and Q∗ (t) from Theorem 1.9 take place. Moreover, it is easy to see that P (t) ≤ Xβ (t) ≤ Q(t) and P ∗ (t) ≤ Xβ (t) ≤ Q∗ (t) for all t ∈ I and β ∈ [0, 1]. This completes the proof of Theorem 1.9.

1.8 Conditions for Global Existence of Solutions We now turn back to the family of equations (1.4) and discuss the problem on existence of solutions Wβ (t) for all t ≥ t0 and β ∈ [0, 1]. The result below holds true. Theorem 1.10 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the mappings Fβ ∈ C(R+ ×Kc (Rn ), Kc (Rn )) and the family of the initial problems DH W = Fβ (t, W ),

W (t0 ) = W0 ,

possesses a local solution for any initial values (t0 , W0 ) ∈ R+ × Kc (Rn );

1.8 Conditions for Global Existence of Solutions

27

(2) there exists a function g(t, w), g ∈ C(R+ × R+ , R), being nondecreasing with respect to w and such that D[Fβ (t, W ), 0 ] ≤ g(t, D[W, 0 ]) for all β ∈ [0, 1] and (t, W ) ∈ R+ × Kc (Rn ); (3) maximal solution r(t, t0 , w0 ) of the comparison equation dw = g(t, w), dt

w(t0 ) = w0 ≥ 0,

exists for all t ≥ t0 . Then the maximal existence interval for the family of solutions Wβ (t) for any β ∈ [0, 1] with the initial values D[W0 , 0 ] ≤ w0 is [t0 , ∞). Proof Let β ∈ [0, 1] and Wβ (t) = W (t, t0 , W0 ) be a solution of the family of equations (1.19) with the initial conditions D[W0 , 0 ] = w0 , which exists on the interval [t0 , a), t0 < a < ∞, where the value a cannot be made larger. According to conditions (2) and (3) of Theorem 1.10 and the principle of comparison we get for m(t) = D[Wβ (t), 0 ] that m(t) ≤ r(t, t0 , D[W0 , 0 ]),

t0 ≤ t < a.

Let (t1 , t2 ) ∈ (t0 , β), t1 < t2 . Then

t1

D[Wβ (t1 ), Wβ (t2 )] = D W0 +

Fβ (s, W (s)) ds, t0



t2 W0 +

Fβ (s, W (s)) ds t0



t2

t2

Fβ (s, W (s)) ds, 0 ≤

=D t1

  D Fβ (s, W (s)), 0 ds

t1

t2 ≤

g(s, D [W (s), 0 ]) ds.

(1.28)

t1

From inequality (1.28) we obtain t2 D[Wβ (t1 ), Wβ (t2 )] ≤

g(s, r(s, t0 , w0 ))ds = r(t2 , t0 , w0 ) − r(t1 , t0 , w0 ) t1

28

1 General Properties of Set-Valued Equations

for all β ∈ [0, 1]. By condition (3) of Theorem 1.10, lim = r(t, t0 , w0 ) exists and t →a −

is finite. If in the limit t → a − is understood as t1 , t2 → a − , then by the Cauchy convergence criterion we get that lim Wβ (t, t0 , w0 ) exists for all β ∈ [0, 1]. t →a −

Let

Wβ (a, t0 , w0 ) = lim Wβ (t, t0 , w0 ). t →a −

Consider the initial problem DH W = Fβ (t, W ), W (a) = Wβ (a, t0 , w0 ).

(1.29)

According to condition (1) of the theorem, the family of initial problems (1.29) possesses a local solution, and therefore the solution Wβ (t, t0 , w0 ) can be extended beyond the boundary of a. This contradicts the assumption made above. Theorem 1.10 is proved.

1.9 Approximate Solution of the Family Equations The family of equations (1.4) is an approximation of the set of equations (1.1). It is of interest to consider the problem on error estimation of approximation of solutions X(t) for system (1.1) by the set of solutions Wβ (t) for Eq. (1.4). Definition 1.5 The family of functions Wβ (t) ∈ C(R+ , Kc (Rn )) is an ε-approximate solution to the set of equations (1.1) if, given ε > 0, there exists β ∗ ∈ [0, 1] such that D[X(t), Wβ ∗ (t)] ≤ ε

(1.30)

for all t ≥ t0 . The following result holds. Theorem 1.11 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the mappings Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )) and for all α ∈ I the mapping F ∈ C(R+ × Kc (Rn ) × I, Kc (Rn )); (2) there exists a function gβ (t, w) ∈ C(R+ × R+ , R) such that D[Fβ (t, W ), F (t, X, α)] ≤ gβ (t, D[W, X]) for all β ∈ [0, 1] and any α ∈ I;

1.10 Euler Solutions for the Family of Equations (1.4)

29

(3) there exists a maximal solution r(t, t0 , w0 ) of the family of initial problems dw = gβ (t, w), dt

w(t0 ) = w0 ,

for all t ≥ t0 ; (4) there exists at least one value of β ∗ ∈ [0, 1], for which 0 < r(t, t0 , w0 ) < ε when all t ≥ t0 . Then Wβ ∗ (t) is an ε-approximate solution to the initial problem (1.1) whenever D[W0 , X0 ] ≤ w0 . Proof Under conditions (1)–(3) of Theorem 1.11 we have, by virtue of Theorem 1.2, that D[X(t), Wβ (t)] ≤ r(t, t0 , w0 ) for all t ≥ t0 , β ∈ [0, 1]. When condition (4) of Theorem 1.11 is satisfied, we find that Wβ ∗ (t) ∈ Kc (Rn ) and is an ε-approximate solution for problem (1.1) in the sense of Definition 1.4. Remark 1.4 Estimate (1.30) in Definition 1.4 differs from the traditional one in the definitions of ε-approximate solution (see Lakshmikantham et al. [49] and the bibliography therein) by the fact that in this case the approximate solution for the set of equations (1.1) is a family of functions Wβ (t), which is an exact solution for the regularized equation (1.4).

1.10 Euler Solutions for the Family of Equations (1.4) Consider the initial problem for the family of equations DH W = Fβ (t, W ),

W (t0 ) = W0 ,

(1.31)

where W ∈ Kc (Rn ) and Fβ ∈ C(I × Kc (Rn ), Kc (Rn )) for all β ∈ [0, 1]. We divide the interval [t0 , t0 + d] into the segments T = [t0 , t1 , . . . , tN = t0 + d] and consider on the interval [t0 , t1 ] the initial problem DH W = Fβ (t0 , W0 ),

W (t0 ) = W0 ∈ Kc (Rn ),

30

1 General Properties of Set-Valued Equations

for all β ∈ [0, 1]. This problem possesses the solution W (t) = W (t, t0 , w0 ) for all t ∈ [t0 , t1 ]. At the point t = t1 we compute W (t1 ) = W1 and consider the problem DH W = Fβ (t1 , W1 ),

W1 (t1 ) = W1 ∈ Kc (Rn ),

for which W (t) = W (t, t1 , W1 ) is a solution for all t ∈ [t1 , t2 ]. Proceeding further we get the solution W (t) on the interval [t0 , t0 + d]. Designate the diameter of Euler rectangular by diam T = max{ti − ti−1 : 1 ≤ i ≤ N}. Definition 1.6 The multivalued mapping W (t) is an Euler solution for the family of equations (1.31) if the sequence of solutions for the equations on T converges uniformly for diam T → 0 as N → ∞ for all β ∈ [0, 1]. The following result holds true. Theorem 1.12 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the family of multivalued mappings Fβ ∈ C(I × Kc (Rn ), Kc (Rn )) and there exists a function gβ (t, w), nondecreasing with respect to w and such that D[Fβ (t, W ), 0 ] ≤ gβ (t, D[W, 0 ]) for all (t, W ) ∈ I × Kc (Rn ); (2) maximal solution r(t, t0 , w0 ) of the family of equations dw = gβ (t, w), dt

w(t0 ) = w0 ,

exists on [t0 , t0 + d]. Then (a) there exists at least one family of mappings Wβ (t), which is an Euler solution of the family of equations (1.31); (b) any Euler solution Wβ (t) of the families of equations (1.31) satisfies the estimate D[Wβ (t), W0 ] ≤ r(t, t0 , w0 ) − w0 for all t ∈ [t0 , t0 + d] and β ∈ [0, 1], whenever w0 = D[W0 , 0 ]. Proof We designate the values Wβ (t) at points ti ∈ T as follows: W0 , W1 , . . ., WN , i.e., Wβ (ti ) = Wi , i = 0, 1, . . . , N − 1. By virtue of condition (1) of Theorem 1.12 we have on any interval (ti , ti+1 )  D[DH Wβ (t), 0 ]T = D[Fβ (ti , Wi ), 0 ] ≤ gβ (ti , D[Wi , 0 ]).

(1.32)

1.10 Euler Solutions for the Family of Equations (1.4)

31

Therefore



t

D[W1 (t), W0 ] = D W0 +

Fβ (t0 , W0 ) ds, W0 t0

t =D

 Fβ (t0 , W0 ) ds, 0

t0

t ≤





t gβ (t0 , D[t0 , W0 ])ds

D Fβ (t0 , W0 ), 0 ds ≤ t0

t0

t gβ (s, r(s))ds = r(t, t0 , D[W0 , 0 ]) − D[W0 , 0 ]

≤ t0

≤ r(t0 + d, t0 , D[W0 , 0 ]) − D[W0 , 0 ] = = const > 0 for all t ∈ [t0 , t1 ] and β ∈ [0, 1]. Proceeding further one can easily obtain the estimate D[Wi (t), W0 ] ≤ r(t0 + d, t0 , D[W0 , 0 ]) − D[W0 , 0 ] =  on [ti , ti+1 ]. Hence it follows that D[Wβ (t), 0 ]T ≤ for all t ∈ [t0 , t0 + d] and β ∈ [0, 1]. Moreover, we get from estimate (1.32) that  D[DH Wβ (t), 0 ]T ≤ gβ (t0 + d, r(t0 + d)) =

dr (t0 + d, D[W0 , 0 ]) = ψ = const > 0. dt

Then for t0 ≤ s ≤ t ≤ t0 + d we obtain  D[Wβ (t), Wβ (s)]T ≤

t

  D Fβ (τ, W (τ )), 0 T dτ

t0

s +

  D Fβ (τ, W (τ )), 0 T dτ

t0

t ≤

s gβ (τ, r(τ ))dτ +

t0

t gβ (τ, r(τ ))dτ =

t0 

= r(t) − r(s) = r (σ )|t − s| ≤ ψ|t − s|

gβ (τ, r(τ ))dτ s

32

1 General Properties of Set-Valued Equations

 for all β ∈ [0, 1] and any s ≤ σ ≤ t. Hence it follows that {Wβ (t)}T satisfies the Lipschitz condition with the constant ψ for all t ∈ [t0 , t0 + d]. Then let the segment [t0 , t0 + d] be divided so that diam T → 0 as N → ∞. Then any “arc” Wβ (t) satisfies the conditions (ti , ti+1 ), i = 1, 2, . . . , N  Wβ (t0 )T = W0 ,

 D[Wβ (t), W0 ]T ≤

and  D[DH Wβ , 0 ]T ≤ ψ

for t ∈ [t0 , t0 + d]

 for all β ∈ [0, 1]. Hence it follows that the family of arcs {Wβ (t)}T is continuous and uniformly bounded and therefore, by virtue of Ascoli–Arzela theorem, there exists a subsequence which is uniformly convergent to the multivalued mapping Wβ (t) on [t0 , t0 + d] and is absolutely continuous on [t0 , t0 + d]. This completes the proof of Theorem 1.12.

1.11 Invariance of the Set of Euler Solutions Consider the initial problem for the family of regularized equations DH W = Fβ (t, W ),

W (t0 ) = W0 ∈ Kc (Rn ),

(1.33)

where W ∈ Kc (Rn ), Fβ (t, W ) ∈ C(I × Kc (Rn ), Kc (Rn )) for all β ∈ [0, 1]. Assume that in Kc (Rn ) there exists a nonempty closed set  ⊂ Kc (Rn ). For any U ∈ Kc (Rn ) non-intersecting with the set  and any S ∈  let there exist Z ∈ Kc (Rn ) such that U = S + Z. In addition, the set U − S is a Hukuhara difference. Assume further that for any U ∈ Kc (Rn ) there exists S0 ∈  and the distance from S0 ∈  to U is minimal, i.e., D[U, ] = U − S0  = inf U − S0 . S0 ∈

The set S is a projection of U on  and is designated by proj (U ). Any nonnegative product = (U − S)t, t ≥ 0, is the closest normal to  at the point of the set S. The set of all obtained so far forms a normal cone adjacent to  at the point of the set S and is designated by N (S). Recall that the distance from any A ∈ Kc (Rn ) to the zero element 0 ∈ Kc (Rn ) is determined by the formula D[A, 0 ] = A = sup a. a∈A

1.11 Invariance of the Set of Euler Solutions

33

For A, B ∈ Kc (Rn ) the least upper bound of the Cartesian product is designated as (A × B) = sup{(a · b) : a ∈ A, b ∈ B}. It is easy to show that A + B2 ≤ A2 + B2 + 2(A × B). The aim of this section is to establish conditions for weak (exact) invariance of Euler solutions to the family of equations (1.33). We shall give the following definition. Definition 1.7 A pair (, Fβ ) is called weakly invariant if for any W0 ∈  there exists an Euler solution W (t) of the family of equations (1.33) on [t0 , ∞) for all β ∈ [0, 1] such that W (t0 ) = W0 and W (t) ∈  for all t ≥ t0 . The following result holds true. Theorem 1.13 Let the following conditions be satisfied: (1) for any β ∈ [0, 1] the family of multivalued mappings Fβ belongs to C(R+ × Kc (Rn ), Kc (Rn )) and there exists a function gβ (t, w), nondecreasing with respect to w and such that D[Fβ (t, W ), 0 ] ≤ gβ (t, D[W, 0 ]) for all (t, W ) ∈ R+ × Kc (Rn ); (2) maximal solution of the family of equations dw = gβ (t, w), dt

w(t0 ) = w0 ,

exists on [t0 , ∞); (3) there exists an open set Q ⊂ Kc (Rn ), where for all t ≥ t0 an Euler solution to the family (1.33) is found, and for every pair (t, Z) ∈ R+ × Q there exists S ∈ proj (Z) such that 

1 Fβ (t, Z) × (Z − S) ≤ qβ (t, D 2 [Z, ]), 2

where qβ ∈ C([t0 , ∞) × R+ , R) for any β ∈ [0, 1]; (4) there exists a maximal solution r(t) = r(t, t0 , w0 ) of the family of equations du = qβ (t, u), dt for all t ∈ [t0 , ∞) and β ∈ [0, 1]. Then

u(t0 ) = u0 ≥ 0,

34

1 General Properties of Set-Valued Equations

(a) the estimate D 2 [W (t), ] ≤ r(t, t0 , D 2 [W0 , ]) is valid for all t ≥ t0 and β ∈ [0, 1]; (b) for r(t, t0 , 0) ≡ 0 and W0 ∈  the inclusion W (t) ∈  is true for all t ≥ t0 and β ∈ [0, 1], i.e., the pair (Fβ , ) is weakly invariant. Proof If conditions (1)–(2) of Theorem 1.13 are satisfied, there exists at least one family of mappings Wβ (t), which is an Euler solution for the family of equations (1.33). Then at the points ti ∈ T = [t0 , t1 , . . . , tN ] we designate the value W (t) by Wi . According to condition (3) of Theorem 1.13 for each i there exists an element Si ∈ proj (Wi ) such that (Fβ (ti , Wi ) × (Wi − Si )) ≤

1 qβ (ti , D 2 [Wi , ]) 2

D 2 [W1 , ] ≤ Wi − S0 2 , since S0 ∈ . Taking into account that W1 = W0 +Z1 , where Z1 = Fβ (t0 , W0 )(t1 − t0 ) and W0 = S0 + Z0 , we get the estimate D 2 [W1 , ] ≤ Z1 + Z0 2 ≤ Z1 2 + Z0 2 + 2(Z1 × Z0 ) t1 ≤ ψ (t1 − t0 ) + D [W0 , ] + 2 2

2

(DH Wβ (t0 ) × Z0 )dt

2

t0

t1 ≤ ψ (t1 − t0 ) + D [W0 , ] + 2 2

2

(Fβ (t0 , W0 ) × (W0 − S0 ))dt

2

t0

≤ ψ 2 (t1 − t0 )2 + D 2 [W0 , ] + qβ (t0 , D 2 [W0 , ])(t1 − t0 ) for all β ∈ [0, 1]. Similar estimates are valid for any point ti ∈ T and consequently the estimate D 2 [Wi , ] ≤ ψ 2 (ti − ti−1 )2 + D 2 [Wi−1 , ] + qβ (ti−1 , D 2 [Wi−1 , ])(ti − ti−1 ) ≤ D 2 [W0 , ] + ψ2

i i   (tj − tj −1 )2 + qβ (tj −1 , D 2 [Wj −1 , ])(tj − tj −1 ) j =1

j =1

≤ D 2 [W0 , ] + ψ 2 diam T

i  (tj − tj −1 ) j =1

1.11 Invariance of the Set of Euler Solutions

+

i 

35

qβ (tj −1 , D 2 [Wj −1 , ])(tj − tj −1 ) ≤ D 2 [W0 , ]

j =1

+ ψ 2 diam T(d − t0 ) +

i 

qβ (tj −1 , D 2 [Wj −1 , ])(tj − tj −1 ),

j =1

(1.34) where d ∈ (t0 , ∞), holds true. From inequality (1.34) it is easy to get the integral inequality t D [W (t), ] ≤ D [W0 , ] + 2

qβ (s, D 2 [W (s), ]) ds

2

(1.35)

t0

for t0 ≤ t ≤ d when considering the coverage of W (t) by the “arcs” Wβ (t) at the nodes of T for any β ∈ [0, 1]. Applying Theorem 1.6.1 from Lakshmikantham et al. [51] to inequality (1.35) we obtain the estimate D 2 [W (t), ] ≤ r(t, t0 , D 2 [W0 , ])

for all t ≥ t0 .

(1.36)

If r(t, t0 , 0) ≡ 0, then (1.36) implies that W (t) ∈  for all t ≥ t0 whenever W0 ∈ . Hence the pair (, Fβ ) is weakly invariant. We shall cite the following definition. Definition 1.8 A pair (, Fβ ) is called strictly invariant if any Euler solution of the family (1.33) exists on [t0 , ∞) and satisfies condition W (t) ∈  for all t ≥ t0 and β ∈ [0, 1] whenever W (t0 ) = W0 ∈ . The following result holds. Theorem 1.14 For the family of equations (1.33) assume that (1) conditions (1) and (2) of Theorem 1.13 are satisfied; (2) there exists a set C ∈ Kc (Rn ), for which A = B + C for any A, B ∈ Kc (Rn ), and the generalized Lipschitz condition is satisfied: (Fβ (t, A) × C) − (Fβ (t, B) × C) ≤ Lβ C2 , where Lβ = const > 0 0 for all β ∈ [0, 1]; (3) for all β ∈ [0, 1] the inequality (Fβ (t, B) × C) ≤ 0 holds true. Then the pair (, Fβ ) is strictly invariant.

36

1 General Properties of Set-Valued Equations

Proof Let Wβ (t) be a family of Euler solutions existing on [t0 , ∞) and Wβ (t0 ) = W0 ∈ . According to Theorem 1.12 there exists a constant > 0 such that Wβ (t0 )|T = W0 ,

D[Wβ (t), W0 ] ≤

for all β ∈ [0, 1]. If Wβ (t) ∈ B[W0 , ] for all β ∈ [0, 1] and S ∈ proj (Wβ ), then we get the estimate D[S, W0 ] ≤ D[S, Wβ ] + D[Wβ , W0 ] ≤ 2D[Wβ , W0 ] ≤ 2 . Hence it follows that S ∈ B[W0 , 2 ]. For any Wβ ∈ B[W0 , ] and S ∈ proj (Wβ ) we get Wβ − S ∈ N (S). Consequently (Fβ (t, W ) × (W − S)) ≤

1 2 D [W, ] 2

for all β ∈ [0, 1]. This inequality is a partial case of condition (3) of Theorem 1.12 L with the function qβ (t, W ) = 2β W , therefore

D 2 [W (t), ] ≤ D 2 [W0 , ] exp

Lβ (t − t0 ) 2



for all β ∈ [0, 1] and t ≥ t0 . By assumption W0 ∈ , so we have W (t) ∈  for all t ≥ t0 and β ∈ [0, 1], i.e., the pair (, Fβ ) is strictly invariant.

1.12 Deviation of Trajectories from the Equilibrium State In this section for nonlinear equations with generalized derivative the deviation of trajectories from the equilibrium state is established in terms of pseudo-linear representation of nonlinear integral inequalities. The family of equations (1.1) is represented as DH U = Fβ (t, U ) + G(t, U, α),

(1.37)

where G(t, U, α) = F (t, U, α) − Fβ (t, U ) for all α ∈ I. Further we shall suppose that Fβ ∈ C(I × Kc (Rn , Kc (Rn ) for all β ∈ [0, 1] and G ∈ C(I × Kc (E) × I, Kc (Rn ), I ⊆ [t0 , a], G(t, 0, α) = 0 for all t ≥ t0 . Let Fβ (t, U ) and G(t, U, α) be such that there exist continuous functions f (t) and m(t) for all t ∈ I , for which: H1 . D[Fβ (t, U ), 0 ] ≤ f (t)D[U, 0 ] for all β ∈ [0, 1]; H2 . D[G(t, U, α), 0 ] ≤ m(t)D n [U, 0 ];

1.12 Deviation of Trajectories from the Equilibrium State

37

t H3 . (t0 , t) = 1 − (n − 1)D n−1 [U0 , 0 ] m(s) × t0 

s exp (n − 1) f (τ )dτ ds > 0 for all α ∈ I, where n > 1. t0

The following result holds true. Theorem 1.15 For the family of equations (1.37) assume that hypotheses H1 –H3 are fulfilled for all (t, s) ∈ [t0 , a]. Then the deviations of the set of solutions U (t) of the family of equations (1.37) from the state 0 ∈ Kc (Rn are estimated by the inequality t D[U (t), 0 ] ≤ D[U0 , 0 ] exp

f (s)ds

t0



 t s 1 n−1 × 1 − (n − 1)D [U0 , 0 ] m(s) exp (n − 1) f (τ )dτ ds − n−1 t0

t0

(1.38) for all t ∈ [t0 , a], β ∈ [0, 1] and α ∈ I. Proof The family of equations (1.37) is represented in the equivalent form t U (t) = U (t0 ) +

t Fβ (s, U (s)) ds +

t0

(1.39)

G(s, U (s), α) ds. t0

Let z(t) = D[U (t), 0 ]. Then z(t0 ) = D[U0 , 0 ], and by virtue of the properties of the metric D we get D[U (t), 0 ] ≤ D[U0 , 0 ]

 t +D

Fβ (s, U (s)) ds + t0

t ≤ D[U0 , 0 ] +

t

G(s, U (s), α) ds , 0 t0





t

D Fβ (s, U (s))ds, 0 + t0



D [G(s, U (s), α)ds, 0 ] . t0

(1.40)

38

1 General Properties of Set-Valued Equations

In view of hypotheses H1 and H2 inequality (1.38) yields the estimate D[U (t), 0 ] ≤ D[U0 , 0 ] t +



f (s)D[U (s), 0 ] + m(s)D n [U (s), 0 ] ds

t0

or t



z(t) ≤ z(t0 ) +

f (s)z(s) + m(s)zn (s) ds

(1.41)

t0

for all t ∈ [t0 , a]. Then we rewrite the inequality (1.41) as z(t) ≤ z(t0 ) +

t 

 f (s) + m(s)zn−1 (s) z(s)ds.

t0

Applying the Gronwall–Bellman lemma to this inequality we arrive at the estimate t   n−1 f (s) + m(s)z (s) ds . z(t) ≤ z(t0 ) exp

(1.42)

t0

It follows from (1.42) that

z

n−1

(t) ≤ z

n−1

t   n−1 f (s) + m(s)z (s) ds . (t0 ) exp (n − 1) t0

Multiplying both parts of this inequality by the negative expression −(n − 1)m(t) exp

t − (n − 1)

m(s)z t0

n−1

(s) ds ,

(1.43)

1.12 Deviation of Trajectories from the Equilibrium State

39

we obtain −(n − 1)m(t)z

n−1

t − (n − 1)

(t) exp

m(s)zn−1 (s) ds t0

≥ −(n − 1)z

n−1



t (t0 )m(t) exp (n − 1) f (s) ds . t0

It is easy to see that d dt



exp



t − (n − 1)

m(s)z

n−1

(s) ds

t0

(1.44)



t n−1 ≥ −(n − 1)z (t0 )m(t) exp (n − 1) f (s) ds . t0

Integrating this inequality between t0 and t we get exp

t − (n − 1)

m(s)zn−1 (s) ds t0

t ≥ 1 − (n − 1)zn−1 (t0 )

(1.45)



s m(s) exp (n − 1) f (τ ) dτ ds.

t0

t0

Under hypothesis H3 inequality (1.45) implies the estimate

t n−1 exp (n − 1) m(s)z (s) ds ≤ −1 (t, t0 )

for all t ∈ [t0 , a].

t0

In view of estimate (1.43) we get from (1.44) that  zn−1 (t0 ) exp z

n−1



t

(n − 1) f (s)ds t0

(t) ≤ 1 − (n − 1)zn−1 (t0 )

t t0



s



m(s) exp (n − 1) f (τ )dτ ds t0

.

(1.46)

40

1 General Properties of Set-Valued Equations

From (1.46) and (1.42), taking into account the designation z(t) = D[U (t), 0 ], one obtains estimate (1.38). This completes the proof of Theorem 1.15. Corollary 1.7 If conditions of hypotheses H1 –H2 are satisfied and 0 < n < 1, then the estimate (1.38) becomes D[U (t), 0 ] ≤ D[U0 , 0 ] exp 1 − (n − 1)D n−1 [U0 , 0 ] t ×

1

n−1 s m(s) exp (n − 1) f (τ ) dτ ds .

t0

t0

Further we shall assume the families of systems (1.1) and (1.37) as follows. H4 . There exist functions mi (t), i = 1, 2, . . . , n, n > 1, continuous on I and such that D[Fβ (t, U ), 0 ] ≤ m1 (t)D[U, 0 ] for all β ∈ [0, 1] and t ∈ [t0 , a]; n  H5 . D[G(t, U, α), 0 ] ≤ mi (t)D i [U, 0 ] for all α ∈ I, t ∈ [t0 , a] and i = i=2

2, . . . , n; H6 . For any (t, s) ∈ [t0 , a] t

t 1 − (n − 1)

D

i−1

[U0 , 0 ]mi (s) exp

t0

(n − 1)m1 (τ ) dτ ds > 0.

t0

The following result holds. Theorem 1.16 Assume that for the family of equations (1.37) the conditions of hypotheses H4 –H6 are fulfilled. Then the deviations of the set of solutions U (t) of the family of equations (1.37) from the state 0 ∈ Kc (E) are estimated by the inequality t D[U (t), 0 ] ≤ D[U0 , 0 ] exp

m1 (s)ds 1 − (n − 1)

t0

×

t  n t0

s D

i−1

[U0 , 0 ]mi (s) exp

i=2

1

− n−1 (n − 1)m1 (τ )dτ ds

t0

for all t ∈ [t0 , a], β ∈ [0, 1] and α ∈ I. The proof of this theorem is similar to that of Theorem 1.15. Further we assume as follows.

1.12 Deviation of Trajectories from the Equilibrium State

41

H7 . There exist integrable nonnegative functions h1 (t) and h2 (t) and constants p > 1 and q ≥ 1 such that for all (t, x) ∈ R+ × Rn D[Fβ (t, U ), 0 ] ≤ h1 (t)D [U, 0 ], p

tk+1 h1 (s)ds > 0; tk

D[G(t, U, α), 0 ] ≤ h2 (t)D [U, 0 ], q

tk+1 h2 (s)ds > 0 tk

for any (tk < tk+1 ) ∈ R+ , k = 0, 1, 2, . . . and all α ∈ I. Under conditions H7 we get from (1.39) D[U (t), 0 ] ≤ D[U0 , 0 ] t +

 h1 (s)D p [U (s), 0 ] + h2 (s)D q [U (s), 0 ] ds

t0

for all t ≥ t0 . Let us show that the following result is valid. Theorem 1.17 Assume that for system (1.37) the conditions of hypothesis H7 are fulfilled and

t t p−1 q−1 [U0 , 0 ] h1 (s) ds + D [U0 , 0 ] h2 (s) ds > 0 1 − (p + q − 2) D t0

t0

(1.47) for all t ∈ [t0 , a]. Let us show that the following result is valid

D[U (t), 0 ] ≤ D[U0 , 0 ] 1 − (p + q − 2) D p−1 [U0 , 0 ] t ×

h1 (s)ds + D t0

for all t ∈ [t0 , a].

1

− p+q−2

t q−1

[U0 , 0 ]

h2 (s)ds t0

(1.48)

42

1 General Properties of Set-Valued Equations

Proof Under condition H7 the correlation (1.39) implies the inequality t D[U (t), 0 ] ≤ D[U0 , 0 ] +

 h1 (s)D p−1 [U (s), 0 ]

t0

+ h2 (s)D q−1 [U (s), 0 ] D[U (s), 0 ] ds. Applying the Gronwall–Bellman lemma to this inequality we arrive at the estimate

t D[U (t), 0 ] ≤ D[U0 , 0 ] exp

 h1 (s)D p−1 [U (s), 0 ]

t0

+ h2 (s)D q−1 [U (s), 0 ] ds



(1.49)

for all t ∈ [t0 , a]. In terms of estimate (1.49) we get the following inequalities: D p−1 [U (t), 0 ] ≤ D p−1 [U0 , 0 ]

 t  p−1 q−1 × exp (p − 1) h1 (s)D [U (s), 0 ] + h2 (s)D [U (s), 0 ] ds , t0

D

q−1

[U (t), 0 ] ≤ D q−1 [U0 , 0 ]

 t  × exp (q − 1) h1 (s)D p−1 [U (s), 0 ] + h2 (s)D q−1 [U (s), 0 ] ds . t0

(1.50) Let p > 1 and q > 1. Then estimates (1.50) can be represented as

D p−1 [U (t), 0 ] ≤ D p−1 [U0 , 0 ] exp (p + q − 2) t ×

  p−1 q−1 [U (s), 0 ] + h2 (s)D [U (s), 0 ] ds , h1 (s)D

t0

q−1 q−1 [U (t), 0 ] ≤ D [U0 , 0 ] exp (p + q − 2) D t × t0

  h1 (s)D p−1 [U (s), 0 ] + h2 (s)D q−1 [U (s), 0 ] ds .

(1.51)

1.12 Deviation of Trajectories from the Equilibrium State

43

Multiplying the first inequality of system (1.51) by −(p+q −2)h1(t) and the second one by −(p + q − 2)h2 (t) we arrive at

−D p−1 [U (t), 0 ]h1 (t)(p + q − 2) exp − (p + q − 2) t ×

 h1 (s)D p−1 [U (s), 0 ] + h2 (s)D q−1 [U (s), 0 ] ds



t0

≥ −D p−1 [U0 , 0 ](p + q − 2)h1 (t),

q−1 −D [U (t), 0 ]h2 (t)(p + q − 2) exp − (p + q − 2) t ×

 h1 (s)D p−1 [U (s), 0 ] + h2 (s)D q−1 [U (s), 0 ] ds



t0

≥ −D q−1 [U0 , 0 ](p + q − 2)h2 (t). It is easy to see that

t  d exp − (p + q − 2) h1 (s)D p−1 [U (s), 0 ] dt t0

+ h2 (s)D

q−1

 [U (s), 0 ] ds ≥ −D p−1 [U0 , 0 ]

× (p + q − 2)h1 (t) − D q−1 [U0 , 0 ](p + q − 2)h2 (t). Integrating inequality (1.52) between t0 and t we get the estimate

t

exp − (p + q − 2)

 h1 (s)D p−1 [U (s), 0 ]

t0



+ h2 (s)D q−1 [U (s), 0 ] ds ≥ 1 − D p−1 [U0 , 0 ](p + q − 2) t ×

t h1 (s)ds − D

t0

q−1

[U0 , 0 ](p + q − 2)

h2 (s)ds. t0

(1.52)

44

1 General Properties of Set-Valued Equations

In view of condition (1.47) we have

t  h1 (s)D p−1 [U (s), 0 ] exp (p + q − 2) t0

+ h2 (s)D

q−1

[U (s), 0 ] ds ≤ 1 − (p + q − 2)





−1 t t p−1 q−1 × D [U0 , 0 ] h1 (s)ds + D [U0 , 0 ] h2 (s)ds . t0

t0

Further, taking into account estimate (1.49), we find that (D[U (t), 0 ])

p+q−2

(D[U0 , 0 ])

−(p+q−2)

≤ 1 − (p + q − 2)

−1 t t p−1 q−1 [U0 , 0 ] h1 (s)ds + D [U0 , 0 ] h2 (s)ds . × D t0

t0

Hence follows estimate (1.48). This completes the proof of Theorem 1.17.

1.13 Notes and References The modern theory of set equations originates from the works by Marchaud [59, 60] and Zaremba [123]. These papers contained new ideas in the generalization of the notion of derivative in the theory of ordinary differential equations (see Michael [93], Bridgland [15], etc.). Later on, in the papers by Wazewski [121], Filippov [27], and Kikuchi [39], fundamental results for differential inclusions were obtained. In particular, a relationship between the theory of multivalued equations and the theory of optimal motion control was established. However, the question remained open as to what is meant by the derivative of the multivalued mapping in view of nonlinearity of the space Kc (Rn ). The answer to this question was provided in 1967, by the wellknown paper of Hukuhara [34]. Therein the operation of sets subtraction and the notion of the derivative and the integral for multivalued mappings were introduced. In 1969, these results enabled one to consider the differential equation with the Hukuhara derivative and to show that its solution is a multivalued mapping (see Pinto et al. [101], de Blasi and Iervolino [22], Stefanini [112], etc.). The development of qualitative analysis of set equations with generalized derivative is one of the modern directions of general equations theory. In the subsequent papers, many researchers (see the monograph by Lakshmikantham et al.

1.13 Notes and References

45

[49], Plotnikov and Skripnik [102], and bibliography therein) established conditions for the existence and uniqueness of solutions and convergence of successive approximations of solutions to this kind of equations and addressed some other problems. In Sect. 1.2 some data from the multivalued analysis are provided (see Aubin [8], Deimling [23, 24], Lakshmikantham et al. [49], Plotnikov and Skripnik [102]). Sections 1.3–1.11 are based on the results by Martynyuk and MartynyukChernienko [84, 88] and Martynyuk [74, 78]. The monotone iterative technique discussed in Sect. 1.7 is adopted from Lakshmikantham and Vatsala [45], Ladde et al. [42]. Section 1.12 is due to Martynyuk [81]. Here, the pseudo-linear representation of nonlinear integral inequalities is employed, the idea of which goes back to El Alami (see Louartassi et al. [56], N’Doye [98], Martynyuk [77] and the bibliography therein). The classical theory of trajectories goes back to the papers by Darboux [21], Ricci et al. [105], Synge [113] and others. For some applications of this theory, see the papers by Belen’ky [11], Ovsyannikov et al. [99] and others.

Chapter 2

Analysis of Continuous Equations

For the set of equations with generalized derivative, sufficient conditions are established for various types of trajectory boundedness and for stability of a stationary set of trajectories. To this end, the scalar and vector Lyapunov functions constructed in terms of an auxiliary matrix-valued function are employed.

2.1 Introduction In this chapter the application of the comparison principle and the direct Lyapunov method in terms of auxiliary matrix-valued functions is proposed for solution of the problems under consideration. In Sect. 2.2 the estimates of the set of solutions to nonlinear dynamics equations are presented. In Sect. 2.3 a matrix-valued Lyapunov function is introduced and its application in the analysis of families of equations is indicated. In Sect. 2.4 the problem on stability of stationary solution to set equations is formulated. In Sect. 2.5 the main theorems of generalized direct Lyapunov method for set equations are presented. In Sect. 2.6 the application of strengthened Lyapunov function for the stability analysis of set trajectories is discussed. In Sect. 2.7 the theorems on boundedness of the set of solutions are proved. In Sect. 2.8 the systems of equations with Hukuhara derivative are investigated on the product of convex compacts. In Sect. 2.9 we present a theorem on Hyers–Ulam–Rassias stability of the set of trajectories. The concluding section provides bibliographical details and comments to this chapter.

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_2

47

48

2 Analysis of Continuous Equations

2.2 Ideas with Many Applications We consider a family of differential equations in the form DH X = F (t, X, α),

X(t0 ) = X0 ∈ Kc (Rn ),

(2.1)

where DH X is a Hukuhara derivative of the set X, F ∈ C(R+ × Kc (Rn ) × I, Kc (Rn )), α ∈ I is a parameter characterizing the uncertainties of parameters of the family of differential equations (2.1). The mapping X ∈ C 1 (I, Kc (Rn )) is a solution of the family of equations (2.1) on I if it is differentiable in the Hukuhara sense and satisfies Eq. (2.1) on I for any α ∈ I. Since X(t) is a continuously differentiable mapping in the Hukuhara sense, we have t X(t) = X0 +

DH X(s) ds,

t ∈ I,

(2.2)

t0

and according to (2.1) t X(t, α) = X0 +

t ∈ I.

F (s, X(s), α) ds,

(2.3)

t0

It is obvious that the mapping X(t, α) is a solution of the initial problem (2.1) on I if and only if it satisfies (2.3) on I for any α ∈ I. We introduce the mappings Fm (t, X) = co



F (t, X, α)

(2.4)

F (t, X, α).

(2.5)

α∈I

and FM (t, X) = co

 α∈I

Consequently, Fm (t, X) is a minimal closed convex shell of the family of sets F (t, X, α), and FM (t, X) is a maximal closed convex shell of the family of sets F (t, X, α). Further, it is assumed that Fm (t, X) and FM (t, X) belong to the space Kc (Rn ). We shall consider the family of mappings Fβ (t, X) constructed by the formula Fβ (t, X) = FM (t, X)β + (1 − β)Fm (t, X),

β ∈ [0, 1].

(2.6)

2.2 Ideas with Many Applications

49

Taking into account the notations (2.4)–(2.6), along with the initial problem (2.1) we shall consider the initial problem for the family of differential equations DH Y = Fβ (t, Y ),

Y (t0 ) = Y0 ∈ Kc (Rn ),

(2.7)

where Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )) for all β ∈ [0, 1]. In the investigation of the initial problem (2.7), the principle of comparison is of importance in the form given below. Theorem 2.1 For the family of equations (2.7), let the following conditions hold: (1) Fβ ∈ C(I × Kc (Rn ), Kc (Rn )) and there exists a monotone function g(t, ω) nondecreasing in ω, g ∈ C(I × R+ , R), such that D[Fβ (t, Y ), Fβ (t, Z)] ≤ g(t, D[Y, Z]) for all Y, Z ∈ Kc (Rn ) and any value of β ∈ [0, 1]; (2) there exists a maximal solution r(t; t0 , ω0 ) of the scalar equation dω = g(t, ω), dt

ω(t0 ) = ω0 ≥ 0,

on I ; (3) the initial conditions (t0 , Y0 ), (t0 , Z0 ) for two sets of solutions Y (t), Z(t) of the families of equations (2.7) are such that D[Y0 , Z0 ] ≤ ω0 . Then, for all t ∈ I we have the estimate D[Y (t), Z(t)] ≤ r(t; t0 , ω0 ). Proof Let m(t) = D[Y (t), Z(t)] so that m(t0 ) = D[Y0 , Z0 ] ≤ ω0 by condition (3) of Theorem 2.1. Taking into account relation (2.2), we have for Eq. (2.7) that t Y (t) = Y0 +

Fβ (s, Y (s)) ds, t0

t ∈ I,

β ∈ [0, 1].

50

2 Analysis of Continuous Equations

For two solutions Y (t) and Z(t) with the initial conditions D[Y0 , Z0 ] ≤ ω0 we obtain the estimate

t

m(t) = D Y0 +

Fβ (s, Y (s)) ds, Z0 +



t0

t

t Fβ (s, Y (s)) ds, Y0 + t0

t

t Fβ (s, Z(s)) ds, Z0 +

t0

 Fβ (s, Z(s)) ds

t0

+ D Y0 +



(2.8)

Fβ (s, Z(s)) ds t0

t =D

Fβ (s, Z(s)) ds

t0

≤ D Y0 +



t

t Fβ (s, Y (s)) ds,

t0

 Fβ (s, Z(s)) ds + D[ Y0 , Z0 ].

t0

Taking into account condition (1) of Theorem 2.1, from inequality (2.8) we find the estimate t m(t) ≤ m(t0 ) +

D[Fβ (s, Y (s)), Fβ (s, Z(s))] ds t0

t ≤ m(t0 ) +

g(s, D[Y (s), Z(s)]) ds

(2.9)

t0

t = m(t0 ) +

g(s, m(s)) ds,

t ∈ I,

t0

for all β ∈ [0, 1]. Applying Theorem 1.6.1 from the monograph by Lakshmikantham et al. [51] to the inequality (2.9) and taking into account condition (2) of Theorem 2.1, we find that m(t) ≤ r(t; t0 , ω0 ) for all t ∈ I . Theorem 2.1 is proved. We shall present some results related to the existence and uniqueness of the solutions to the initial problem (2.7). Theorem 2.2 For the initial problem (2.7), let the following conditions be satisfied: (1) there exists a constant M0 > 0 such that for Fβ ∈ C(I × B(Y0 , b), Kc (Rn )), when all β ∈ [0, 1], the inequality D[Fβ (t, Y ), ] ≤ M0 is satisfied, where B(Y0 , b) = {Y ∈ Kc (Rn ) : D[Y, Y0 ] ≤ b};

2.2 Ideas with Many Applications

51

(2) there exist a function g(t, ω), nondecreasing in ω for all t ∈ I , and a constant M1 > 0 such that g ∈ C(I × [0, 2b], R), g(t, ω) ≤ M1 for all (t, ω) ∈ I × [0, 2b] and ω(t) = 0 is a unique solution to the initial problem dω = g(t, ω), dt

ω(t0 ) = 0 on I ;

(3) for all (t, Y ) ∈ I × B(Y0 , b) the estimate D[Fβ (t, Y ), Fβ (t, Z)] ≤ g(t, D[Y, Z]) holds for all β ∈ [0, 1]. Then the successive approximations t Yn+1 (t) = Y0 +

Fβ (s, Yn (s)) ds,

n = 0, 1, 2, . . . ,

t0

exist on the interval I0 = [t0 , t0 + ], where  = min(a, b/M), M = max(M0 , M1 ), as continuous functions that converge uniformly to the solution of the initial problem (2.7) on I0 . The proof of this theorem is based on the application of the comparison principle and the Ascoli–Arzela theorem. In contrast to Theorem 2.3.1 from the monograph by Lakshmikantham et al. [49], condition (3) of Theorem 2.2 must be satisfied for all β ∈ [0, 1]. Further we need the notion of partial ordering in the space (Kc (Rn ), D). We denote by K(Kc (Rn )) some subset of Kc (Rn ) consisting of the sets X ∈ Kc (Rn ) such that any element u ∈ X is a nonnegative (positive) vector for which ui ≥ 0 (ui > 0) for all i = 1, 2, . . . , n. Thus, K is a cone in Kc (Rn ), and K 0 is its nonempty interior. If there exists a set Z ∈ Kc (Rn ) such that for any X and Y ∈ Kc (Rn ), the inclusion Z ∈ K(Kc (Rn )) is fulfilled and X = Y + Z, then X ≥ Y (X > Y ). Let R(t) be a solution of the family of differential equations (2.7) under the initial conditions (t0 , Y0 ), defined on some interval [t0 , a) for any value β ∈ [0, 1]. The mapping R(t) is a maximal solution of the family of differential equations (2.7) if for any other solution Y (t) with the initial conditions (t0 , Y0 ) defined on the interval [t0 , a) ˜ for any value β ∈ [0, 1] the following inequality is true: Y (t) ≤ R(t) for all t ∈ [t0 , a) ∩ [t0 , a). ˜ Here a, a˜ are finite numbers or +∞.

52

2 Analysis of Continuous Equations

The following assertion holds. Theorem 2.3 For the initial problem (2.7) let the following conditions be satisfied: (1) the mapping Fβ (t, Y ), Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )), is monotone nondecreasing in Y for each t ∈ R+ and each β ∈ [0, 1], i.e. Fβ (t, Y ) ≤ Fβ (t, Z) for each β ∈ [0, 1] and t ∈ R+ , whenever Y ≤ Z; (2) for any Z, W ∈ C 1 (R+ , Kc (Rn )), the inequalities DH Z < Fβ (t, Z) and DH W ≥ Fβ (t, W ) are fulfilled for all β ∈ [0, 1] and t ∈ R+ ; (3) at the initial time t0 ∈ R+ , the inequality Z(t0 ) < W (t0 ) is satisfied. Then, for all t ≥ t0 , the following estimate holds true: Z(t) < W (t). Remark 2.1 If, in the conditions of Theorem 2.3, condition (2) is replaced by the inequalities (2 ) DH Z ≤ Fβ (t, Z) and DH W > Fβ (t, W ) for all t ∈ R+ and all β ∈ [0, 1], then the assertion of Theorem 2.3 will not change. We now present conditions for the global existence of solution to family equation (2.7). Theorem 2.4 For Eq. (2.7), let the following conditions be satisfied: (1) there exists a scalar function g(t, ω), nondecreasing in ω for each t ∈ R+ , such that for the family of mappings Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )) for any β ∈ [0, 1] the inequality D[Fβ (t, Y ), ] ≤ g(t, D[Y, ]) holds true for all (t, Y ) ∈ R+ × Kc (Rn ), where g ∈ C(R2+ , R+ ); (2) the maximal solution r(t; t0 , ω0 ) of the equation dω = g(t, ω), dt

ω(t0 ) = ω0 ≥ 0,

exists for all t ∈ [t0 , ∞); (3) for any β ∈ [0, 1] the family of mappings Fβ (t, Y ) satisfies the existence conditions for the local solution of the initial problem (2.7) at any (t0 , Y0 ) ∈ R+ × Kc (Rn ) such that D[Y0 , ] ≤ ω0 . Then any solution Y (t; t0 , Y0 ) of the family of equations (2.7) exists on the interval [t0 , ∞).

2.3 Lyapunov-Like Functions and Their Applications

53

Proof Consider the solution Y (t) = Y (t; t0 , Y0 ) of equations (2.7) with the initial conditions D[Y0 , ] = ω0 existing on the interval [t0 , a), t0 < a < ∞, and assume that the value of a cannot be increased. Let m(t) = D[Y (t), ]. Then, by Theorem 2.1 we have the estimate m(t) ≤ r(t; t0 , ω0 ) for all t0 ≤ t < a. For any t1 , t2 , such that t0 < t1 < t2 < a, it is easy to obtain the relations

t1

D[Y (t1 ), Y (t2 )] = D Y0 +

Fβ (s, Y (s)) ds, Y0 + t0

t2

Fβ (s, Y (s)) ds,  ≤ t1

Fβ (s, Y (s)) ds t0



t2 =D



t2

D[Fβ (s, Y (s)), ] ds

(2.10)

t1

t2 ≤

g(s, D[Y (s), ]) ds t1

for all β ∈ [0, 1]. Taking into account condition (1) of Theorem 2.4 and inequality (2.10), we get t2 D[Y (t1 ), Y (t2 )] ≤

g(s, r(s; t0 , ω0 )) ds = r(t2 ; t0 , ω0 ) − r(t1 ; t0 , ω0 ).

(2.11)

t1

Estimate (2.11) yields that limt →a − Y (t; t0 , Y0 ) exists, since limt →a − r(t; t0 , ω0 ) exists and is finite according to the assumption t1 , t2 → a − . Let Y (a; t0, Y0 ) = limt →a − Y (t; t0 , Y0 ). We consider the initial problem DH Y = Fβ (t, Y ),

Y (a) = Y (a; t0, Y0 ),

for all β ∈ [0, 1].

According to condition (3) of Theorem 2.4, the solution Y (t; t0 , Y0 ) can be extended to the interval exceeding [t0 , a), i.e. the solution Y (t; t0 , Y0 ) with the initial values D[Y0 , ] ≤ ω0 exists on [t0 , ∞).

2.3 Lyapunov-Like Functions and Their Applications Together with the family of differential equations (2.1) we shall consider the matrixvalued function S(t, X) = [ uij (t, X) ],

i, j = 1, 2,

(2.12)

54

2 Analysis of Continuous Equations

with the elements uij (t, X) constructed in terms of auxiliary families of equations: (a) if β = 0, then the element u11 (t, X) is associated with the equation DH X = Fm (t, X),

X(t0 ) = X0 ∈ Kc (Rn );

(b) if β = 1, then the element u22 (t, X) is associated with the equation DH X = FM (t, X),

X(t0 ) = X0 ∈ Kc (Rn );

(c) if 0 < β < 1, then the element u12 (t, X) = u21 (t, X) is associated with the family of equations DH X = Fβ (t, X),

X(t0 ) = X0 ∈ Kc (Rn ).

By means of the vector θ ∈ R2+ we construct the scalar function V (t, X, θ ) = θ TS(t, X)θ,

(2.13)

for which the total derivative by virtue of equation (2.7) is calculated by the formula D + V (t, A, θ ) = lim sup



V (t + h, A + hFβ (t, A), θ )   − V (t, A, θ ) h−1 : h → 0+

(2.14)

for any A ∈ Kc (Rn ) and β ∈ [0, 1]. We shall now present a result of the principle of comparison with function (2.13) for Eq. (2.1). Theorem 2.5 For the family of equations (2.1), let the following conditions be satisfied: (1) there exist a function S ∈ C(R+ × Kc (Rn ), R2×2 ), a vector 2 θ ∈ R+ and a constant L > 0 such that V (t, X, θ ) ∈ C(R+ × Kc (Rn ) × R2+ , R+ ) and |V (t, A, θ ) − V (t, B, θ )| ≤ LD[A, B] for any A, B ∈ Kc (Rn ), t ∈ R+ ; (2) there exists a function g(t, ω), g ∈ C(R2+ , R), such that  D + V (t, A, θ )(2.1) ≤ g(t, V (t, A, θ )) for all t ∈ R+ , A ∈ Kc (Rn ), θ ∈ R2+ ;

2.3 Lyapunov-Like Functions and Their Applications

55

(3) the maximum solution r(t; t0 , ω) of the equation dω = g(t, ω), dt

ω(t0 ) = ω0 ≥ 0,

exists for all t ∈ [t0 , a). Then, if the solution X(t) = X(t; t0 , X0 ) of the family equation (2.7) exists on the interval [t0 , a) with the initial conditions under which V (t0 , X0 , θ ) ≤ ω0 , we have that the estimate V (t, X(t), θ ) ≤ r(t; t0 , ω0 )

(2.15)

is valid for all t ∈ [t0 , a). Proof Let the initial conditions (t0 , X0 ) ∈ R+ ×Kc (Rn ) be such that V (t0 , X0 , θ ) ≤ ω0 and the solution X(t) = X(t; t0 , X0 ) of Eq. (2.1) exists on the interval [t0 , a). For the function m(t) = V (t, X(t), θ ) and arbitrarily small h > 0 we find   m(t + h) − m(t) ≤ LD X(t + h), X(t) + hFβ (t, X(t), α)  + V t + h, X(t) + hFβ (t, X(t)), θ − V (t, X(t), θ ) for all β ∈ [0, 1]. Hence we get the inequality D + m(t) = lim sup h→0+

+ L lim sup h→0+

 1 [m(t + h) − m(t)] ≤ D + V (t, X(t), θ )(2.1) h

1 [D [X(t + h), X(t) + hF (t, X(t), α)]] ≤ g(t, m(t)), h

m(t0 ) ≤ ω0 , since D[DH X(t), F (t, X(t))] ≡ 0. Therefore, by Theorem 1.5.2 from the monograph by Lakshmikantham et al. [51], it follows that for all t ∈ [t0 , a) the estimate (2.15) is true. Further we shall consider the vector function L(t, X, θ ) = AS(t, X)θ,

(2.16)

where A is a constant (2 × 2)-matrix and present a result of the principle of comparison with the function (2.16) for Eq. (2.1). Theorem 2.6 For the family of equations (2.1), let the following conditions be satisfied: (1) there exist function (2.12), a vector θ ∈ R2+ and a constant (2×2)-matrix A such that the function (2.16) satisfies the conditions L ∈ C(R+ ×Kc (Rn )×R2+ , R2+ ) and  1 , X2 ], | L(t, X1 , θ ) − L(t, X2 , θ ) | ≤ B D[X

56

2 Analysis of Continuous Equations

where B is a constant (2 × 2)-matrix with nonnegative elements and the  1 , X2 ] = (D[X1 , X ], D[X2 , X ])T, D  : Kc (Rn ) × Kc (Rn ) → R2+ , vector D[X 1 2   n X1 , X2 ∈ Kc (R ); (2) there exists a vector function G(t, ω) which is nondecreasing in ω for each t ∈ R+ , G ∈ C(R+ × R2+ , R2 ), and such that  D + L(t, A, θ )(2.1) ≤ G (t, L(t, A, θ )) for all (t, A, θ ) ∈ R+ × Kc (Rn ) × R2+ ; (3) the maximum solution R(t) = R(t; t0 , ω0 ) of the comparison system dω = G(t, ω), dt

ω(t0 ) = ω0 ≥ 0,

exists for all t ∈ [t0 , a). Then, if the solution X(t) = X(t; t0 , X0 ) of the family of equations (2.1) exists on the interval [t0 , a) with the initial conditions under which L(t0 , X0 , θ ) ≤ ω0 , the component-wise estimate L(t, X(t), θ ) ≤ R(t; t0 , ω0 ) is valid for all t ∈ [t0 , a). The proof of Theorem 2.6 is similar to that of Theorem 2.5, and therefore is omitted here. Further we shall apply Theorems 2.5 and 2.6 in order to establish sufficient stability conditions for the stationary solution 0 to the family of equations (2.1).

2.4 Stability of the Set of Stationary Solutions We introduce the notion of stability of stationary solution for the family of differential equations (2.1). The solution X(t) of the family of equations (2.1) satisfies the initial condition X(t0 ) = X0 ,

X0 ∈ Kc (Rn ),

if, for t0 ∈ R+ , the value of X(t) at the point t0 is equal to X0 . If Fβ (t, 0 ) = 0 , then X(t) = 0 is a set of equilibrium states or a set of stationary solutions to the family of equations (2.1). Since t → diam(X(t)) is a nondecreasing function t → ∞, the norm X(t) = diam(X(t)) for t ≥ t0 cannot be immediately applied in the definitions of stability of solutions to the family of equations (2.1). Therefore, we introduce additional hypotheses on the initial problem (2.1).

2.4 Stability of the Set of Stationary Solutions

57

H1 . For the family of equations (2.1), there exists a set of stationary solutions 0 ∈ Kc (Rn ), i.e. F (t, 0 , α) = 0 for all t ∈ R+ , α ∈ I. H2 . For any initial values X0 , Y0 ∈ Kc (Rn ), there exists a set W0 ∈ Kc (Rn ) such that Y0 = X0 + W0 . The set W0 is called the Hukuhara difference. H3 . The family of equations (2.1) possesses a unique solution X(t) = X(t, t0 , X0 − Y0 ) = X(t, t0 , W0 ) for all t ≥ t0 . We note that if for the sets under consideration the Hukuhara difference exists, then it is uniquely defined (see Hukuhara [36] and the bibliography therein). Further, in the statement of the problem on stability of solutions to equation (2.1), assumptions H1 –H3 are taken into account. Definition 2.1 The stationary solution 0 of the family of equations (2.1) is: (a) stable if for any t0 ∈ R+ and ε > 0 there exists δ = δ(t0 , ε) > 0 such that the inequality D[W0 , 0 ] < δ implies the estimate D[X(t; t0 , W0 ), 0 ] < ε for all t ≥ t0 ; (b) attractive if for any t0 ∈ R+ there exists α(t0 ) > 0 and for any ξ > 0 there exists τ (t0 , W0 , ξ ) ∈ R+ such that the inequality D[W0 , 0 ] < α(t0 ) implies the estimate D[X(t; t0 , W0 ), 0 ] < ξ for any t ≥ t0 + τ (t0 , W0 , ξ ); (c) asymptotically stable if the conditions (a) and (b) of Definition 2.1 are fulfilled simultaneously; (d) strictly stable if for any ε > 0 there exists a δ = δ(ε) > 0 such that for any set of trajectories X(t) = X(t, t0 , W0 ) the conditions D[X(t1 ), 0 ] ≤ δ and t1 ≥ t0 imply that D[X(t), 0 ] < ε for all t ≥ t0 . Other types of stability, attraction, and asymptotic stability of the set of stationary solutions 0 of the family of equation (2.1) are defined in the same way as above. Example 2.1 (cf. Lakshmikantham et al. [49]) equations DH X = (−eα )X,

Consider the set of differential

X(0) = X0 ∈ Kc (R),

(2.17)

where α ∈ [0, 1] is an uncertain parameter. Since the values of solutions to equations (2.17) are interval functions, Eq. (2.17) can be rewritten as [x1 , x2 ] = (−eα )X = [(−eα )x2 , (−eα )x1 ],

( ) =

d , dt

where X(t) = [x1 (t), x2 (t)] and X(0) = [x10, x20 ]. From this correlation we have the system of equations x1 = −eα x2 ,

x1 (0) = x10,

x2 = −eα x1 ,

x2 (0) = x20,

58

2 Analysis of Continuous Equations

whose solution are the functions 1 1 [x10 + x20 ]e−t + [x10 − x20 ]et , 2 2 1 1 x2 (t) = [x20 + x10 ]e−t + [x20 − x10 ]et 2 2

x1 (t) =

(2.18)

for all t ≥ 0 and α = 0. If, for a given set X0 ∈ Kc (R), there exist sets Y0 , W0 ∈ Kc (R) such that X0 = Y0 +W0 , then there exists a Hukuhara difference X0 −Y0 = W0 . Let X0 = [x10, x20 ], 1 1 Y0 = ([x10 − x20 ], [x20 − x10 ]), then W0 = ([x10 + x20 ], [x20 + x10 ]). 2 2 If x10 = −x20 , then from (2.18) we have for t ≥ t0 1 (−[x20 − x10 ], [x20 − x10 ])et + 2 1 ([x10 + x20 ], [x10 + x20 ])e−t ; 2 1 (b) X(t, Y0 ) = ([x10 − x20 ], [x20 − x10 ])et ; 2 (c) X(t, W0 ) = ([x10 + x20], [x10 + x20 ])e−t . (a) X(t, X0 ) =

Hence it follows that if the Hukuhara difference exists, for the initial values of problem (2.17) the zero solution is stable, while in the expressions (a) and (b) undesirable components are found.

2.5 Theorems on Stability Different type stability conditions for the stationary solution 0 to Eq. (2.1) will be established in terms of function (2.13) and its total derivative along the set of solutions to equation (2.1) calculated by formula (2.14). Theorem 2.7 For the family of differential equations (2.1), let conditions of hypotheses H1 –H3 be satisfied and, in addition, (1) there exist matrix-valued function (2.12) and a vector θ ∈ R2+ such that function (2.13) satisfies the local Lipschitz condition |V (t, X1 , θ ) − V (t, X2 , θ )| ≤ LD[X1 , X2 ], where L > 0 for all (t, X) ∈ R+ × S0 (ρ0 );

2.5 Theorems on Stability

59

(2) there exist comparison vector functions ψ1 and ψ2 of the class K (see Hahn [31]) and constant symmetric positive definite (2 × 2)-matrices A1 and A2 such that ψ1T (D[X, 0 ]) A1 ψ1 (D[X, 0 ]) ≤ V (t, X, θ ) ≤ ≤ ψ2T (t, D[W0 , 0]) A2 ψ2 (t, D[W0 , 0]) for all (t, X) ∈ R+ × S0 (ρ0 ); (3) for all (t, X) ∈ R+ × S0 (ρ0 ) the inequality  D + V (t, X, θ )(2.1) ≤ 0 holds true. Then the set of stationary solutions 0 of the family of equations (2.1) is stable. Proof We transform condition (2) of Theorem 2.7 to the form λm (A1 )ψ 1 (D[X, 0 ]) ≤ V (t, X, θ ) ≤ λM (A2 )ψ 2 (t, D[W0 , 0 ]) ,

(2.19)

where λm (A1 ) and λM (A2 ) are the minimal and the maximal eigenvalues of the matrices A1 and A2 , and ψ 1 and ψ 2 are of the class K so that ψ 1 (D[X, 0 ]) ≤ ψ1T (D[X, 0 ]) ψ1 (D[X, 0 ]) , ψ 2 (t, D[W0 , 0 ]) ≥ ψ2T (t, D[W0 , 0 ]) ψ2 (t, D[W0 , 0 ]) in the domain of values (t, X) ∈ R+ × S0 (ρ0 ). Let the values ε > 0 and t0 ∈ R+ be given. We choose the value δ = δ(t0 , ε) > 0 from the condition λM (A2 )ψ 2 (t0 , δ) < λm (A1 )ψ 1 (ε). We shall show that, for the given choice of the value δ(t0 , ε) > 0 and under conditions of Theorem 2.7, the stationary solution X(t; t0 , 0 ) of Eq. (2.1) is stable. If this is not so, then there must exist a solution X(t) = X(t; t0 , X0 ) and a value t1 > t0 such that D[X(t1 ), 0 ] = ε

and D[X(t), 0 ] ≤ ε < H0 ,

H0 < H,

for all t0 ≤ t ≤ t1 , whenever D[W0 , 0 ] < δ. From Theorem 2.5 and condition (3) of Theorem 2.7, it follows that V (t, X(t), θ ) ≤ V (t0 , X0 , θ ),

t0 ≤ t ≤ t1 .

(2.20)

60

2 Analysis of Continuous Equations

In view of estimate (2.19) we find from inequality (2.20) that λm (A1 )ψ 1 (ε) = λm (A1 )ψ 1 (D[X(t1 ), 0 ]) ≤ V (t1 , X(t1 ), θ ) ≤ V (t0 , W0 , θ ) ≤ λM (A2 )ψ 2 (t0 , D[W0 , 0 ]) ≤ λM (A2 )ψ 2 (t0 , δ) < λm (A1 )ψ 1 (ε). The contradiction obtained proves Theorem 2.7. Example 2.2 Consider a family of equations DH X = λ(t)X,

X(t0 ) = X0 ∈ Kc (Rn ),

(2.21)

where λ(t) ∈ L1 (R+ ), λ(t) > 0 is a real function on R+ . From (2.21) we find that t X(t) = X0 exp

t ≥ t0 ,

λ(s) ds ,

(2.22)

t0

for any X(t) ∈ Kc (Rn ). We get stability conditions for the state 0 of the family of equations (2.21). Relation (2.22) implies t D[X(t), 0 ] ≤ D[X0 , 0 ] exp

λ(s) ds .

(2.23)

t0

The state 0 is stable if and only if the function

t

λ(s) ds is bounded for all t ≥ t0 .

t0

Actually, the boundedness of this function implies the existence of a constant k > 0 such that t λ(s) ds ≤ k < +∞ for all t ≥ t0 . t0

For any ε > 0 choose δ ≤ kε . Therefore, if D[X0 , 0 ] < δ, then D[X(t), 0 ] < kδ = ε1 . Due to the arbitrariness of ε the assertion is proved. Remark 2.2 The fact of stability of the state 0 in a real system does not mean that the description of the system by the family of equation (2.14) is always acceptable. Namely, let in system (2.14) t0 = 0 and the function (see Zubov [123])  λ(t) =

ln 10 for t ∈ [0, 10], 0

for t > 10.

2.5 Theorems on Stability

61

Then D[X(t), 0 ] ≤

 D[X0 , 0 ]10t D[X0 , 0

]1010

for t ∈ [0, 10], for t > 10.

This relation provides that the state 0 is stable with the function λ(t), but under the initial conditions D[X0 , 0 ] = 10−5 , which is unreal to measure, the deviation of D[X(t), 0 ] may reach the magnitude 105, which seems impossible in a real system. Hence, the stability of the state 0 stands to point under certain additional restrictions on the estimate of the values ε and δ. Theorem 2.8 For the family of equations (2.1) let the conditions of hypotheses H1 – H3 , and conditions (1) and (2) of Theorem 2.7 be fulfilled as well as the additional condition  (3 ) D + V (t, X, θ )(2.1) ≤ −γ V (t, X, θ ) for all (t, X) ∈ R+ × S0 (ρ0 ). Then the set of stationary solutions 0 of the family of equations (2.1) is asymptotically stable. Proof By the conditions of Theorem 2.8 it follows that the stationary solution 0 of Eq. (2.1) is stable. Let ε = ρ0 and δ0 = δ(t0 , ρ0 ). Then, according to Theorem 2.7, the inequality D[W0 , 0 ] < δ0 implies the estimate D[X(t), 0 ] < ρ0 for all t ≥ t0 , where X(t) = X(t; t0 , W0 ) is a solution of equations (2.1). From condition (3 ) of Theorem 2.8 it follows that V (t, X(t), θ ) ≤ V (t0 , W0 , θ ) exp [−γ (t − t0 )] for all t ≥ t0 . For the given ε > 0 we choose τ = τ (t0 , ε) by the formula τ (t0 , ε) =

1 λM (A2 )ψ 2 (t0 , δ0 ) ln + 1. γ λm (A1 )ψ 1 (ε)

It can be easily verified that the inequality λm (A1 )ψ 1 (D[X(t), 0 ]) ≤ V (t, X(t), θ ) ≤ ≤ λM (A2 )ψ 2 (t0 , δ0 ) exp [−γ (t − t0 )] < λm (A1 )ψ 1 (ε) is valid. This completes the proof of Theorem 2.8. Further, for the value 0 < η < ρ0 we shall consider the set S(ρ0 ) ∩ S c (η) and suppose that the function V (t, X, θ ) is defined in the domain of values (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η).

62

2 Analysis of Continuous Equations

Theorem 2.9 For the family of equations (2.1) assume that conditions of hypotheses H1 –H3 are fulfilled and, in addition: (1) function (2.13) is such that |V (t, X1 , θ ) − V (t, X2 , θ )| ≤ LD[X1 , X2 ], where L > 0 for all (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η); (2) there exist comparison vector functions ψ1 and ψ2 ∈ of the class K and constant symmetric positive definite matrices A1 and A2 such that ψ1T (D[X, 0 ])A1 ψ1 (D[X, 0 ]) ≤ V (t, X, θ ) ≤ ψ2T (D[W0 , 0 ]) A2 ψ2 (D[W0 , 0 ]) for all (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η); (3) for all (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η) the inequality holds true  D + V (t, X, θ )(2.1) ≤ 0. Then the set of stationary solutions 0 of the family of equations (2.1) is uniformly stable. Proof Suppose that the values 0 < ε < ρ0 and t0 ∈ R+ are given. We choose δ = δ(ε) so that λM (A2 )ψ 2 (δ) < λm (A1 )ψ 1 (ε), and show that in this case the stationary solution 0 of Eq. (2.1) is uniformly stable. If this is not the case, then a solution X(t) to Eq. (2.1) will be found for which the relations D[X(t1 ), 0 ] = δ,

D[X(t2 ), 0 ] = ε

and

δ ≤ D[X(t), 0 ] ≤ ε < ρ0 are fulfilled for all t1 ≤ t ≤ t2 . Let η = δ, then condition (3) of Theorem 2.9 provides the inequality V (t2 , X(t2 ), θ ) ≤ V (t1 , X(t1 ), θ ) and by condition (2) of Theorem 2.9 we arrive at the inequality λm (A1 )ψ 1 (ε) = λm (A1 )ψ 1 (D[Y (t2 ), 0 ]) ≤ V (t2 , X(t2 ), θ ) ≤ V (t1 , X(t1 ), θ ) ≤ λM (A2 )ψ 2 (D[X(t1 ), 0 ]) = λM (A2 )ψ 2 (δ) < λm (A1 )ψ 1 (ε). The contradiction obtained proves Theorem 2.9.

2.5 Theorems on Stability

63

Theorem 2.10 For the family of equations (2.1) let all conditions of hypotheses H1 –H3 , conditions (1) and (2) of Theorem 2.9 and condition (3 ) be fulfilled, and there exist a comparison function ψ3 ∈ of the class K such that  D + V (t, X, θ )(2.1) ≤ −ψ3 (D[X(t), 0 ]) for all (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η). Then the set of stationary solutions 0 of the family of equations (2.1) is uniformly asymptotically stable. Proof If the conditions of Theorem 2.10 are satisfied, then all conditions of Theorem 2.9 are fulfilled and the stationary solution 0 of Eq. (2.7) is uniformly stable. We shall assume that ε = ρ0 , and choose δ0 = δ(ρ0 ) > 0. Moreover, the condition D[W0 , 0 ] < δ0 implies the estimate D[Y (t), 0 ] < ρ0 for all t ≥ t0 uniformly in t0 ∈ R+ . We shall show that there exists t0 ≤ t ∗ ≤ t0 + τ , where τ =1+

λM (A2 )ψ 2 (δ0 ) , ψ3 (δ)

so that D[X(t ∗ ), 0 ] < δ. Suppose this is not so, and D[X(t ∗ ), 0 ] ≥ δ for t0 ≤ t ∗ ≤ t0 + τ . From condition (3 ) of Theorem 2.8 we have the estimate t V (t, X(t), θ ) ≤ V (t0 , W0 , θ ) −

ψ3 (D[X(s), 0 ]) ds t0

for all t0 ≤ t ≤ t0 + τ . In view of the choice of the value τ , we find 0 ≤ V (t0 + τ, X(t0 + τ ), θ ) ≤ λM (A2 )ψ 2 (δ0 ) − ψ3 (δ)τ < 0. This inequality contradicts the estimate of the function V (t, X, θ ) from condition (2) of Theorem 2.9. Therefore, there exists t ∗ such that the condition D[W0 , 0 ] < δ0 implies the estimate D[X(t), 0 ] < δ for all t ≥ t ∗ + τ . This completes the proof of Theorem 2.10. We shall further apply function (2.16) and the comparison Theorem 2.6. Also, we shall present a general scheme of obtaining stability conditions for the set of stationary solutions to the family of equations (2.1). An analogue of this approach in the theory of ordinary differential equations is the technique of proving stability theorems for zero solution of perturbed motion equations in terms of Lyapunov vector function.

64

2 Analysis of Continuous Equations

We shall prove the following assertion: Theorem 2.11 For the family of equations (2.1) let all conditions of hypotheses H1 –H3 be satisfied and, in addition: (1) there exist a function L(t, X(t), θ ) and a vector function G(t, ω), G ∈ C(R+ × R2+ , R2 ), G(t, 0) = 0, which is nondecreasing in ω for all t ∈ R+ , such that conditions (1)–(3) of Theorem 2.6 hold for all (t, X) ∈ R+ × S0 (ρ0 ); (2) there exist comparison vector functions ψ1 and ψ2 , of the class K and constant symmetric positive definite (2×2)-matrices A1 and A2 such that for the function L0 (t, X, θ ) =

2 

Li (t, X, θ )

i=1

the two-sided inequality   ψ1T(D[X, 0 ])A1 ψ1 (D[X, 0 ]) ≤ V0 (t, X, θ )   ≤ ψ2T(D[X, 0 ])A2 ψ2 (D[X, 0 ]) holds for all (t, X) ∈ R+ × S0 (ρ0 ); (3) zero solution of the comparison system dω = G(t, ω), dt

ω(t0 ) = ω0 ≥ 0,

possesses certain type of stability. Then the set of stationary solutions 0 of the family of equations (2.1) possesses the same type of stability. The proof of Theorem 2.11 is similar to the proof of Theorem 2.10, and therefore is omitted here. Note that if the comparison vector function G(t, ω) = G(ω) does not depend on t ∈ R+ , then under certain additional conditions, Theorem 17 from Martynyuk [72] (see also Martynyuk and Obolenskij [82]) can be applied to the comparison system dω = G(ω), dt

ω(t0 ) = ω0 ≥ 0.

This provides a criterion of uniform asymptotic stability for the set of stationary solutions 0 of the family of equations (2.1). 

Example 2.3 Consider the system of equations (2.1), where F (t, X, α) = F (t, X), F ∈ C(R+ × Kc (Rn ), Kc (Rn )) and F (t, 0 ) = 0 for all t ∈ R+ . We shall show ∞ that if a function c(t), c(s) ds < +∞, exists so that 0

D[F (t, X), 0 ] ≤ c(t)D[X, 0 ],

2.6 Application of Strengthened Lyapunov Function

65

then the set of stationary solutions 0 ∈ Kc (Rn ) of the set of equations DH X = F (t, X),

X(t0 ) = X0 ,

is uniformly stable. In fact, for the function V (t, X) = D[X, 0 ] we have 1 D[X, 0 ] ≤ V (t, X) ≤ 2D[X, 0 ], 2 |V (t, X) − V (t, X∗ )| ≤ D[X, X∗ ] for all (t, X), (t, X∗ ) ∈ R+ × Kc (Rn ); V (t + h, X + hF (t, X)) = D[X + hF (t, X), 0 ] ≤ ≤ D[X, 0 ] + hD[F (t, X), 0 ] ≤ D[X, 0 ] + hc(t)D[X, 0 ]. Hence D + V (t, X) ≤ c(t) D[X, 0 ] = G(t, V (t, X)). For this inequality, the comparison equation is of the form dw = c(t) w, dt

w(t0 ) = w0 ,

and its solution w = 0 is uniformly stable under the above conditions. Therefore, the set of stationary solutions 0 is uniformly stable by virtue of Theorem 2.11.

2.6 Application of Strengthened Lyapunov Function If it is difficult to verify the fulfillment of condition (2) of Theorem 2.9, then the weakening of requirements to function (2.13) can be compensated by the introducing into consideration of a strengthening function defined in the domain R+ × S(ρ0 ) ∩ S c (η) for 0 < η < ρ0 , where S c (η) denotes the complement of the set S(η). In the result below, this idea is realized in combination with the principle of comparison. Theorem 2.12 For the family of equations (2.1) let conditions of hypotheses H1 – H3 be satisfied and, moreover: (1) there exists function (2.13) such that V ∈ C(R+ × S(ρ0 ), R+ ), |V (t, X1 , θ ) − V (t, X2 , θ )| ≤ LD[X1 , X2 ], where L > 0, and 0 ≤ V (t, X, θ ) ≤ a(t, D[W0 , 0 ]), where a ∈ C(R+ × S(ρ0 ), R+ ), a(t, ·) is of the class K for any value t ∈ R+ ;

66

2 Analysis of Continuous Equations

(2) for all (t, X) ∈ R+ × S(ρ0 ) the inequality  D + V (t, X, θ )(2.1) ≤ g1 (t, V (t, X, θ )) is satisfied, where g1 ∈ C(R2+ , R) and g1 (t, 0) = 0 for all t ∈ R+ ; (3) for any 0 < η < ρ0 there exists a function Vη ∈ C(R+ × S(ρ0 ) ∩ S c (η), R) such that |Vη (t, X1 ) − Vη (t, X2 )| ≤ Lη D[X1 , X2 ], where Lη > 0 is a constant, and for the function V0 (t, X, θ ) = V (t, X, θ ) + Vη (t, X) the two-sided inequality ψ1T(D[X, 0 ])A1 ψ1 (D[X, 0 ]) ≤ V0 (t, X, θ ) ≤ ψ2T(D[W0 , 0 ])A2 ψ2 (D[W0 , 0 ])

(2.24)

is satisfied, where ψ1 and ψ2 are comparison vector functions of the class K, and A1 and A2 are constant symmetric positive definite matrices; (4) for all (t, X) ∈ R+ × S(ρ0 ) ∩ S c (η) the inequality  D + V0 (t, X, θ )(2.1) ≤ g2 (t, V0 (t, X, θ )) is true, where g2 ∈ C(R2+ , R), g2 (t, 0) = 0 for all t ∈ R+ ; (5) the zero solution ω1 = 0 of the comparison equation dω1 = g1 (t, ω1 ), dt

ω1 (t0 ) = ω10 ≥ 0,

(2.25)

is stable; (6) the zero solution ω2 = 0 of the comparison equation dω2 = g2 (t, ω2 ), dt

ω2 (t0 ) = ω20 ≥ 0,

(2.26)

is stable uniformly in t0 ∈ R+ . Then the set of stationary solutions 0 of the family of equation (2.1) is stable. Proof Estimates (2.24) from condition (3) of Theorem 2.12 are reduced to λm (A1 )ψ 1 (D[X, 0 ]) ≤ V0 (t, X, θ ) ≤ λM (A2 )ψ 2 (D[W0 , 0 ]),

(2.27)

where λm (A1 ) > 0 and λM (A2 ) > 0 are the minimum and the maximum eigenvalues of the matrices A1 and A2 , respectively, and the functions ψ 1 and ψ 2 are of the class K.

2.6 Application of Strengthened Lyapunov Function

67

Let the values 0 < ε < ρ0 and t0 ∈ R+ be given. The uniform stability of the zero solution ω2 = 0 of Eq. (2.26) implies that, given λm (A1 )ψ 1 (ε) > 0, there exists a δ0 = δ0 (ε) such that the condition 0 < ω20 < δ0 yields ω2 (t; t0 , ω20 ) < λm (A1 )ψ 1 (ε),

t ≥ t0 ,

where ω2 (t; t0 , ω20 ) is any solution of equation (2.26). Given ε > 0, for the function ψ 2 of the class K from condition (2.27) we can indicate a δ2 = δ2 (ε) > 0 such that 1 δ0 . 2

λM (A2 )ψ 2 (δ2 )
0 and t0 ∈ R+ one can find a δ ∗ = 2  1  δ ∗ t0 , δ0 > 0 such that the condition 2 0 < ω10 < δ ∗

(2.29)

yields ω1 (t; t0 , ω10 )
0 so that the inequalities D[W0 , 0] < δ1

a(t0 , D[W0 , 0 ]) < δ ∗

will be fulfilled simultaneously. Then we choose δ = min(δ1 , δ2 ) and show that the condition D[W0 , 0 ] < δ implies D[X(t), 0 ] < ε

(2.30)

68

2 Analysis of Continuous Equations

for all t ≥ t0 for any solution X(t) of the family of equations (2.1). If it does not, then the solutions X(t) of Eq. (2.1) and the moments of time t1 , t2 > t0 must exist so that D[X(t1 ), 0 ] = δ2 ,

D[X(t2 ), 0 ] = ε,

δ2 ≤ D[X(t), 0 ] ≤ ε < ρ0

for t1 ≤ t ≤ t2 . Let η = δ2 . Since 0 < δ2 < ρ0 , conditions (3) of Theorem 2.12 are satisfied for the function Vη (t, X). For the function m(t) = V (t, X(t), θ ) + Vη (t, X(t)), condition (4) of Theorem 2.12 provides the inequality D + m(t) ≤ g2 (t, m(t)),

t1 ≤ t ≤ t2 ,

which implies the estimate V (t2 , X(t2 ), θ ) + Vη (t2 , X(t2 )) ≤ r2 (t2 ; t1 , ω20 ),

(2.31)

where ω20 = V (t1 , X(t1 ), θ ) + Vη (t1 , X(t1 )) and r2 (t2 ; t1 , ω20 ) is a maximal solution to the comparison equation (2.26). Note that for the function V (t, X(t), θ ) the estimate V (t1 , X(t1 ), θ ) ≤ r1 (t1 ; t0 , ω10 )

(2.32)

holds true, where ω10 = V (t0 , W0 , θ ) and r1 (t1 ; t0 , ω10 ) is a maximal solution of the comparison equation (2.25). In view of inequalities (2.30) and (2.29) we get from estimate (2.32) that V (t1 , X(t1 ), θ )
0 such that for all t ∈ (ti , ti ) ω(ti , X(ti )) =

σ σ , ω(ti , X(ti )) = σ and ω(t, X(t)) ≥ 2 2

(a)

σ σ and ω(t, X(t)) ≥ . 2 2

(b)

or ω(ti , X(ti )) = σ, ω(ti , X(ti )) =

 Assume that D + ω(t, X(t))(2.1) ≤ M, where M > 0 is a constant.

70

2 Analysis of Continuous Equations

It follows from condition (a) that σ σ = σ − = ω(ti , X(ti )) − ω(ti , X(ti )) ≤ M(ti − ti ). 2 2 Hence we find that for any i the estimate ti − ti ≥ Theorem 2.13 implies

σ 2M

is true. Condition (2 ) of

ti

V (t, X(t), θ ) ≤ r1 (t; t0 , ω10 ) −

n   i=1

c (ω(s, X(s))) ds.

ti

Since ω10 = V (t0 , W0 , θ ) ≤ a(t0 , D[W0 , 0 ]) < a(t0 , δ0 ) < δ ∗ (ρ0 ) for all t ≥ t0 , according to inequality (2.29) we obtain the estimate ω1 (t; t0 , ω10 ) < 12 δ0 (ρ0 ) for all t ≥ t0 and 0 ≤ V (t, X(t), θ ) ≤

σ  σ 1 δ0 (ρ0 ) − c n. 2 2 2M

(2.35)

As n→∞, inequality (2.35) is a contradiction. This proves that lim sup ω(t, X(t)) t →∞

= 0, and consequently the set of stationary solutions 0 of the family of equations (2.1) is asymptotically stable.

2.7 Theorems on Boundedness The theorems by Yoshizawa (see [122]) are the classical results of application of the direct Lyapunov method in the problem on boundedness of solutions to the system of ordinary differential equations. We shall turn back to the family of differential equations (2.1) and formulate boundedness conditions for their solution. Suppose that the family of equations DH X = F (t, X),

X(t0 ) = X0 ∈ Kc (Rn ),

(2.36)

where X ∈ Kc (Rn ) and F ∈ C(R+ × Kc (Rn ), Kc (Rn )), possesses the solution X(t) = X(t, t0 , W0 ), defined for all t ≥ t0 , where W0 = X0 − V0 for any X0 , V0 ∈ Kc (Rn ). Definition 2.2 The set of solutions X(t) of the family of equations (2.36) is: (B1 ) equi-bounded if for any α > 0 and t0 ∈ R+ there exists β(t0 , α) > 0 such that the condition W0  < α implies X(t) < β(t0 , α) for all t ≥ t0 ; (B2 ) uniformly bounded if the value β in the definition B1 does not depend on t0 ;

2.7 Theorems on Boundedness

71

(B3 ) quasi-equiultimate bounded with the boundary B if for any α > 0 and t0 ∈ R+ there exist B > 0 and τ = τ (t0 , α) > 0 such that the condition W0  < α implies X(t) < B for all t ≥ t0 + τ ; (B4 ) quasi-uniformly ultimate bounded if the value τ in the definition B3 does not depend on t0 ; (B5 ) equi-ultimate bounded if the conditions of definitions B1 and B3 are satisfied simultaneously; (B6 ) uniformly ultimate bounded if the conditions of definitions B2 and B4 are satisfied simultaneously. Further we shall apply function (2.13) and establish boundedness conditions for the set of solutions to the family of equations (2.36). Theorem 2.14 For Eq. (2.36), let the following conditions be fulfilled: (1) there exist a function S ∈ C(R+ × Kc (Rn ), R2×2 ), a vector θ ∈ R2+ and a constant L > 0 such that V (t, X, θ ) ∈ C(R+ × Kc (Rn ) × R2+ , R+ ) and |V (t, X1 , θ ) − V (t, X2 , θ )| ≤ LD[X1 , X2 ] for all (t, X) ∈ R+ × Kc (Rn ); (2) there exist comparison vector functions ψ1 , ψ2 ∈ of the class KR and constant symmetric positive definite (2 × 2)-matrices A1 and A2 such that ψ1T(X)A1 ψ1 (X) ≤ V (t, X, θ ) ≤ ψ2T(t, X)A2 ψ2 (t, X) for all (t, X) ∈ R+ × Kc (Rn ); (3) for all (t, X) ∈ R+ × Kc (Rn ) the inequality  D + V (t, X, θ )(2.36) ≤ 0 is satisfied. Then the set of solutions X(t) of the family of equations (2.36) is equi-bounded. Proof Condition (2) of Theorem 2.14 is transformed as: λm (A1 )ψ 1 (X) ≤ V (t, X, θ ) ≤ λM (A2 )ψ 2 (t, X),

(2.37)

where ψ 1 , ψ 2 are of the class KR and such that ψ 1 (X) ≤ ψ1T(X)ψ1 (X)

and

ψ 2 (t, X) ≥ ψ2T(t, X)ψ2 (t, X). Now, given α > 0 and t0 ∈ R+ , we choose β = (t0 , α) > 0 so that λM (A2 )ψ 2 (t0 , α) < λm (A1 )ψ 1 (β).

(2.38)

72

2 Analysis of Continuous Equations

Let us show that for such a choice of β the solution X(t) is equi-bounded. If this is not the case, then there exist a solution X(t) = X(t, t0 , W0 ) and a value t1 > t0 such that X(t1 ) = β and X(t) ≤ β for t0 ≤ t ≤ t1 . It follows from condition (3) of Theorem 2.14 that V (t, X(t), θ ) ≤ V (t0 , W0 , θ ),

t0 ≤ t ≤ t1 .

(2.39)

According to condition (2.37), we find from inequalities (2.38) and (2.39) that λm (A1 )ψ 1 (β) = λm (A1 )ψ 1 (X(t)) ≤ V (t, X(t), θ ) ≤ V (t0 , W0 , θ ) ≤ λM (A2 )ψ 2 (t0 , W0 )

(2.40)

< λM (A2 )ψ 2 (t0 , α) < λm (A1 )ψ 1 (β). The contradiction obtained proves the assertion of Theorem 2.14. Further, for the set S(ρ) = {X ∈ Kc (Rn ) : D[X, ] < ρ}, where  is a zero element of the set Kc (Rn ), we will consider the complement of S c (ρ) and assume that the magnitude ρ can take arbitrarily large values. We shall present a result on uniform boundedness of the set of solutions X(t) to Eq. (2.36). Theorem 2.15 For the family of equations (2.36), assume that the following conditions are fulfilled: (1) there exist a function S ∈ C(R+ × S c (ρ), R2×2 ), a vector θ ∈ R2+ and a constant L > 0 such that V (t, X, θ ) ∈ C(R+ × S c (ρ) × R2+ , R+ ) and |V (t, X1 , θ ) − V (t, X2 , θ )| ≤ LD[X1 , X2 ] for all (t, X) ∈ R+ × S c (ρ); (2) there exist comparison vector functions ψ1 , ψ3 of the class KR and constant symmetric positive definite (2 × 2)-matrices A1 and A2 such that ψ1T(X)A1 ψ1 (X) ≤ V (t, X, θ ) ≤ ψ3T(X)A2 ψ3 (X) for all (t, X) ∈ R+ × S c (ρ); (3) for all (t, X) ∈ R+ × S c (ρ), the inequality  D + V (t, X, θ )(2.36) ≤ 0 is fulfilled. Then the set of solutions X(t) of the family of equations (2.36) is uniformly bounded. Proof As in the proof of Theorem 2.13, we transform condition (2) of Theorem 2.15 to the form (2.37) and choose the value β = β(α) from the condition λM (A2 )ψ 3 (α) < λm (A1 )ψ 1 (β).

2.8 Differential Equations on Product of Convex Spaces

73

If it turns out that α > ρ, then in the estimates of the form of (2.39), (2.40) the set S c (ρ) is considered. If 0 < α ≤ ρ, then the value β = β(ρ) is considered and the same arguments as in the proof of Theorem 2.14 are employed. Theorem 2.16 For the family of equations (2.36) assume that condition (1) of Theorem 2.14 is fulfilled and (2 ) there exists a constant η > 0 such that  D + V (t, X, θ )(2.36) ≤ −ηV (t, X, θ ) for all (t, X) ∈ R+ × Kc (Rn ). Then if condition (3) of Theorem 2.14 holds for X ≥ B, where 0 < B < +∞, then the solution X(t) is equi-ultimate bounded. Proof By the conditions of Theorem 2.14 it follows that the solution X(t) is equibounded. Therefore, the condition W0  < α implies X(t) < β for all t ≥ t0 . Condition (2 ) of Theorem 2.16 implies the estimate V (t, X(t), θ ) ≤ V (t0 , W0 , θ ) exp [ −η(t − t0 )]

(2.41)

for all t ≥ t0 . Let τ=

1 λM (A2 )ψ 2 (t0 , α) ln η λm (A1 )ψ 1 (B)

and for t ≥ t0 + τ the condition X(t) ≥ B. We get from inequality (2.41) that λm (A1 )ψ 1 (B) ≤ λm (A1 )ψ 1 (X(t)) ≤ V (t, X(t), θ ) < < λM (A2 )ψ 2 (t0 , α) exp [−ητ ] = λm (A1 )ψ 1 (B).

(2.42)

Contradiction (2.42) completes the proof of Theorem 2.16. Note that the nonuniform boundedness conditions for the solutions of equation (2.36) can be obtained in terms of the strengthened Lyapunov function in the same way as in the stability analysis of the set 0 of stationary solutions to this equation.

2.8 Differential Equations on Product of Convex Spaces We consider the set of differential systems DH X = F (t, X),

X(t0 ) = X0 ∈ Kcm (Rn ),

(2.43)

74

2 Analysis of Continuous Equations

where X ∈ Kcm (Rn ), the mapping F ∈ C(R+ × Kcm (Rn ), Kcm (Rn )), Kcm (Rn ) = Kc (Rn ) × Kc (Rn ) × . . . × Kc (Rn ) m times and for any i ∈ [1, m] we have Xi ∈ Kc (Rn ). Suppose that Xi and Yi are some sets from the family Kcm (Rn ). We designate by D[Xi , Yi ] the Hausdorff distance between these sets for any i ∈ [1, m] and use the formula D0 [X, Y ] = eT D[Xi , Yi ], e = (1, . . . , 1)T ∈ Rm +, to define the distance between the sets (X, Y ) ∈ Kcm (Rn ). The pair (D0 , Kcm (Rn )) forms a metric space. Together with the metric D0 [X, Y ] we consider the vector distance D : Kcm (Rn ) → Rm + , i.e. D[X, Y ] = (D[X1 , Y1 ], . . . , D[Xm , Ym ]) . In addition, we assume that any pair (D, Kcm (Rn )) forms some other metric space. For the set of differential systems (2.43) we consider the matrix-valued function   U (t, X) = Uij (t, X) ,

i, j ∈ [1, m],

(2.44)

whose elements Uij (t, X) are such that Uij (t, X) ∈ C(R+ × Kcm (Rn ), R+ ) for i = j and Uij (t, X) ∈ C(R+ × Kcm (Rn ), R) for i = j ∈ [1, m]. In terms of function (2.44) we construct the vector function L(t, X, b) = U (t, X)b,

b ∈ Rm +,

(2.45)

and assume that L ∈ C(R+ × Kcm (Rn ), Rm + ) and L(t, X, b) = 0 for all t ∈ R+ , if only X = 0 ∈ Kcm (Rn ). For function (2.45), the derivative by virtue of the set of differential systems (2.43) is considered in the form D + L(t, X, b) = D + U (t, X)b,

(2.46)

where  D + U (t, A) = lim sup [U (t + h, A + hF (t, A)) − U (t, A)]h−1 :  for any A ∈ Kcm (Rn ). h → 0+ We shall first establish the main theorem of the principle of comparison with vector function (2.45) for the set of differential systems (2.43). Theorem 2.17 For the set of systems (2.43) suppose that a matrix-valued function (2.44) is constructed and the function L(t, X, b) satisfies the following conditions: (1) L ∈ C(R+ × Kcm (Rn ), Rm + ), L(t, X, b) = 0 for all t ∈ R+ , provided that X = 0;

2.8 Differential Equations on Product of Convex Spaces

75

(2) there exists an m × m-matrix A with nonnegative elements such that |L(t, X1 , b) − L(t, X2 , b)| ≤ AD[X1 , X2 ] for all (t, X) ∈ R+ × Kcm (Rn ); m (3) for function (2.46) there exists a function G ∈ C(R+ × Rm + , R+ ), G(t, 0) = 0, G(t, w) is quasimonotone with respect to w, such that

D + L(t, X, b) ≤ G(t, L(t, X, b)) for all (t, X, b) ∈ R+ × Kcm (Rn ) × Rm +; (4) for all t ≥ t0 there exists a maximal solution R(t) = R(t, t0 , w0 ) of the vector system dw = G(t, w), dt

w(t0 ) = w0 ≥ 0.

(2.47)

Then, along any solution X(t) = X(t, t0 , X0 ) of Eq. (2.43) existing for t ≥ t0 , the following estimate is satisfied: L(t, X(t), b) ≤ R(t)

for all t ≥ t0 .

(2.48)

Proof Suppose that the set of differential systems (2.43) possesses the solution X(t) for all t ≥ t0 for the initial values X0 ∈  ⊂ Kcm (Rn ). For the function L(t, X(t), b) = g(t), we have g(t0 ) = L(t0 , X0 , b) ≤ w0 . Due to condition (2) of Theorem 2.17 for an arbitrary small h > 0 we get g(t + h) − g(t) = L(t + h, X(t + h), b) − L(t, X(t), b) ≤ AD[X(t + h), X(t) + hF (t, X(t))] + L(t + h, X(t) + hF (t, X(t)), b) − L(t, X(t), b). Hence it follows that   D + g(t) = lim sup [g(t + h) − g(t)]h−1 : h → 0+ ≤ D + L(t, X(t), b) + A lim sup {(D[X(t + h),  X(t) + hF (t, X(t))]) h−1 : h → 0+ .

(2.49)

Since DH X exists by assumption, the relation X(t + h) = X(t) + Z(t) is true, where Z(t) = Z(t, h) is the Hukuhara difference for an arbitrary small h > 0.

76

2 Analysis of Continuous Equations

Taking this relation into account, we transform the second term in estimate (2.49) as: D[X(t + h), X(t) + hF (t, X(t))] = D[X(t) + Z(t), X(t) + hF (t, X(t))] = D[Z(t), hF (t, X(t))] = D[X(t + h) − X(t), hF (t, X(t))]. Hence it follows that

 1 X(t + h) − X(t) D[X(t + h), X(t) + hF (t, X(t))] = D , F (t, X(t)) , h h and further   lim sup [D[X(t + h), X(t) + hF (t, X(t))]] h−1 : h → 0+   

X(t + h) − X(t) −1 + , F (t, X(t)) h : h → 0 = lim sup D h

(2.50)

= D[DH X(t), F (t, X(t))] = 0 along any solution X(t) = X(t, t0 , X0 ) of system (2.43). In view of relation (2.50), we have from condition (3) of Theorem 2.17 and inequality (2.49) that D + g(t) ≤ G(t, g(t)),

g(t0 ) ≤ w0 .

(2.51)

If condition (4) of Theorem 2.17 is satisfied, for inequality (2.51) the comparison system (2.47) is considered and, by Theorem 3.1.2 from the monograph by Laksmikantham et al. [51], the estimate (2.48) is established. This completes the proof of Theorem 2.17. Theorem 2.17 allows us to derive a general scheme of establishing sufficient conditions under which certain dynamic properties of the stationary solution  ∈ Kcm (Rn ) of the set of differential systems (2.43) follow from the corresponding dynamical properties of the zero solution of comparison system (2.47). Theorem 2.18 Assume that (1) for the set of differential systems (2.43), matrix-valued function (2.44) is constructed and for function (2.45) conditions (1)–(2) of Theorem 2.17 are satisfied;

2.8 Differential Equations on Product of Convex Spaces

(2) for the function L0 (t, X, b) =

m 

77

Li (t, X, b) there exist symmetric constant

i=1

m × m matrices 1 and 2 , and comparison vector functions b(D0 [X, ]) and a(D0 [X, ]), a and b are of the Hahn class K, such that bT (D0 [X, ]) 1 b(D0 [X, ]) ≤ L0 (t, X, b) ≤ a T (D0 [X, ]) 2 a(D0 [X, ])

for all (t, X) ∈ R+ × Kcm (Rn ); (2.52)

(3) for the function D + L(t, X, b), defined by expression (2.46), there exists a quasimonotone function G(t, w), nondecreasing with respect to w, G ∈ m C(R+ × Rm + , R+ ), G(t, 0) = 0, such that D + L(t, X, b) ≤ G(t, L(t, X, b)) for all (t, X) ∈ R+ × S(H ), where   S(H ) = X ∈ Kcm (Rn ) : D0 [X, ] < H ; (4) for the mappings F ∈ C(R+ × S(H ), Kcm (Rn )) there exists a set  ∈ Kcm (Rn ) such that F (t, ) =  for all t ∈ R+ . Then, if the matrices 1 and 2 are positive definite, then the stability properties of the stationary solution  of the sets of systems (2.43) follow from the stability properties of the zero solution of comparison system (2.47). Proof Let λm ( 1 ) and λM ( 2 ) be the minimum and the maximum eigenvalue of the matrices 1 and 2 , respectively. Estimate (2.52) is transformed as λm ( 1 )β(D0 [X, ]) ≤ L0 (T , X, b) ≤ λM ( 2 )α(D0 [X, ]),

(2.53)

where α and β are of the Hahn class K such that bT (D0 [X, ])b(D0[X, ]) ≥ β(D0 [X, ]) and a T (D0 [X, ])a(D0[X, ]) ≤ α(D0 [X, ]). Assume that the zero solution of comparison system (2.47) is equiasymptotically stable, i.e. it is equi-stable and attractive. Let the values (t0 , ε) : t0 ∈ R+ and 0 < ε < H be given. The equistability property of the state w = 0 of system (2.47) provides that, given λm ( 1 )β(ε) > 0, there exists a δ1 = δ1 (t0 , ε) > 0 such that the condition eT w0 < δ1 implies eT w(t, t0 , w0 ) < λm ( 1 )β(ε)

for all t ≥ t0 ,

(2.54)

78

2 Analysis of Continuous Equations

where w(t, t0 , w0 ) is any solution of system (2.47). Let w0 = L(t0 , X0 , b). We choose δ = δ(t0 , ε) > 0 so that λM ( 2 )α(δ) < δ1 (t0 , ε). Suppose that D0 [X0 , ] < δ. We shall show that for any solution X(t) = X(t, t0 , X0 ), the estimate D0 [X(t), ] < ε holds for all t ≥ t0 . Consider the solution X(t) with the initial values X0 ∈ Kcm (Rn ) for which D0 [X0 , ] < δ. Let there exist a value t1 > t0 such that D0 [X(t1 ), ] = ε

and D0 [X(t), ] ≤ ε < H

for all

t0 ≤ t ≤ t1 .

By Theorem 2.17 we have the estimate L(t, X(t), b) ≤ R(t, t0 , w0 )

for all t0 ≤ t ≤ t1 ,

where R(t, t0 , w0 ) is the maximum solution of comparison system (2.47). According to estimate (2.53) we have L0 (t0 , X0 , b) ≤ λM ( 2 )α(D0 [w0 , ]) < λM ( 2 )α(δ) < δ1 (t0 , ε). From estimate (2.53) and inequalities (2.54) we get λm ( 1 )β(D0 [X(t1 ), ]) ≤ L0 (t1 , X(t1 ), b) ≤ R0 (t1 , t0 , w0 ) < λm ( 1 )β(ε),

(2.55)

where R0 (t, t0 , w0 ) = eT R(t, t0 , w0 ), e = (1, . . . , 1)T ∈ Rm + . The contradiction (2.55) shows that there is no t1 ∈ R+ for which D0 [X(t1 ), ] = ε. This proves the equistability of stationary solution  ∈ Kcm (Rn ) of the set of systems (2.43). Assume further that the trivial solution of the comparison vector system (2.47) is attractive. Let ε = H and  δ = δ(t0 , H ). For some 0 < η < H , for the given λm ( 1 )β(η) > 0 and t0 ∈ R+ , one finds δ1∗ = δ1 (t0 ) > 0 and τ = τ (t0 , η) > 0 such that the condition eT w0 < δ1∗ implies the estimate eT w(t) < λm ( 1 )β(η) for all t ≥ t0 + τ (t0 , η). Let w0 = L(t0 , X0 , b). We choose δ0∗ = δ0 (t0 ) > 0 so that the inequality λM ( 2 )α(δ0∗ ) < δ1∗ is satisfied. Then, assume that D0 [w0 , ] < δ0 , where δ0 = min(δ1∗ , δ0∗ ). Besides, D0 [X(t), ] < H for all t ≥ t0 , and estimate (2.48) is true for all t ≥ t0 . Suppose now that there exists a sequence {tk }, tk ≥ t0 + τ , tk → ∞, and D0 [X(tk ), ] ≥ η, where X(t) is a solution of the set of differential systems (2.43) with the initial condition D0 [X0 , ] < δ0 . From estimate (2.55) we have λm ( 1 )β(η) ≤ L0 (tk , X(tk ), b) ≤ R0 (tk , t0 , w0 ) < λm ( 1 )β(η).

2.9 Hyers–Ulam–Rassias Stability of the Set of Equations

79

The obtained contradiction shows that there exists no sequence {tk }, for which D0 [X(tk ), ] ≥ η as tk → ∞. This completes the proof of Theorem 2.18. The effective application of Theorem 2.18 is associated with the stability analysis of the state w = 0 of comparison system (2.47). We shall indicate one stability criterion for this system. Corollary 2.1 Let conditions (1), (2), and (4) of Theorem 2.18 be satisfied and there exist a constant m × m-matrix P with nonnegative off-diagonal elements such that D + L(T , X, b) ≤ P L(T , X, b)

for all (t, X, b) ∈ R+ × Kcm (Rn ) × Rm +.

If the matrices 1 and 2 in estimate (2.52) are positive definite and for any δ > 0 the system of inequalities m 

Pij j < 0,

i = 1, 2, . . . , m,

j =1

admits the solution 1 , . . . , m such that 0 < j for all j = 1, 2, . . . , m, then the stationary solution  ∈ Kcm (Rn ) of the set of differential systems (2.43) is uniformly asymptotically stable.

2.9 Hyers–Ulam–Rassias Stability of the Set of Equations In this section we present the Hyer–Ulam–Rassias stability (H.U.R. stability) conditions for the family of regularized equations. Consider the family of regularized equations DH U (t) = Fβ (t, U (t)),

U (t0 ) = U0 ∈ Kc (Rn ),

(2.56)

where U ∈ Kc (Rn ), Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )) and β ∈ [0, 1]. For the family of equations (2.56) we formulate the definition of H.U.R. stability as follows. Definition 2.3 Family of equations (2.56) is H.U.R.-stable with respect to the set of functions (t) ∈ Kc (Rn ) if there exists a constant C > 0 such that for every Y (t) ∈ Kc (Rn ), for which DH Y − Fβ (t, Y ) ⊆ (t) for all t ∈ R+ ,

(2.57)

a solution U (t) ∈ Kc (Rn ) of the family of equations (2.56) is found so that D[Y (t), U (t)] ≤ C D[ (t), 0 ] for all t ∈ R+ , where 0 ∈ Kc (Rn ) is a zero element of the set Kc (Rn ).

(2.58)

80

2 Analysis of Continuous Equations

Remark 2.3 Definition 2.3 can be changed if, instead of condition (2.57), the condition D[GY (t), 0 ] ≤ D[ (t), 0 ], is considered for all t ∈ R+ , where GY (t) = DH Y − Fβ (t, Y ). It is clear that the set of functions Y (t) ∈ Kc (Rn ) is a solution to the inequality (2.57) if the family of functions GY (t) ∈ Kc (Rn ) is such that D[GY (t), 0 ] ≤ C D[ (t), 0 ] for all t ∈ R+ and DH Y = Fβ (t, Y ) + GY (t) for all t ∈ R+ . Remark 2.4 If the set of functions Y (t) ∈ Kc (Rn ) is a solution of the inequality (2.58), then it is also a solution of the inequality t D[Y (t) − Y0 −

t Fβ (s, Y (s))ds, 0 ] ≤ C

0

D[ (s), 0 ]ds 0

for all t ∈ R+ . Further conditions for H.U.R.-stability of the family of equations (2.56) are established. Theorem 2.19 Assume that (1) for any β ∈ [0, 1] the mappings Fβ ∈ C(R+ × Kc (Rn ), Kc (Rn )); (2) for the given set of functions (t) ∈ Kc (Rn ) for every Y (t) ∈ ∈ Kc (Rn ) the inclusion DH Y − Fβ (t, Y ) ⊆ (t) for all t ∈ R+ is fulfilled; (3) there exists a function λ(t) ∈ L1 (R+ , R+ ) such that D[Fβ (t, U ), Fβ (t, Y )] ≤ λ(t)D[U (t), Y (t)] for all (t, U, Y ) ∈ R+ × Kc (Rn ) × Kc (Rn );

2.9 Hyers–Ulam–Rassias Stability of the Set of Equations

81

(4) there exists a constant γ > 0 such that t D[ (s), 0 ]ds ≤ γ D[ (t), 0 ] 0

for all t ∈ R+ . Then the family of equations (2.56) is H.U.R.-stable with respect to the set of functions (t) ∈ Kc (Rn ). Proof Let U (t) = U (t, t0 , U0 ) be any solution of the family of equations (2.56) with the initial conditions U0 = Y0 ∈ Kc (Rn ). For the family of equations (2.56) we have t U (t) = Y0 +

(2.59)

Fβ (s, U (s)) ds 0

and under condition (2) of Theorem 2.19 t Y (t) − Y0 −

t Fβ (s, Y (s))ds ≤

0

(s)ds, t ∈ R+ .

(2.60)

0

In view of conditions (2)–(4) of Theorem 2.19, the correlations (2.59) and (2.60) yield

t

D[Y (t),U (t)] ≤ D Y0 +

Fβ (s, U (s)) ds, 0

t Y0 +

Fβ (s, Y (s)) ds + 0

0

t

≤ 0

0

  D Fβ (s, U (s)), Fβ (s, Y (s)) ds +

0

D[ (s), 0 ] ds 0

λ(s)D[Y (s), U (s)] ds + γ D[ (t)0 ] 0

(s) ds

t

t ≤



t Fβ (s, Y (s)) ds +

Fβ (s, U (s)) ds, 0

t

(s) ds

t ≤D



t

(2.61)

82

2 Analysis of Continuous Equations

for all t ∈ R+ . Getting together the first and the last expressions in estimates (2.61) we arrive at t D[Y (t), U (t)] ≤ γ D[ (t), 0 ] +

λs D[Y (s), U (s)] ds. 0

Hence, applying the known integral inequality, we find that D[Y (t), U (t)] ≤ γ D[ (t), 0 ] t

t +

γ D[ (s), 0 ]λ(s) exp

λ(u)du

s

0

 ≤ γ D[ (t), 0 ] 1 −

t

t exp

 λ(u)du

s

0



t

t  = γ D[ (t), 0 ] 1 − exp λ(u) du s

t = γ exp

(2.62)

0

λ(u) du D[ (t), 0 ]

0

∞ ≤ γ exp

λ(u) du D[ (t), 0 ] = C D[ (t), 0 ],

0

 where C = γ exp

∞

 λ(u)du .

0

Therefore, we obtain from (2.62) that D[Y (t), U (t)] ≤ C D[ (t), 0 ] for all t ∈ R+ . This completes the proof of Theorem 2.19. Remark 2.5 If in condition (2) of Theorem 2.19 the set Y (t) ∈ Kc (Rn ) is an approximate solution of the family of equations (2.56), then estimate (2.58) is the estimate of deviations of the approximate solution from the unknown exact solution U (t) ∈ Kc (Rn ) of the family of equations (2.56).

2.10 Notes and References

83

2.10 Notes and References The concepts of stability (see Lyapunov [53]) and/or boundedness of motion (see Yoshizawa [122]) are the key ones in the modern theory of equations. Some results of the development of the theory of motion stability in the late twentieth century were stated in the monographs published in the series “Stability and Control: Theory, Methods and Applications,” Gordon and Breach Science Publishers, Volumes 1–12, 1995–2000 and Taylor and Francis Group,Volumes 13–22, 2003– 2004, edited by A.A. Martynyuk and V. Lakshmikantham. The development of the stability theory for set trajectories of the families of equations is a compelling need. The main methods used to this end are: the generalized direct Lyapunov method, based on auxiliary scalar, vector or matrixvalued functions and the comparison principle, based on differential or integral inequalities. It is known that both methods are well developed for many classes of differential equations (see Corduneanu [2], Szarski [115], Walter [119]). The method used in this chapter is a generalization of the method of matrix Lyapunov functions, which is described in the monograph by Martynyuk [64] (see Shurz [109] for Review of the book) for continuous systems of ordinary differential equations, singularly perturbed systems and systems with random parameters. In the monograph by Martynyuk [66] (see Gajic [28] for Review of the book), for continuous large-scale systems, the method of matrix Lyapunov functions is developed on the basis of mixed and regular decomposition of complex systems. In this case, the hierarchical matrix Lyapunov functions are applied to obtain sufficient conditions for various types of stability and instability of motion. The method of matrix Lyapunov functions was developed in the general theory of equations in connection with the research of large-scale systems and the impossibility to apply the method of vector Lyapunov functions in certain important cases (see Djordjevi`c [25], Martynyuk [69], and others). A new version of the method of matrix Lyapunov functions was developed while motion stability investigation of systems with structural perturbations (see Martynyuk and Miladzhanov [89]). In the monograph by Kats and Martynyuk [38] the method of matrix Lyapunov functions is adapted for the analysis of stability and stabilization of motion of nonlinear systems with random structure. Sections 2.2–2.7 of this chapter are based on the works by Martynyuk [70], Martynyuk and Martynyuk-Chernienko [83–85]. The approach to the trajectories analysis developed in these works is associated with the general theory of set differential equations (see Laksmikantham et al. [49]) and the generalized direct Lyapunov method based on auxiliary matrix-valued functions. In Sect. 2.8 the results of the paper by Martynyuk et al. [91] are used. The notion of Hyers–Ulam–Rassias-stability (H.U.R.-stability) is based on the works by Ulam [115], Hyers [35, 36], and Rassias [103] and has been attracting the

84

2 Analysis of Continuous Equations

attention of many experts for the last two decades (see Rus [106], Corduneanu et al. [20] and the bibliography therein). Unlike the notion of stability in the sense of Lyapunov, the H.U.R.-stability of equation and/or system of equations is not associated with the restrictions on the initial conditions for solutions (motion) of the equations under consideration. The results of Sect. 2.9 are new.

Chapter 3

Discrete-Time Systems with Switching

For the set of difference equations generated by discretization of the set of differential equations with Hukuhara derivative a principle of comparison with matrix Lyapunov function is specified and sufficient stability conditions of certain type are established. The analysis is carried out in terms of a matrix Lyapunov function of special structure. For an essentially nonlinear multiconnected switched difference system, conditions are obtained providing the asymptotic stability of its zero solution for any switching law. An example is presented to demonstrate efficiency of the proposed approaches.

3.1 Introduction In the present chapter we set out a general approach to stability analysis problem for a set of trajectories of difference equations with uncertain parameter values. In Sect. 3.2 we present some general information from metric spaces which is necessary for further investigation. Section 3.3. sets out statement of the problem. Here a procedure of regularization of the family of difference equations with uncertain parameter values is proposed. After the “regularization” procedure is applied to the initial difference system, a theorem of the comparison method and a generalized Lyapunov function are employed to establish sufficient stability conditions for zero solution. Section 3.4 deals with the structure of auxiliary matrix function. The study takes into account auxiliary families of equations obtained in the process of regularization of the original family of equations. Some simple stability criteria for the stationary solution of set equations (3.1) are given in Sect. 3.5. Section 3.6 treats a multi-connected switched difference system. As an application of the general approach, a nonlinear multiconnected (complex) switched difference system is studied.

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_3

85

86

3 Discrete-Time Systems with Switching

Section 3.7 considers the construction of a comparison system. By means of the comparison method, the conditions are obtained under which the zero solution of the system is asymptotically stable for any switching law. In Sect. 3.8 a new approach is developed for constructing a general Lyapunov function for switching systems. Since the proposed function is constructed in terms of auxiliary (2 × 2)-matrix function, this allows one to weaken the requirements to the dynamic properties of subsystems, in terms of which the matrix function elements are found. Finally, Sect. 3.9 provides an example which illustrates some general results of the chapter.

3.2 Preliminaries Further we shall need the following notions and results. Let Kc (Rq ) denote a family of all nonempty compact and convex subsets in the space Rq ; K(Rq ) contain all nonempty compact subsets in Rq , and C(Rq ) be a subset of all nonempty closed subsets in Rq . The distance between nonempty closed subsets A and B of the space Rq is specified by the formula D[A, B] = max {dH (A, B), dH (B, A)} , where dH (B, A) = sup {d(b, A) : b ∈ B} is a Hausdorff separation of the sets A and B, and d(b, A) = inf{b − a : a ∈ A} is a distance from the point b to the set A,  ·  is the Euclidean norm. The pair (C(Rq ), D) is a complete separable metric space, where K(Rq ) and Kc (Rq ) are closed subsets. Let F be a mapping of the domain Q of the space Rq into the metric space (Kc (Rq ), D), i.e., F : Q → Kc (Rq ), which is equivalent to the inclusion F (t) ∈ Kc (Rq ) for all t ∈ Q. Such mappings are called the multivalued mappings of Q into Rq . Let N denote a set of positive integers, N+ = N ∪ {0}, and we designate by Nn0 the set Nn0 = {n0 , n0 + 1, . . . , n0 + k, . . .}, where k ∈ N and n0 ∈ N+ . We recall a result from the theory of classical difference equations, which will be of further need. It is worth noting that, in the statement of this result, vector inequalities are understood component-wise: if a, b ∈ Rq , then the inequality a ≥ b means that ai ≥ bi for all i = 1, . . . , q.

3.3 Statement of the Problem

87

Theorem 3.1 (see Lakshmikantham et al. [51]) Let the function G(n, U ), G : N+ q × R+ → Rq , satisfy the condition G(n, U˜ ) ≤ G(n, Uˆ ) for n ∈ N+ and 0 ≤ U˜ ≤ Uˆ . If n0 ∈ N+ , and Yn ≥ 0,

Yn+1 ≤ G(n, Yn ),

Zn+1 ≥ G(n, Zn )

for any n ≥ n0 , then the inequality Yn0 ≤ Zn0 implies Yn ≤ Zn for all n ≥ n0 . Corollary 3.1 Let the function g(n, r), g : N+ × R+ → R, be nondecreasing with respect to r for every value of n. If n0 ∈ N+ , and yn ≥ 0,

yn+1 ≤ g(n, yn ),

zn+1 ≥ g(n, zn )

for any n ≥ n0 , then the inequality yn0 ≤ zn0 implies yn ≤ zn for all n ≥ n0 . Corollary 3.2 If in the conditions of Corollary 3.1 the function g(n, r) is of the form g(n, r) = r + w(n, r), where |w(n, r)|/r → 0 as r → 0, then the assertion of Corollary 3.1 holds.

3.3 Statement of the Problem Consider a set of discrete-time equations in the form Xn+1 = F (n, Xn , α),

Xn0 = X0 ,

(3.1)

where the mapping F : N+ × Kc (Rq ) → Kc (Rq ) is continuous with respect to Xn for every n, and Xn ∈ Kc (Rq ) for all n ≥ n0 ; α ∈ I ⊂ Rd is the uncertainty parameter. Together with system (3.1) we shall consider the following sets of difference equations: Xn+1 = FM (n, Xn ), where FM (n, Xn ) = co



Xn0 = X0 ,

(3.2)

Xn0 = X0 ,

(3.3)

F (n, Xn , α);

α∈I

Xn+1 = Fm (n, Xn ),

88

3 Discrete-Time Systems with Switching

where Fm (n, Xn ) = co



F (n, Xn , α);

α∈I

Xn+1 = Fβ (n, Xn ),

Xn0 = X0 ,

(3.4)

where Fβ (n, Xn ) = FM (n, Xn )β + Fm (n, Xn )(1 − β), β ∈ [0, 1]. In what follows it is assumed that Fm , FM , and Fβ ∈ Kc (Rq ). We shall establish stability conditions for a stationary solution  ∈ Kc (Rq ) of the set of systems of difference equations (3.1) in terms of the Lyapunov function constructed for Eqs. (3.2)–(3.4).

3.4 Structure of Auxiliary Matrix Function We introduce an auxiliary function U (n, β, Xn ) = [Uij (n, β, Xn )],

i, j = 1, 2,

(3.5)

where the element U11 (n, Xn ) is associated with the set of equations (3.2), U22 (n, Xn ) is associated with the set of equations (3.3), U12 (n, β, Xn ) = U21 (n, β, Xn ) is associated with the set of equations (3.4). In terms of function (3.5) we construct a scalar function V (n, Xn , β, θ ) = θ T U (n, β, Xn )θ,

θ ∈ R2+ ,

(3.6)

and assume that V : N+ × Kc (Rq ) × [0, 1] × R2+ → R+ . The function (3.6) is a Lyapunov function for the set of equations (3.1) if, together with the first difference V (n, Xn , β, θ ) = V (n + 1, Xn+1 , β, θ ) − V (n, Xn , β, θ ),

(3.7)

it solves the problem of stability of a stationary solution  ∈ Kc (Rq ) of the set of equations (3.1). Alongside the set of difference equations (3.1), consider the scalar comparison equation un+1 = g(n, un ),

un0 = u0 ,

(3.8)

3.5 Stability of a Stationary Solution

89

which is associated with the function (3.6) and the first difference (3.7). Here the function g(n, r) is continuous and nondecreasing with respect to r for every n ∈ N+ , and g(n, 0) = 0 for all n ∈ N+ . Theorem 3.2 For the set of equations (3.1) let function (3.6) be constructed and for the first difference (3.7) the estimate  V (n, Xn , β, θ )(3.1) ≤ w(n, V (n, Xn , β, θ ))

(3.9)

hold, where w(n, r) satisfies the condition of Corollary 3.2. If n0 ∈ N+ and V (n0 , Xn0 , β, θ ) ≤ un0 , then V (n + 1, Xn+1 , β, θ ) ≤ un+1 for all n ∈ Nn0 . Proof Denote un+1 = V (n + 1, Xn+1 , β, θ ). By the hypothesis of Theorem 3.2 we have V (n0 , Xn0 , β, θ ) ≤ un0 and, moreover, un+1 ≤ un + w(n, un )

for all n ≥ n0 .

Therefore, in the comparison equation (3.8), g(n, r) = r + w(n, r). Further, according to Corollary 3.2, we have the estimate V (n + 1, Xn+1 , β, θ ) ≤ un+1 for all n ≥ n0 . This completes the proof of Theorem 3.2.

3.5 Stability of a Stationary Solution We recall that the set W0 ∈ Kc (Rq ) is called the Hukuhara difference for the sets X0 , Y0 ∈ Kc (Rq ) if X0 = Y0 + W0 . For the set of equations (3.1) we introduce the following assumptions: H1 . For Eq. (3.1) there exists a set of stationary solutions 0 ∈ Kc (Rq ), i.e., F (n, 0 ) = 0 for all n ∈ N+ . H2 . For any X0 ∈ Kc (Rq ) and Y0 ∈ Kc (Rq ) there exists the Hukuhara difference W0 ∈ Kc (Rq ). Definition 3.1 The stationary solution 0 of the set of equations (3.1) is: (a) stable if for any n0 ∈ N+ and ε > 0 there exists a δ = δ(n0 , ε) > 0 such that the inequality D[W0 , 0 ] < δ implies the estimate D[Xn , 0 ] < ε for all n ≥ n0 , where W0 is the Hukuhara difference for the initial values X0 ∈ Kc (Rq ); (b) attractive if for any n0 ∈ N+ there exists α(n0 ) > 0, and for any ξ > 0 there exists τ (n0 , W0 , ξ ) ∈ N+ such that the inequality D[W0 , 0 ] < α(n0 ) implies the estimate D[Xn , 0 ] < ξ for any n ≥ n0 + τ (n0 , W0 , ξ ); (c) asymptotically stable if it is both stable and attractive. For the set of difference equations (3.1) the following result is valid.

90

3 Discrete-Time Systems with Switching

Theorem 3.3 Assume that for the set of difference equations (3.1) there exist: (a) function (3.6), constant symmetric (2 × 2)-matrices A(θ ) and B(θ ) and comparison vector functions φ1 , φ2 ∈ KR-class of Hahn such that φ1T (D[Xn , 0 ])A(θ )φ1(D[Xn , 0 ]) ≤ V (n, Xn , β, θ ) ≤ φ2T (D[Xn , 0 ])B(θ )φ2 (D[Xn , 0 ])

(3.10)

for all β ∈ [0, 1], n ∈ N+ , Xn ∈ Kc (Rq ) and θ ∈ R2+ ; (b) function w(n, r) specified in Corollary 3.2, such that estimate (3.9) is valid for all β ∈ [0, 1] and n ∈ N+ . Then, if the matrices A(θ ) and B(θ ) are positive definite, the stationary solution 0 ∈ Kc (Rq ) of the set of difference equations (3.1) possesses the same dynamical properties as the zero solution of the comparison equation (3.8). Proof Let λm (A) and λM (B) be the minimal and maximal eigenvalues of the matrices A and B, respectively. From the estimate (3.10) it follows that λm (A)b(D[Xn , 0 ]) ≤ V (n, Xn , β, θ ) ≤ λM (B)a(D[Xn , 0 ]), where the functions a, b ∈ KR-class of Hahn, and such that φ1T (D[Xn , 0 ])φ1 (D[Xn , 0 ]) ≥ b(D[Xn , 0 ]), φ2T (D[Xn , 0 ])φ2 (D[Xn , 0 ]) ≤ a(D[Xn , 0 ]) for all Xn ∈ Kc (Rq ), n ∈ N+ . Further, we assume that the zero solution of the comparison equation (3.8) is asymptotically stable. Let n0 ∈ N+ and ε ∈ (0, H ) be given, where H = const > 0. Moreover, for the values λm (A)b(ε) > 0 and n0 ∈ N+ , there exists a value δ1 = δ1 (n0 , ε) > 0 such that, if 0 < un0 < δ1 , then un+1 < λm (A)b(ε) for all n ≥ n0 . Next, we choose a value δ = δ(n0 , ε) > 0 satisfying the condition λM (B)a(δ) < δ1 (n0 , ε). According to Theorem 3.2, the estimate V (n + 1, Xn+1 , β, θ ) ≤ un+1 holds for all n ≥ n0 , and by virtue of inequality (3.10) we obtain λm (A)b(D[Xn+1 , 0 ]) ≤ V (n + 1, Xn+1 , β, θ ) ≤ un+1 for all n ≥ n0 . Let X0 ∈ Kc (Rq ), and D[X0 , 0 ] < δ. We take un0 = V (n0 , X0 , β, θ ). Then, it is easy to see that un0 ≤ λM (B)a(D[X0 , 0 ]) ≤ λM (B)a(δ) < δ1 .

3.6 Multi-Connected Switched Difference System

91

Hence, λm (A)b(D[Xn+1 , 0 ]) ≤ λm (A)b(ε) for all n ≥ n0 , and therefore, D[Xn+1 , 0 ] < ε for all n ≥ n0 . Further, from the estimate λm (A)b(D[Xn+1 , 0 ]) ≤ V (n + 1, Xn+1 , β, θ ) ≤ un+1

for n ≥ n0

it follows that for the initial conditions X0 ∈ S0 (ρ0 ) we have D[Xn+1 , 0 ] → 0 as n → ∞, whenever un+1 → 0 as n → ∞. This completes the proof.

3.6 Multi-Connected Switched Difference System Consider the system (σ )

Xi (n+1) = Xi (n)+Fi

(Xi (n))+

m 

(σ )

ij (n, X(n)),

i = 1, . . . , m,

(3.11)

j =1

describing the dynamics of a complex (multiconnected) system composed of m interconnected subsystems. Here Xi (n) ∈ Rqi are state vectors, X(n) = T (n))T ; function σ = σ (n), with σ (n) ∈ {1, . . . , S}, defines the (X1T (n), . . . , Xm switching law; n ∈ N+ ; components of the vectors Fi(s) (Xi ) are continuous for all Xi ∈ Rqi homogeneous functions of the order μi > 1; vector functions ij(s) (n, X) are defined for n ∈ N+ , X < H , 0 < H ≤ +∞, and for every fixed n are continuous with respect to X; i, j = 1, . . . , m; s = 1, . . . , S. We assume that the estimates (s)

(s)

ij (n, X) ≤ cij Xj αij hold for n ∈ N+ , X < H , where cij(s) ≥ 0, αij > 0; i, j = 1, . . . , m; s = 1, . . . , S. When σ (n) = s, s ∈ {1, . . . , S}, the subsystem (s)

Xi (n + 1) = Xi (n) + Fi (Xi (n)) +

m 

(s)

ij (n, X(n)),

i = 1, . . . , m,

j =1

is active. Thus, it is assumed that switching between distinct operation modes does not change the orders of homogeneity μi of the interacting subsystems and the degree of influence αij of each subsystem on the other one. For instance, such

92

3 Discrete-Time Systems with Switching (s)

situation takes place in the case when the components of vector functions Fi (Xi ) (s) and ij (n, X) are homogeneous forms, while switching occurs in the coefficients of these forms. From the properties of the right-hand sides of system (3.11) it follows that the system admits the zero solution. We will look for conditions under which this solution is asymptotically stable for any switching law.

3.7 Construction of a Comparison System It is worth mentioning that the methods for Lyapunov functions construction are better developed for continuous-time systems than for discrete-time ones. However, in some cases, the Lyapunov function found for a differential system can be applied to the corresponding difference system as well, see Aleksandrov and Zhabko [4], etc. According to this approach, for every i ∈ {1, . . . , m}, consider the isolated difference subsystems Xi (n + 1) = Xi (n) + Fi(s) (Xi (n)),

s = 1, . . . , S,

(3.12)

and the corresponding subsystems of differential equations (s) Z˙ i (t) = Fi (Zi (t)),

s = 1, . . . , S.

(3.13)

Assumption 3.1 For every i ∈ {1, . . . , m}, the zero solutions of all subsystems (3.13) are asymptotically stable. Remark 3.1 In Aleksandrov and Zhabko [4], it was proved that the fulfillment of Assumption 3.1 implies that the zero solutions of difference subsystems (3.12) are asymptotically stable as well. Assumption 3.2 For every i ∈ {1, . . . , m}, for subsystems (3.13) a common Lyapunov function vi (Zi ) is constructed which is twice continuously differentiable positive definite positive homogeneous function of the degree γi > 2, such that the functions (∂vi (Zi )/∂Zi )T Fi(s) (Zi ), s = 1, . . . , S, are negative definite. Remark 3.2 Sufficient conditions for the existence of a common Lyapunov function for a family of homogeneous systems of differential equations have been obtained in Vassilyev et al. [116]. If Assumption 3.2 is fulfilled, then, see Zubov [124], the inequalities a1i Zi γi ≤ vi (Zi ) ≤ a2i Zi γi ,

∂vi (Zi ) T (s) ∂vi (Zi ) (s) γi −1 , Fi (Zi ) ≤ −a4i Zi γi −1+μi ≤ a3i Zi  ∂Zi ∂Zi

3.7 Construction of a Comparison System

93 (s)

hold for all Zi ∈ Rqi , where a1i , a2i , a3i , a4i are positive constants depending on chosen Lyapunov functions, i = 1, . . . , m, s = 1, . . . , S. Construct the vector Lyapunov function V (Z) = (v1 (Z1 ), . . . , vm (Zm ))T . Consider the first difference of the function with respect to solutions of system (3.11). It can be easily shown, see Aleksandrov and Zhabko [4], that there exists a number H1 ∈ (0, H ) such that the estimates  (σ ) vi (3.11) ≤ ϕi (V (X(n))),

i = 1, . . . , m,

hold for X(n) < H1 and for all n ∈ N+ . Here ϕi(σ ) (V )

=

γi −1+μi γi

−bi(σ )vi

γi −1 γi

+ vi

m 

αij γ dij(σ ) vj j

+ gi (V ),

j =1

bi(σ )

=

(σ ) − a4i a2i

γi −1+μi γi

,

dij(σ )

=

− a3i cij(σ ) a1i

γi −1 γi

α

− γij

a1j

j

,

2α 2μi +γi −2 γi −2 m  γjij γ γ + vi i vj gi (V ) = c vi i

j =1 2μi γi

+ vi

m 

αij (γi −2) γj

vj

j =1

+

m 

αij γi γj

vj

,

(3.14)

j =1

and c is a positive constant, i, j = 1, . . . , m. We see that the system U (n + 1) = U (n) + (σ ) (U (n)),

(3.15)

(σ ) (U ))T , is a comparison where U = (u1 , . . . , um )T , (σ ) (U ) = (ϕ1(σ ) (U ), . . . , ϕm system for (3.11). In fact, there exists a number η > 0 such that if for the components of vectors U˜ and Uˆ the inequalities 0 ≤ u˜ i ≤ uˆ i < η, i = 1, . . . , m, hold, then U˜ + (σ ) (U˜ ) ≤ Uˆ + (σ ) (Uˆ ). Hence (see Theorem 3.1), the asymptotic stability in Rm + of the zero solution of (3.15) implies the asymptotic stability of the zero solution of system (3.11). In system (3.15), switching occurs between the subsystems from the family

U (n + 1) = U (n) + (s) (U (n)),

s = 1, . . . , S.

(3.16)

94

3 Discrete-Time Systems with Switching

3.8 Construction of a Common Lyapunov Function Let us investigate now the stability of the zero solution of the comparison system (3.15). As it was mentioned in the Introduction, to prove the asymptotic stability uniform with respect to the switching law, it is sufficient to construct a common Lyapunov function for the family of subsystems (3.16) satisfying the conditions of the discrete counterpart of the Lyapunov asymptotic stability theorem. Assumption 3.3 The inequality system αij hj ≥ μi hi

max c(s) s=1,...,S ij

for

= 0,

i, j = 1, . . . , m,

(3.17)

admits a positive solution. Remark 3.3 Assumption 3.3 means that the orders of the right-hand sides of the isolated subsystems (3.12) are, in a certain sense, less or equal to those of the functions characterizing interconnections between the subsystems. Assumption 3.4 The inequality system (s) μ

− bi ξi i +

m 

(s) α

dij ξj ij < 0,

i = 1, . . . , m,

s = 1, . . . , S,

(3.18)

j =1

admits a positive solution. Remark 3.4 The assumption on the existence of positive numbers ξ1 , . . ., ξm satisfying inequalities (3.18) is a well-known Martynyuk–Obolenskij condition of stability for the corresponding Wazewskij systems (s) μ

z˙ i (t) = −bi zi i (t) +

m 

(s) α

dij zj ij (t),

i = 1, . . . , m,

s = 1, . . . , S,

j =1

(see Aleksandrov and Platonov [3], Martynyuk and Obolenskij [82] and Martynyuk [68, 72]). Theorem 3.4 If Assumptions 3.1–3.4 are fulfilled, then the zero solution of system (3.11) is asymptotically stable for any switching law. Proof Let h1 , . . . , hm and ξ1 , . . . , ξm be positive solutions of inequalities (3.17) γi −1+μi

γ and (3.18), respectively. Denote fi (ui ) = ui i , h˜ i = hi (γi − 1 + μi ), ξ˜i = γi −1+μi ξi , i = 1, . . . , m. Without loss of generality, we assume that h˜ i < 1, i = 1, . . . , m. Find λ > 0 such that

(s) μ −bi ξi i

+

m  j =1

(s) α

dij ξj ij ≤ −λ,

i = 1, . . . , m,

s = 1, . . . , S.

3.8 Construction of a Common Lyapunov Function

95

Choose a positive number δ. Let L(δ) = max

γi − 1 + μi μiγ−1 δ i . i=1,...,m γi

max fi (ui ) = max

i=1,...,m 0≤ui ≤δ

It is worth noting that L(δ) → 0 as δ → 0. Construct a common Lyapunov function for subsystems (3.16) in the form

(U ) = max V

i=1,...,m

fi (ui ) ξ˜i

1/h˜ i (3.19)

,

(U ) is continuous for U ∈ Rm see [24]. Function V + and positive definite. Choose some s ∈ {1, . . . , S}, and consider a solution U (n) = (u1 (n), . . ., um (n))T of s-th subsystem from the family (3.16), which remains in the region Q = {U ∈ Rm : U = 0, 0 ≤ ui ≤ δ, i = 1, . . . , m} for n = n0 , . . . , n1 , where n0 ≥ 0, n0 < n1 ≤ +∞. Consider the first difference of function (3.19) with respect to the solution. We obtain   = V (U (n + 1)) − V (U (n)) V (s) = max

i=1,...,m

fi (ui (n + 1)) ξ˜i

1/h˜ i

− max

i=1,...,m

fi (ui (n)) ξ˜i

1/h˜ i

for n0 ≤ n < n1 . For every n ∈ {n0 , . . . , n1 }, denote by An a subset of {1, . . . , m} such that

fi (ui (n)) ξ˜i fi (ui (n)) ξ˜i

1/h˜ i

1/h˜ i

(U (n)) =V

for i ∈ An ,

(U (n)) 1. If

fr (ur (n)) ξ˜r

1/h˜ r



fi (ui (n)) ≥ω ξ˜i

1/h˜ i ,

then, for sufficiently small values of δ, we obtain that the following inequalities hold:

1

˜ ˜  fi (ui (n) + ηin ui (n)) h˜ i −1 fr (ur (n)) hi /hr   ˜ V (s) ≤ b1 L(δ) ξ˜i ξ˜r

˜

1 fr (ur (n)) 1/hr − 1− ω ξ˜r 1

˜ ˜  fr (ur (n)) hi /hr  fi (ui (n) + ηin ui (n)) − fi (ui (n))  h˜ i −1 ˜ ≤ b2 L(δ)   ξ˜r ξ˜i

˜



˜

fi (ui (n)) 1/hi −1 1 fr (ur (n)) 1/hr + − 1− ω ξ˜i ξ˜r



˜ fr (ur (n)) 1/hr 1  1 1 ˜ 1− ≤ b3 L(δ) − 1 − ≤− V (U (n)), ω 2 ω ξ˜r where b˜1 , b˜2 , b˜3 are positive constants. In the case when

fr (ur (n)) ξ˜r

1/h˜ r



fi (ui (n)) ≤ω ξ˜i

1/h˜ i ,

3.8 Construction of a Common Lyapunov Function

97

and the value of δ is sufficiently small, the relations ˜

1/h −1  fi i (ui (n) + ηin ui (n))fi (ui (n) + ηin ui (n))   V (s) ≤ 1/h˜ h˜ i ξ˜i i 

˜ ˜ fr (ur (n)) hi /hr (s) (s) ˜ × −bi fi (ui (n)) + bi ξi ξ˜r ⎞⎞ ⎛ αij

˜ ˜ γi −1 m  fr (ur (n)) hi /hr⎝ (s) ˜ γj −1+μj γi −1+μi (s) + dij ξ˜j + b(δ)⎠⎠ −bi ξi + ξ˜i ξ˜r

 −

j =1

fr (ur (n)) ξ˜r

1/h˜ i −1

+

fi



1/h˜ r

−

fi (ui (n)) ξ˜i

1/h˜ i 

≤ (U (n))

(ui (n) + ηin ui (n))fi (ui (n) + ηin ui (n)) 1/h˜ h˜ i ξ˜i i γ −1

× ξi i

h˜ i (U (n)) (−λ + b(δ))V

are valid. Here b(δ) > 0, and b(δ) → 0 as δ → 0, whereas (s)

(U (n)) =  ×  −

L(δ)bi h˜ i

fr (ur (n)) ξ˜r fr (ur (n)) ξ˜r



fi (ui (n) + ηin ui (n)) ξ˜i

h˜ i /h˜ r

1/h˜ r

− −

fi (ui (n)) ξ˜i

fi (ui (n)) ξ˜i

1/h˜ i −1

h˜ i /h˜ i 

1/h˜ i  .

By using the inequality x ζ − y ζ ≥ ζy ζ −1 (x − y) which holds for all x ≥ y ≥ 0 and ζ > 1, we obtain (U (n)) 1 ≤ h˜ i



˜

˜  fi (ui (n) + ηin ui (n)) 1/hi −1 fi (ui (n)) 1/hi −1 − ξ˜i ξ˜i 

˜ ˜

˜ ˜  fr (ur (n)) hi /hr fi (ui (n)) hi /hi × − ξ˜r ξ˜i

Lbi(s)

98

3 Discrete-Time Systems with Switching

 1−h˜ i 

˜ ˜ 1 fr (ur (n)) (1−hi )/hr 1 ˆ bL(δ) − ≤ ω ξ˜r h˜ i 

˜ ˜

˜ ˜  fr (ur (n)) hi /hr fi (ui (n)) hi /hi × − , ξ˜r ξ˜i where bˆ = const > 0.   (U (n)) < 0 for 0 < δ < δ0 , Hence, there exists a δ0 > 0 such that V (s) U (n) ∈ Q and for all s = 1, . . . , S. Thus, function (3.19) and its first difference with respect to any subsystem from the family (3.16) satisfy in Rm + all the conditions of the discrete counterpart of the Lyapunov asymptotic stability theorem. Therefore, the zero solution of the comparison system (3.15) is asymptotically stable in Rm + for any switching law, and, as a consequence, we obtain that the zero solution of multiconnected system (3.11) is asymptotically stable uniformly with respect to the switching law. This completes the proof.

3.9 Example Let system (3.11) be of the form ⎧ (σ ) 3 (σ ) α 3 ⎪ ⎪ ⎨x1 (n + 1) = x1 (n) − p11 x1 (n) + x2 (n) + p12 x3 (n), (σ ) 3 x2 (n), x2 (n + 1) = x2 (n) − x13 (n) − p22 ⎪ ⎪ (σ ) ⎩x (n + 1) = x (n) − p x 5 (n) + p(σ ) x 3 (n). 3

3

33

3

21

(3.20)

2

(s) (s) (s) (s) (s) , p22 , p33 , p12 , p21 are Here x1 (n), x2 (n), x3 (n) are scalar variables; n ∈ N+ ; p11 (s) (s) constant coefficients with pii(s) > 0, i = 1, 2, 3, p12 = 0, p21 = 0, s = 1, . . . , S; α is a positive rational with the odd denominator. System (3.20) can be treated as a complex system describing the interaction of the following two (m = 2) subsystems

 (σ ) 3 x1 (n + 1) = x1 (n) − p11 x1 (n) + x23 (n), (σ ) 3 x2 (n) x2 (n + 1) = x2 (n) − x13 (n) − p22

and (σ )

x3 (n + 1) = x3 (n) − p33 x35 (n). (σ )

(σ )

The terms p12 x3α (n) and p21 x23 (n) characterize the interaction between the subsystems.

3.9 Example

99

In this case, the isolated subsystems of differential equations (3.13) are of the form  (s) z˙ 1 (t) = −p11 z13 (t) + z23 (t), s = 1, . . . , S, (3.21) (s) z˙ 2 (t) = −z13 (t) − p22 z23 (t), and (s) 5 z3 (t), z˙ 3 (t) = −p33

s = 1, . . . , S.

(3.22)

It is easy to verify that v1 (z1 , z2 ) = 14 (z14 + z24 ) is a common Lyapunov function for subsystems (3.21), while v2 (z3 ) = 12 z32 is a common Lyapunov function for subsystems (3.22). Thus, here μ1 = 3, μ2 = 5, γ1 = 4, γ2 = 2. Using the vector Lyapunov function V (Z) = (v1 (z1 , z2 ), v2 (z3 ))T , construct a comparison system for (3.20). We get ⎧ 3/2 α/2 (σ ) 3/4 ⎪ u1 (k + 1) = u1 (k) − b1(σ )u1 (k) + d12 u1 (k)u2 (k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ + g1 (u1 (k), u2 (k)), 3/4 (σ ) (σ ) 1/2 ⎪ ⎪ u2 (k + 1) = u2 (k) − b2 u32 (k) + d21 u2 (k)u1 (k) ⎪ ⎪ ⎪ ⎪ ⎩ + g2 (u1 (k), u2 (k)),

where (s) (s) 8p11 p22 (s) b 1 = ) 2   , (s) (s) 2 p11 + p22 (s)

(s)

d12 = 2(α+3)/2|p12 |,

(s)

(s)

b2 = 8p33 ,

(s)

(s)

d21 = 4|p21 |,

s = 1, . . . , S, and functions g1 (u1 , u2 ) and g2 (u1 , u2 ) are determined by the formula (3.14). Applying Theorem 3.4, we obtain that the zero solution of system (3.20) is asymptotically stable for any switching law if one of the following conditions is fulfilled: (i) α > 5; (ii) α = 5, and  min

s=1,...,S

(s)

b1

(s)

d12



 min

s=1,...,S

(s)

b2

(s)

d21

 > 1.

100

3 Discrete-Time Systems with Switching

3.10 Notes and References It is known that discrete-time systems are widely used in mathematical modeling of processes, the state of which is available for measurement at discrete instants of time. The general stability theory of classical discrete-time systems is welldeveloped (see Aleksandrov and Zhabko [4], Lakshmikantham et al. [51], LaSalle [52] and the references therein). An unsolved problem is the problem of constructing an appropriate Lyapunov function satisfying special properties needed for the zero solution stability to be established. In the monographs by Martynyuk [66, 69], for discrete-time systems a version of the generalized direct Lyapunov method is developed based on the matrix-valued functions. Namely, sufficient conditions for various types of stability are established in terms of matrix-valued functions for the class of linear discrete-time systems, and a method of their construction is indicated. In addition, in the book the method of Lyapunov hierarchical matrix functions is developed, which allows one to weaken essentially the requirements for the dynamic properties of the subsystems being part of a complex system. Discrete-time large-scale systems under structural perturbation were investigated in the book by Martynyuk and Miladzhanov [89] via the matrix-valued Lyapunov functions. Various dynamic properties of discrete-time systems have been investigated in many works (see Liberzon [54], Liberzon et al. [55], Lukyanova and Martynyuk [57], Martynyuk [65], etc.). Such systems appear in modeling of mechanical systems, electric power systems, technological processes and intelligent control systems with logic-based controllers, etc. This chapter is based on the papers by Aleksandrov et al. [7] and Martynyuk [76]. In the cases, when stability conditions for the comparison equation (3.8) can be obtained in an explicit form, Theorem 3.3 solves the problem of stability of the stationary solution 0 of the set of discrete-time equations (3.1). Theorem 3.4 permits one to reduce the problem of asymptotic stability of the zero solution of multi-connected switched system to that of the existence of positive solutions for the system of inequalities of special form. The conditions of solvability for such systems have been investigated in the papers by Aleksandrov and Platonov [2, 3]. The main ideas of this work were developed in other papers. Namely, in the work by Aleksandrov et al. [5], for certain classes of nonlinear switched difference systems, approaches to the construction of switched comparison equations are proposed. On the basis of the dwell-time method, conditions on switching law providing the asymptotic stability of the zero solution of the comparison equations are derived. An example is presented to demonstrate efficiency of the obtained results. The paper by Aleksandrov et al. [6] addresses the stability problem for a set of switched nonlinear difference equations with parametric uncertainties. For the

3.10 Notes and References

101

corresponding family of subsystems, a regularization procedure is suggested, and a multiple Lyapunov function is constructed. With the aid of the Lyapunov function, classes of switching signals are determined for which the asymptotic stability of a stationary solution of a given set of equations may be guaranteed. An application of the proposed approach to the stability analysis of multi-connected switched difference systems by nonlinear approximation is presented. An example is given to illustrate the main results.

Chapter 4

Qualitative Analysis of Impulsive Equations

In this chapter, for the family of impulsive equations, a heterogeneous matrix-valued Lyapunov-like function is considered, the comparison principle is formulated, and the stability conditions for the set of stationary solutions are established. In addition, for a class of impulsive equations with uncertain parameter the monotone iterative technique for constructing a set of solutions is adapted.

4.1 Introduction In recent years, the method of matrix Lyapunov-like functions, which is a generalization of the classical Lyapunov direct method based on matrix-valued functions, has been significantly developed (see, for example, Martynyuk [64, 66, 69] and the references therein). Parallel to the development of the method for different classes of new equations, the structure of the matrix-valued Lyapunov functions remains of great importance and attracts an increasing attention. It is well known that the components of the matrix-valued functions depend on the system of equations under consideration, as well as on compositions of its subsystems. A natural subject for the investigation by means of the multi-component Lyapunovlike functions with different components is the class of impulsive systems. The main goal of the present chapter is to apply the method of heterogeneous matrix-valued Lyapunov-like functions and to establish efficient stability conditions for the families of impulsive equations. A generalization of the direct Lyapunov method is proposed. The chapter is organized as follows. In Sect. 4.2 some results for the families of impulsive equations are presented. In Sect. 4.3 the construction of heterogeneous matrix-valued Lyapunov-like functions is given and a comparison theorem in terms of such functions is derived. Section 4.4 dwells on the stability analysis of the families of impulsive equations. The stability definitions are introduced and sufficient conditions for the stability of stationary solutions are proved. In Sect. 4.5 a monotone iterative technique for the families of impulsive equations with

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_4

103

104

4 Qualitative Analysis of Impulsive Equations

uncertain parameter is developed. In the concluding Sect. 4.6, some comments and bibliography pertinent to the chapter are presented.

4.2 Preliminaries Let Rn be the n-dimensional Euclidean space with norm || · ||, and let R+ = [0, ∞). We denote by Kc (Rn ) the space of all nonempty compact convex subsets of Rn and assume that the sequence {tk } is defined on R+ so that 0 < t1 < t2 < . . . < tk < . . . and lim tk = ∞. In the further presentation, we will use the following k→∞

notations: 1. If F ∈ P C (R+ × Kc (Rn ), Kc (Rn )), then the function F : R+ × Kc (Rn ) → Kc (Rn ) is continuous on (tk−1 , tk ]× Kc (Rn ) for each k = 1, 2, . . . and for each X ∈ Kc (Rn ) the following relation: lim

(t,Y )→(tk+ ,X)

F (t, Y ) = F (tk+ , X)

is satisfied for k = 1, 2, . . . . 2. If g ∈ P C(R+ × R+ , R), then the function g : (tk−1 , tk ] × R+ → R is continuous for each w ∈ R+ and the limit lim g(t, z) = g(tk+ , w) exists.

(t,z)→(tk+,w) n (R )), then the function is differentiable on each interval

3. If ∈ + ×Kc of the type (tk−1 , tk ), k = 1, 2, . . . . 4. The set X : I → Kc (Rn ) is differentiable in the sense of Hukuhara at t0 ∈ I if there exists DH X(t0 ) ∈ Kc (Rn ) such that both limits P C 1 (R

lim{[X(t0 + 0) − X(t0 )]θ −1 : θ → 0+ } and lim{[X(t0 ) − X(t0 − 0)]θ −1 : θ → 0+ } exist and are equal to DH X(t0 ), where I = [t0 , t0 + a], t0 ≥ 0, a > 0. We consider the initial value problem for the families of impulsive differential equations DH X = F (t, X),

t = tk ,

X(tk+ ) = Ik (X(tk )),

t = tk ,

X(t0 ) = X0 ∈ Kc (R ), n

(4.1) (4.2)

4.2 Preliminaries

105

where X ∈ Kc (Rn ), F ∈ P C(R+ × Kc (Rn ), Kc (Rn )), Ik : Kc (Rn ) → Kc (Rn ) for each k = 1, 2, . . . , and the sequence {tk } is defined above. The solution of the family of equations (4.1)–(4.2) is a function X(t; t0 , X0 ) which is piecewise continuous on [t0 , ∞), left-continuous on each subinterval (tk , tk+1 ], k = 1, 2, . . . , and is given by ⎧ ⎪ X0 (t; t0 , X0 ), t0 ≤ t ≤ t1 ; ⎪ ⎪ ⎪ ⎨X (t; t , X+ ), t1 < t ≤ t2 ; 1 1 1 X(t; t0 , X0 ) = ⎪ ........... ⎪ ⎪ ⎪ ⎩ tk < t ≤ tk+1 , Xk (t; tk , Xk+ ), where Xk (t; tk , Xk+ ) is the solution of the set differential equation (4.1)–(4.2). Together with the problem (4.1)–(4.2) we will consider the impulsive scalar differential equation dw = g(t, w), dt

t = tk ,

w(tk+ ) = k (w(tk )),

t = tk ,

(4.3) (4.4)

w(t0 ) = w0 , where g ∈ P C(R2+ , R), g(t, w) is nondecreasing in w for all t ∈ R+ , k : R+ → R for each k = 1, 2, . . . and k (w) is nondecreasing in w. The maximal solution of (4.3)–(4.4) is a function r(t; t0 , w0 ), defined by ⎧ ⎪ r0 (t; t0 , r0 ), t0 ≤ t ≤ t1 , ⎪ ⎪ ⎪ ⎨r (t; t , r + ), t1 < t ≤ t2 , 1 1 1 r(t; t0 , w0 ) = ⎪ .......... ⎪ ⎪ ⎪ ⎩ rk (t; tk , rk+ ), tk < t ≤ tk+1 , and satisfies the inequality w(t; t0 , w0 ) ≤ r(t; t0 , w0 ),

t = tk ,

(4.5)

for all t ∈ R+ and every solution w(t; t0 , w0 ) of (4.3)–(4.4) with the initial conditions rk+ = k (rk−1 (tk )), k = 1, 2, . . . .

106

4 Qualitative Analysis of Impulsive Equations

4.3 Comparison Principle Together with the family of impulsive differential equations (4.1)–(4.2) we will consider the regularized equation DH Y = Fβ (t, Y ),

Y (t0 ) = Y0 ∈ Kc (Rn ),

(4.6)

where Fβ (t, Y ) = βF (t, Y ) + (1 − β)Iβ (Y (tk )), k = 1, 2, . . . , 0 ≤ β ≤ 1, and the elements of the matrix-valued function U (t, X) = [Vij (t, ·)],

i, j = 1, 2,

(4.7)

are chosen as follows: V11 : R+ × Kc (Rn ) → R+ is related to Eq. (4.1), V22 : R+ × Kc (Rn ) → R+ is related to Eq. (4.2), V12 = V21 : R+ × Kc (Rn ) → R is related to Eq. (4.6). The scalar function V (t, X, ψ) = ψ T U (t, X)ψ,

ψ ∈ R2+ ,

(4.8)

is called a heterogeneous Lyapunov function if it, together with the derivative D + V (t, X, ψ), allows us to solve the problem on stability (instability) of the set stationary solutions 0 of the family of impulsive differential equations (4.1)–(4.2). We will say that the function V (t, X, ψ) belongs to the class F0 if the following conditions are satisfied: (a) V (t, X, ψ) is continuous in (tk−1 , tk ] × Kc (Rn ) for each k = 1, 2, . . . and for each X ∈ Kc (Rn ) the following limit: lim

(t,X)→(tk+,W )

V (t, X, ψ) = V (tk+ , W, ψ),

k = 1, 2, . . .

exists; (b) for each A, B ∈ Kc (Rn ) and all t ∈ R+ the following estimate |V (t, A, ψ) − V (t, B, ψ)| ≤ LD[A, B] holds, where L is the Lipschitz constant. Remark 4.1 If in the matrix-valued function (4.7) we have V12 = V21 = 0, then L(t, X, ψ) = U (t, X)ψ, ψ ∈ R2+ is a heterogeneous vector Lyapunov function, i.e., a vector function with heterogeneous components.

4.3 Comparison Principle

107

Remark 4.2 If V12 = V21 = V22 = 0 and Kc (Rn ) = Rn in the matrixvalued function (4.7), then the function V (t, X, ψ) = V (t, X) is a generalized Lyapunov function for the family of equations (4.1)–(4.2), which was applied in Lakshmikantham et al. [49] and some others. Further, we will give a comparison theorem in terms of the heterogeneous function (4.8) for the family of equations (4.1)–(4.2). Theorem 4.1 Assume that for the family of impulsive differential equations (4.1)– (4.2) the following conditions are satisfied: (a) there exists the heterogeneous function (4.8) of the class F0 ; (b) there exists a function g(t, w), nondecreasing in w, g ∈ P C(Rn+ , R), such that D + V (t, X, ψ)|(4.1) ≤ g(t, V (t, X, ψ)),

t = tk ,

for each X ∈ Kc (Rn ), where D + V (t, X, ψ)|(4.1) = lim sup{[V (t + θ, X + θ F (t, X), ψ) − V (t, X, ψ)]θ −1 : θ → 0+ }; (c) there exist functions k (w), k = 1, 2, . . . , nondecreasing in w, such that V (tk+ , X(tk+ ), ψ) ≤ k (V (tk+ , X(tk+ ), ψ)),

t = tk ,

for each k = 1, 2, . . . ; (d) there exists a maximal solution of the scalar impulsive equation (4.3)–(4.4) for all t ∈ R+ . Then, if the solution X(t) = X(t; t0 , X0 ) of the family of impulsive differential equations (4.1)–(4.2) exists on [t0 , a) and V (t0 , X0 , ψ) ≤ w0 , the estimate V (t, X(t), ψ) ≤ r(t; t0 , w0 )

(4.9)

holds for all t ∈ [t0 , a). The proof of Theorem 4.1 is similar to the proof of Theorem 5.2.3 in Lakshmikantham et al. [51] for each interval of the type [tk−1 , tk ], k = 1, 2, . . . . Theorem 4.1 has a number of corollaries that can be useful in the study of particular families of equations of the form of (4.1)–(4.2). Corollary 4.1 Assume that in the conditions (b), (c) of Theorem 4.1 the functions g(t, V (t, X, ψ)) and Ik (V (tk , X(tk ), ψ)) are such that: (a) g(t, V (t, X, ψ)) = 0 and k (V (tk , X(tk ), ψ)) = 0 for all k = 1, 2, . . . , then the estimate (4.9) has the form V (t, X(t), ψ) ≤ V (t0+ ; w0 , ψ),

t ≥ t0 ;

108

4 Qualitative Analysis of Impulsive Equations

(b) g(t, V (t, X, ψ)) = 0, k (V (tk , X(tk ), ψ)) = dk V (tk , X(tk ), ψ), and dk ≥ 0 for all k = 1, 2, . . . , then the estimate (4.9) has the form V (t, X(t), ψ) ≤ V (t0+ , w0 , ψ)

*

dk ,

t ≥ t0 ;

t0 ≤tk ≤t

(c) g(t, V (t, X, ψ)) = −βV (t, X, ψ), β > 0 and

k (V (tk , X(tk ), ψ)) = dk V (tk , X(tk ), ψ) for all k = 1, 2, . . . , then the estimate (4.9) has the form V (t, X(t), ψ) ≤ V (t0+ , w0 , ψ)

*

dk e−β(t −t0 ) ,

t ≥ t0 .

t0 ≤tk ≤t

4.4 Stability Analysis For the family of impulsive equations (4.1)–(4.2), we introduce the following assumptions: H1 . For the family of impulsive equations (4.1)–(4.2) the relations F (t, 0 ) = 0 and Ik (0 ) = 0 hold, where 0 ∈ Kc (Rn ) is a set of stationary solutions of (4.1)–(4.2) for k = 1, 2, . . . . H2 . The solution X(t) of set impulsive equations (4.1)–(4.2) exists and is unique for t ≥ 0. H3 . For every Y0 ∈ Kc (Rn ) the Hukuhara difference X0 − Y0 = W0 , W0 ∈ Kc (Rn ) exists. Under the conditions H1 –H3 we consider the stability problem for the set of stationary solutions X(t, t0 , 0 ) = 0 of the family of impulsive equations (4.1)– (4.2) for all t ≥ t0 . Definition 4.1 The set of stationary solutions X(t, t0 , 0 ) = 0 of the family of impulsive equations (4.1)–(4.2) is said to be: (a) stable if and only if for every t0 ∈ R and every ε > 0 there exists a δ = δ(t0 , ε) > 0 such that D[W0 , 0 ] < δ implies D[X(t, t0 , W0 ), 0 ] < ε for all t ≥ t0 ; (b) attractive if and only if for every t0 ∈ R there exists an α(t0 ) > 0 and for every ξ > 0 there exists τ (t0 ; W0 , ξ ) ∈ R+ such that D[W0 , 0 ] < α(t0 ) implies D[X(t; t0 , W0 ), 0 ] < ξ for all t ≥ t0 + τ (t0 , W0 , ξ ); (c) asymptotically stable if it is stable and attractive. On the basis of Theorem 4.1 for the family of impulsive equations, the stability conditions for the sets of stationary solutions 0 can be formulated as follows. Theorem 4.2 Assume that for the family of impulsive differential equations (4.1)– (4.2) the following conditions are satisfied:

4.4 Stability Analysis

109

(a) there exists the heterogeneous function (4.8), comparison functions ϕ¯i ∈ K, i = 1, 2, and constant (2 × 2)-matrices Ai = Ai (ψ), i = 1, 2, such that (a1 ) |V (t, X1 , ψ) − V (t, X2 , ψ)| ≤ LD[X1 , X2 ], L > 0, for all (t, X) ∈ R+ × Kc (Rn ); (a2 ) ϕ¯1T (X)A1 ϕ¯ 1 (X) ≤ V (t, X, ψ) ≤ ϕ¯2T (X)A2 ϕ¯ 2 (X) for all (t, X) ∈ R+ × Kc (Rn ); (b) there exist functions g(t, u), g ∈ C(R2+ , R) and k : R+ → R+ , k = 1, 2, . . . , such that (b1) D + V (t, X, ψ) ≤ g(t, V (t, X, ψ)), t = tk , where g(t, 0) = 0 for all (t, X) ∈ R+ × S(β); (b2) for every X ∈ S(β), β > β0 > 0, and all k the following assertions take place: X+Ik (X) ∈ S(β) and V (tk , Ik (X(tk )), ψ) ≤ k (V (tk , X(tk ), ψ)), where k (u) is a function, nondecreasing in u, k (0) = 0 for all k = 1, 2, . . . . Then, if the matrices A1 , A2 from (a2 ) are positive definite and the zero solution of the scalar comparison equations (4.3)–(4.4) is stable (attractive, asymptotically stable), then the set of stationary solutions 0 of the family of impulsive equations (4.1)–(4.2) is stable (attractive, asymptotically stable). Proof Under the conditions of Theorem 4.2, condition (a2 ) is transformed to λm (A1 )ϕ1 (X) ≤ V (t, X, ψ) ≤ λM (A2 )ϕ2 (X), where λm (A1 ), λM (A2 ) are the minimal and maximal eigenvalues of the matrices A1 and A2 , respectively, and ϕ1 (X) ≤ ϕ T1 (X)ϕ 1 (X),

ϕ2 (X) ≤ ϕ T2 (X)ϕ 2 (X)

for all X ∈ S(β), ϕ1 , ϕ2 ∈ K. Let 0 < ε < β ∗ = min(β, β0 ) and t0 ∈ R+ be given. Let the zero solution of (4.3)–(4.4) be stable. Then, for λm (A1 )ϕ1 (ε) > 0 there exists a δ1 (t0 , ε) > 0 such that 0 ≤ w0 < δ1 implies w(t, t0 , w0 ) < λm (A1 )ϕ1 (ε) for all t ≥ t0 , where w(t, t0 , w0 ) is an arbitrary solution of the scalar impulsive equation (4.3)–(4.4). Let w0 = λM (A2 )ϕ2 (D[W0 , 0 ]) and choose δ2 = δ2 (ε) such that λM (A2 )ϕ2 (δ2 ) < δ1 . Next, we will choose δ = min(δ1 , δ2 ) and show that, if D[W0 , 0 ] < δ, then for all t ≥ t0 we will have D[X(t), 0 ] < ε, where X(t) = X(t, t0 , X0 ) is an arbitrary solution of the family of equations (4.1)–(4.2). Suppose that this is not true. Then, for the solution X(t, t0 , X0 ) with D[W0 , 0 ] < δ, there exists a t ∗ > t0 such that tk < t ∗ < tk+1 for some k, for which D[X(t ∗ ), 0 ] ≥ ε and D[X(t), 0 ] < ε for t0 ≤ t ≤ tk . Since 0 < ε < β ∗ , then from condition (b2) of Theorem 4.2, we have D[X(tk+ ), 0 ] = D[Ik (X(tk )), 0 ] < β0

110

4 Qualitative Analysis of Impulsive Equations

and D[X(tk ), 0 ] < ε. Hence, there exists a tˆ such that tk < tˆ < t ∗ and ε ≤ D[X(tˆ), 0 ] < β0 . We denote m(t) = V (t, X(t), ψ), and consider the behavior of the function m(t) on the interval [t0 , tˆ]. From Theorem 4.1, we have that V (t, X(t), ψ) ≤ r(t, t0 , w0 )

for t0 ≤ t ≤ tˆ,

where w0 = λM (A2 )ϕ2 (D[W0 , 0 ]) and r(t, t0 , w0 ) is the maximal solution of the scalar equation (4.3)–(4.4). From the estimate (4.9) we obtain λm (A1 )ϕ1 (D[X(tˆ), 0 ]) ≤ V (tˆ, X(tˆ), ψ) ≤ r(tˆ, t0 , w0 ) < λm (A1 )ϕ1 (ε), which contradicts the existence of a tˆ ∈ [tk , tk+1 ] for any k. The contradiction obtained shows that the set of stationary solutions 0 of the family of impulsive equations (4.1)–(4.2) is stable. The proofs of the attractivity and asymptotic stability are similar. Theorem 4.2 has a number of corollaries, which are given below. Corollary 4.2 If in Theorem 4.2, the functions g(t, V (t, X, ψ)) and k (V (tk , X(tk ), ψ)) have the form given in conditions (a), (b) of Corollary 4.1, then the set of stationary solutions 0 of the family of impulsive equations (4.1)–(4.2) is stable. Corollary 4.3 If in Theorem 4.2, the functions g(t, V (t, X, ψ)) and k (V (tk , X(tk ), ψ)) have the form given in condition (c) of Corollary 4.1, then the set of stationary solutions 0 of the family of impulsive equations (4.1)–(4.2) is asymptotically stable. As in the case of Lyapunov vector functions with homogeneous components, for the function (4.9) the algorithms of constructing stability conditions for the set of stationary solutions in terms of dynamic properties of a comparison system are similar to the above approach. Transform the family of equations (4.1), (4.2) and (4.6) as DH X = F1 (t, x) + G1 (t, X, Z), Y (tk+ ) = Ik1 (Y (tk )) + G2 (t, X, Z), DH Z = Fβ (t, Z),

t = tk , t = tk .

(4.10) (4.11) (4.12)

4.4 Stability Analysis

111

where k = 1, 2, . . . and 0 < β < 1. It is assumed that there exist the Hukuhara differences G1 (·) = F (t, X) − Fβ (t, Z) ∈ Kc (Rn ), G2 (·) = Ik (Y (tk )) − Fβ (t, Z) ∈ Kc (Rn ). The components Vij (·) of the matrix-valued function are chosen so that V11 (X) = D[X, 0 ],

V11 (X) ≥ γ11 D 2 [X, 0 ],

V22 (Y ) = D[Y, 0 ],

V22 (Y ) ≥ γ22D 2 [Y, 0 ],

V21 (Z) = V12 (Z) = D[X, 0 ]D[Y, 0 ],

(4.13)

V12 (X, Y ) ≥ γ12D[X, 0 ]D[Y, 0 ], where the constants γii > 0 for i = 1, 2 and γ12 is an arbitrary constant. The scalar function V (X, Y ) is constructed by the formula V (X, Y ) = ψ T

V11 (X) V12 (X, Y ) ψ, V21 (X, Y ) V22 (Y )

(4.14)

where ψ ∈ R2+ , ψi > 0 and (X, Y ) ∈ Kc (Rn ). We introduce the notations



γ γ ψ1 0 , G = 11 12 B= 0 ψ2 γ21 γ22 and transform the lower estimate of the function (4.14) to the form Vm (X, Y ) ≥ uT B T GBu,

(4.15)

where u = (D[X, 0 ], D[Y, 0 ])T . The generalized derivative of the function (4.14) is calculated by the formula +

D V (X, Y ) = ψ

T

D + V11 (X) D + V12 (X, Y ) ψ, D + V21 (X, Y ) D + V22 (Y )

(4.16)

where D + V12(X, Y ) = D + V21 (X, Y ) and D + V11 (X) = lim sup{[V11(A + hF (t, A)) − V (A)] h−1 : h → 0+ } D + V22 (Y ) = V22 (k, Yk ) = V22 (k + 1, Ik+1 (Yk+1 )) − V22 (k, Ik (Yk )) is the first-order difference for V22 (Y ).

112

4 Qualitative Analysis of Impulsive Equations

Suppose that there exist constants ρij , i = 1, 2, j = 1, 2, . . . , 8 such that  D + V11 (X)(F ) ≤ ρ11 D 2 [X, 0 ], 1  + D V22 (Y )(I ) ≤ ρ21 D 2 [Y, 0 ], k1

 D V11 (X)(G ) ≤ ρ12 D 2 [X, 0] + ρ13 D[X, 0 ]D[Y, 0 ], 1  + D V22 (Y )(G ) ≤ ρ22 D 2 [Y, 0 ] + ρ23 D[X, 0 ]D[Y, 0 ], 2  + D V12 (X, Y )(F ) ≤ ρ14 D 2 [X, 0 ] + ρ15 D[X, 0 ]D[Y, 0 ], 1  +  D V12 (X, Y ) (I ) ≤ ρ24 D 2 [Y, 0 ] + ρ25 D[X, 0 ]D[Y, 0 ], k1  + D V12 (X, Y )(G ) ≤ ρ16 D 2 [X, 0 ] + ρ17 D[X, 0 ]D[Y, 0 ] +

(4.17)

1

 D + V12 (X, Y )

+ ρ18 D 2 [Y, 0 ], (G2 )

≤ ρ26 D 2 [X, 0 ] + ρ27 D[X, 0 ]D[Y, 0 ] + ρ28 D 2 [Y, 0 ].

From (4.17) we obtain the following upper estimate for the total derivative of the function (4.16) D + V (X, Y ) ≤ uT Su,

(4.18)

where the matrix

σ11 σ12 S= , σ21 σ22

σ12 = σ21

has the elements σ11 = β12 (ρ11 + ρ12 ) + 2β1 β2 (ρ14 + ρ16 + ρ26 ), σ22 = β22 (ρ21 + ρ22 ) + 2β1 β2 (ρ18 + ρ24 + ρ28 ), σ12 =

1 2 (β ρ13 + β22 ρ23 ) + β1 β2 (ρ15 + ρ25 + ρ17 + ρ27 ). 2 1

We summarize the exposition by the following theorem. Theorem 4.3 If for the family of equations (4.10)–(4.12), the functions Vij (·), i, j = 1, 2, satisfy conditions (4.13) and (4.17), the matrix G > 0 (positive definite) and the matrix S < 0 (negative definite), then the state 0 of the family of equations (4.10)–(4.11) is asymptotically stable.

4.5 Monotone Iterative Technique

113

Proof From the conditions of Theorem 4.3 it follows that the function V (X, Y ) is positive definite, and its total derivative (4.18) is negative definite with respect to the family of equations (4.10)–(4.12). Hence, according to Theorem 4.2, the state 0 of the family of equations (4.10)–(4.12) is asymptotically stable. Remark 4.3 To satisfy the conditions of Theorem 4.3, it is necessary that γii > 0, i = 1, 2, and σ11 < 0, σ22 < 0. From the last conditions, it follows that the stability of the state 0 of the family of equations (4.10)–(4.12) can also occur even in the case when the family of equations DH X = F1 (t, X), Y (tk+ ) = Ik1 (Y (tk )) may be unstable. This means that the mappings G1 and G2 can stabilize the equilibrium states of (4.10)–(4.11) to asymptotically stable states. Such situation is observed in the theory of hybrid systems (see, for example, Martynyuk [79] and the references therein), when the connections between heterogeneous subsystems stabilize the equilibrium state of the entire system whereas the individual subsystems can be unstable.

4.5 Monotone Iterative Technique Consider the family of uncertain impulsive equations DH X = F (t, X, α),

t = tk ;

X(tk+ ) = Ik (X(tk ), α),

(4.19)

t = tk ;

(4.20)

X(t0 ) = X0 ∈ Kc (R ), n

where X ∈ Kc (Rn ), F ∈ P C(R+ × Kc (Rn ) × I, Kc (Rn )), Ik : Kc (Rn ) × I → Kc (Rn ), α ∈ I is an uncertain parameter. We transform the family (4.19)–(4.20) into the form DH X = Fβ (t, X) + G(t, X, α),

t = tk ,

X(tk+ ) = Ikβ (X(tk )) + Jkβ (X(tk ), α),

(4.21)

t = tk .

Here Fβ (t, X) = Fm (t, X)β + FM (t, X)(1 − β), where Fm (t, X) = co

 α∈I

F (t, X, α),

FM (t, X) = co

 α∈I

F (t, X, α);

(4.22)

114

4 Qualitative Analysis of Impulsive Equations

G(t, X, α) = F (t, X, α) − Fβ (t, X); Ikβ (X(tk )) = Ikm (X(tk ))β + IkM (X(tk ))(1 − β), where 

Ikm (X(tk )) = co

Ik (X(tk ), α),

α∈I



IkM (X(tk )) = co

Ik (X(tk ), α);

α∈I

Jkβ (X(tk ), α) = Ik (X(tk ), α) − Ikβ (X(tk )). We assume that the sequence 0 < t1 < t2 < . . . < tk < . . . is such that lim tk = T for k → ∞. Next, we consider the nonlinear approximation of the regularized family of equations (4.21)–(4.22) DH U = Fβ (t, U ),

t = tk ,

U (tk+ ) = Ikβ (U (tk )),

t = tk .

(4.23) (4.24)

We will prove the following lemma. Lemma 4.1 Assume that for the family of equations (4.23)–(4.24) the following conditions are satisfied: (1) there exist functions V , W ∈ P C 1 (R+ , Kc (Rn )), Fβ ∈ P C(R+ × Kc (Rn ), Kc (Rn )) and Fβ (t, U ) is a family monotone nondecreasing in U for all t and each β ∈ [0, 1]; (2) the following inequalities: DH V ≤ Fβ (t, V ), V (tk+ ) ≤ Ikβ (V (tk )), DH W ≥ Fβ (t, V ),

t = tk , t = tk ; t = tk ,

W (tk+ ) ≥ Ikβ (W (tk )),

t = tk ,

k = 1, 2, . . .

hold; (3) Ikβ : Kc (BR n ) → Kc (Rn ), Ikβ (U ) is nondecreasing in U for each k and β ∈ [0, 1]; (4) for each β ∈ [0, 1] there exists a constant Lβ > 0 such that Fβ (t, X) ≤ Fβ (t, Y ) + Lβ (X − Y ) for all X, Y ∈ Kc (Rn ) such that X ≥ Y . Then V (0) ≤ W (0) implies V (t) ≤ W (t) for all t ≥ 0 and β ∈ [0, 1].

4.5 Monotone Iterative Technique

115

Proof Let J1 = [0, t1 ] and V (0) ≤ W (0). We apply Theorem 5.2.1 from Lakshmikantham et al. [49] to the continuous part of the family of equations (4.23), and obtain V (t) ≤ W (t) for all t ∈ [0, t1 ] and β ∈ [0, 1]. Hence, V (t1 ) ≤ W (t1 ), and since I1β (U ) is a nondecreasing function, we have V (t1+ ) ≤ I1β (V (t1 )) ≤ I1β (W (t1 )) ≤ W (t1+ ). Therefore, V (t1+ ) ≤ W (t1+ ). Repeating the same procedure for the interval J2 = (t1 , t2 ], we obtain the estimate V (t) ≤ W (t) for t ∈ (t1 , t2 ) and each value of β ∈ [0, 1]. This proves Lemma 4.1. Corollary 4.4 Let V , W ∈ P C 1 (R+ × Kc (Rn ), Kc (Rn )) and there exist functions p, q ∈ C(R+ , Kc (Rn )) such that DH V ≤ p(t)

for t = tk ,

V (tk+ ) ≤ q(tk )

for t = tk ;

DH W ≥ p(t)

for t = tk ,

W (tk+ ) ≥ q(tk )

for t = tk .

and

Then the assertion of Lemma 4.1 remains valid. In the following we will consider the family of equations (4.21)–(4.22). Definition 4.2 Let V , W ∈ P C 1 (J, Kc (Rn )). The sets V , W are: (a) a pair of upper and lower solutions of the family of equations (4.21)–(4.22) of the type I if DH V ≤ Fβ (t, V ) + G(t, W, α),

t = tk ,

V (tk+ ) ≤ Ikβ (V (tk )) + Jkβ (W (tk ), α),

t = tk ,

V (0) ≤ X0 , and DH W ≥ Fβ (t, W ) + G(t, V , α),

t = tk ,

W (tk+ ) ≥ Ikβ (W (tk )) + Jkβ (V (tk ), α), W (0) ≥ X0 ;

t = tk ,

116

4 Qualitative Analysis of Impulsive Equations

(b) a pair of upper and lower solutions of the family of equations (4.21)–(4.22) of the type II if DH V ≤ Fβ (t, W ) + G(t, V , α), V (tk+ )

t = tk ,

≤ Ikβ (W (tk )) + Jkβ (V (tk ), α),

t = tk ,

V (0) ≤ X0 , and DH W ≥ Fβ (t, V ) + G(t, W, α), W (tk+ )

t = tk ,

≥ Ikβ (V (tk )) + Jkβ (W (tk ), α),

t = tk ,

W (0) ≥ X0 , for each β ∈ [0, 1]. We will show that for the family of equations (4.21)–(4.22) a theorem similar to Theorem 5.3.1 from Lakshmikantham et al. [49] holds. Theorem 4.4 Assume that: (1) there exists a pair of upper and lower solutions of the family of equations (4.21)–(4.22) V , W ∈ P C 1 (J, Kc (Rn )) of the type I and Vβ (t) < Wβ (t) for t ∈ J; (2) there exists a β ∗ ∈ [0, 1] such that the mapping Fβ ∈ C(J ×Kc (Rn ), Kc (Rn )), Fβ (t, X) is nondecreasing in X for all t ∈ J and every β ∈ (0, β ∗ ); (3) for any α ∈ I the mapping G ∈ C(J × Kc (Rn ) × I, Kc (Rn )) and G(t, X, α) is nonincreasing in X for all t ∈ J ; (4) the mappings Ikβ (X) are continuous and nondecreasing in X for each β ∈ (0, β ∗ ); (5) the mappings Jkβ (X, α) are continuous and nonincreasing in X for each α ∈ I and β ∈ (0, β ∗ ); (6) the mappings (Fβ , G) and (Ikβ , Jkβ ) map the bounded sets into the bounded sets of Kc (Rn ) for each β ∈ (0, β ∗ ) and α ∈ I. Then, there exist monotone sequences {Vn (t)} and {Wn (t)} in Kc (Rn ) such that Vn → P (t) and Wn (t) → Q(t) for n → ∞, the pair (P (t), Q(t)) is a type I pair of upper and lower solutions of the family of equations (4.21)–(4.22), such that DH P = Fβ (t, P ) + G(t, Q, α), P (tk+ ) = Ikβ (P (tk )) + Jkβ (Q(tk ), α), P (0) = X0 ,

t = tk ; t = tk ;

(4.25)

4.5 Monotone Iterative Technique

117

and DH Q = Fβ (t, Q) + G(t, P , α),

t = tk ;

Q(tk+ ) = Ikβ (Q(tk )) + Jkβ (P (tk ), α),

t = tk ;

(4.26)

Q(0) = X0 for t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Proof For any n ≥ 0, consider the families of equations DH Vn+1 = Fβ (t, Vn ) + G(t, Vn , α),

t = tk ,

Vn+1 (tk+ ) = Ikβ (Vn (tk )) + Jkβ (Wn (tk ), α),

t = tk ,

(4.27)

Vn+1 (0) = X0 and DH Wn+1 = Fβ (t, Wn ) + G(t, Vn , α), Wn+1 (tk+ ) = Ikβ (Wn (tk )) + Jkβ (Vn (tk ), α),

t = tk , t = tk ,

(4.28)

Wn+1 (0) = X0 , where V (0) ≤ X0 ≤ W (0) and β ∈ (0, β ∗ ). From the conditions of Theorem 4.4 it follows that there exist solutions Vn+1 (t) and Wn+1 (t) of the families of equations (4.27), (4.28), respectively, for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Let V0 (t) = V (t) and W0 (t) = W (t) for t ∈ J . Theorem 4.4 will be proved if we show that V0 ≤ V1 ≤ V2 ≤ . . . ≤ Vn ≤ Wn ≤ . . . ≤ W2 ≤ W1 ≤ W0

(4.29)

for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. From the assumptions made, it follows that V0 and W0 form a pair of upper and lower solutions of the type I of the family of equations (4.21)–(4.22). We set Vn = V0 and Wn = W0 in (4.27), (4.28), and obtain the mappings V1 (t) and W1 (t) for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I that are solutions of the families of equations (4.27) and (4.28). We will show that the following inequalities are true: (a) V0 ≤ V1 , (b) V1 ≤ W1 , and (c) W1 ≤ W0 for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I.

118

4 Qualitative Analysis of Impulsive Equations

In order to prove the assertion (a) we consider the family of equations (4.27) for n = 0: DH V1 = Fβ (t, V0 ) + G(t, W0 , α),

t = tk ,

V1 (tk+ ) = Ikβ (V0 (tk )) + Jkβ (W0 (tk ), α),

t = tk ,

(4.30)

V1 (0) = X0 . From condition (1) of Theorem 4.4, we have DH V0 ≤ Fβ (t, V0 ) + G(t, W0 , α),

t = tk ,

V0 (tk+ ) ≤ Ikβ (V0 (tk )) + Jkβ (W0 (tk ), α),

t = tk ,

(4.31)

V0 (0) ≤ X0 . From (4.30) and (4.31) it follows that V0 (t) ≤ V1 (t)

for t ∈ (tk−1 , tk ].

Since V0 (tk+ ) ≤ V1 (tk+ ) for each t = tk , we have V0 (t) ≤ V1 (t) for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. To prove the assertion (b) consider the families of equations (4.27) and (4.28) for n = 0. From the conditions (2)–(5) of Theorem 4.4, for k = 1, 2, . . . , β ∈ (0, β ∗ ) and α ∈ I we have DH V1 ≤ Fβ (t, W0 ) + G(t, W0 , α), V1 (tk+ )

t = tk ,

≤ Jkβ (W0 (tk )) + Jkβ (W0 (tk ), α),

t = tk ,

V1 (0) = X0 , and DH W1 ≥ Fβ (t, W0 ) + G(t, W0 , α), W1 (tk+ ) ≥ Ikβ (W0 (tk )) + Jkβ (W0 (tk ), α),

t = tk , t = tk ,

W1 (0) = X0 . From the above inequalities we obtain V1 (t) ≤ W1 (t)

for t ∈ J.

Repeating the proof from the case (a) for the given family of equations we obtain W1 (t) ≤ W0 (t)

for all t ∈ J.

4.5 Monotone Iterative Technique

119

Hence V0 (t) ≤ V1 (t) ≤ W1 (t) ≤ W0 (t)

for all t ∈ J.

Let for some j > 1, we have Vj −1 (t) ≤ Vj (t) ≤ Wj (t) ≤ Wj −1 (t)

for t ∈ J.

We will show that for every t ∈ J the inequalities Vj (t) ≤ Wj +1 (t) ≤ Wj (t) are true. Consider the families of equations DH Vj = Fβ (t, Vj −1 ) + G(t, Wj −1 , α),

Vj +1 (t) ≤

t = tk ,

Vj (tk+ ) = Ikβ (Vj −1 (tk )) + Jkβ (Wj −1 (tk ), α),

t = tk ,

(4.32)

Vj (0) = X0 , and DH Vj +1 = Fβ (t, Vj ) + G(t, Wj , α),

t = tk ,

Vj +1 (tk+ ) = Ikβ (Vj (tk )) + Jkβ (Wj (tk ), α),

t = tk ,

(4.33)

Vj (0) = X0 for all β ∈ (0, β ∗ ) and α ∈ I. From (4.32)–(4.33) and conditions (2)–(5) of Theorem 4.4, we have DH Vj +1 ≥ Fβ (t, Vj −1 ) + G(t, Wj −1 , α),

t = tk ,

Vj +1 (tk+ ) ≥ Ikβ (Vj −1 (tk )) + Jkβ (Wj −1 (tk ), α),

t = tk ,

(4.34)

Vj +1 (0) = X0 . Hence, Vj (t) ≤ Vj +1 (t) for all t ∈ J . In a similar way, it is easy to show that Wj +1 (t) ≤ Wj (t) for t ∈ J . Next, we will show that Vj +1 (t) ≤ Wj +1 (t) for all t ∈ J . For this purpose, in the families of equations (4.27) and (4.28) we set n = j and from conditions (2)–(5) of Theorem 4.4, we obtain the families of inequalities DH Vj +1 ≤ Fβ (t, Wj ) + G(t, Wj , α),

t = tk ,

Vj +1 (tk+ ) ≤ Ikβ (Wj (tk )) + Jkβ (Wj (tk ), α),

t = tk ,

Vj +1 (0) = X0 and DH Wj +1 ≥ Fβ (t, Wj ) + G(t, Wj , α), Wj +1 (tk+ ) ≥ Ikβ (Wj (tk )) + Jkβ (Wj (tk ), α), Wj +1 (0) = X0 ,

t = tk , t = tk ,

120

4 Qualitative Analysis of Impulsive Equations

from which it follows that Vj +1 (t) ≤ Wj +1 (t)

for all t ∈ J.

Thus, we have sequences of mappings {Vn (t)}, {Wn (t)} that are piecewise continuous and satisfy the inequalities (4.29). From the construction of these mappings it follows that they are uniformly bounded for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Since the sequences {Vn (t)}, {Wn (t)} are equi-continuous on each of the subintervals [tk , tk+1 ], applying the Arzela–Ascoli theorem, we can show that the sequence {Vn (t)} converges to the mapping P (t) and the sequence {Wn (t)} converges to the mapping Q(t) uniformly on [tk , tk+1 ]. Since Ikβ , Jkβ are continuous for any k = 1, 2, . . . , we have that lim Vn (tk+ ) = lim {Ikβ (Vn−1 (tk )) + Jkβ (Wn−1 (t), α)}

n→∞

n→∞

and, therefore, P (tk+ ) = Ikβ (P (tk )) + Jkβ (Q(tk ), α), Q(tk+ ) = Ikβ (Q(tk )) + Jkβ (P (tk ), α) for all β ∈ (0, β ∗ ) and α ∈ I. Next, from the families of integral equations t Vn+1 (t) = X0 +

[Fβ (s, Vn (s)) + G(s, Wn (s), α)] ds, 0

t Wn+1 (t) = X0 +

[Fβ (s, Wn (s)) + G(s, Xn (s), α)] ds 0

for n → ∞ it follows that in each subinterval [tk , tk+1 ] we have the families of equations (4.26) and (4.27). We will show that the pair (P (t), Q(t)) is a pair of the minimal and maximal mappings of the family of equations (4.19), (4.20). Let X(t) be an arbitrary solution of (4.19), (4.20) such that V0 ≤ X(t) ≤ W (t) for all t ∈ J . Then V0 ≤ P (t) ≤ X(t) ≤ Q(t) ≤ W0

(4.35)

for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Let for some n we have Vn (t) ≤ X(t) ≤ Wn (t) for all t ∈ J . In view of the monotonicity of the functions Fβ and G for each t ∈ J , β ∈ (0, β ∗ ) and α ∈ I, and monotonicity of the functions Ikβ , Jkβ for each k,

4.6 Notes and References

121

we obtain DH X ≥ Fβ (t, Vn ) + G(t, Wn , α),

t = tk ,

X(tk+ ) ≥ Ikβ (Vn (tk )) + Jkβ (Wn (tk ), α),

t = tk ,

X(0) ≥ X0 and DH Vn+1 = Fβ (t, Vn ) + G(t, Wn , α), Vn+1 (tk+ )

= Ikβ (Vn (tk )) + Jkβ (Wn (tk ), α),

t = tk , t = tk ,

Vn (0) = X0 , from which it follows that Vn+1 (t) ≤ X(t) for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Similarly, we can prove that Wn+1 (t) ≥ X(t) for all t ∈ J , β ∈ (0, β ∗ ) and α ∈ I. Since these estimates are true for any n, letting n → ∞, we get (4.35). This completes the proof of Theorem 4.4.

4.6 Notes and References An impulsive system is an example of a two-component system consisting of a continuous and discrete component. In this connection, such systems are sometimes called the hybrid systems (see, for example, Martynyuk [79], Wang [121] and the references therein). The theory of impulsive dynamic systems draws the attention of many specialists because of the fact that these systems are a natural basis for mathematical modeling of many real processes that are characterized by sudden changes in the state vector. The Lyapunov stability of such systems has been studied quite fully and a generalized presentation of the results obtained is published in a number of papers (see Akhmetov and Zafer [1], Milman and Myshkis [94], etc.) and books (see Lakshmikantham et al. [46], Martynyuk [69], Stamova [110], Stamova and Stamov [111], Bainov and Simeonov [4], Haddad et al. [30], Pandit et al. [100], and the references therein). However, the stability theory of stationary solutions of families of impulsive equations is not fully developed (see Martynyuk [73]). Some excellent results on the families of impulsive equations that generalize the corresponding results for the families of ordinary differential equations have been given in the book of Lakshmikantham et al. [49].

122

4 Qualitative Analysis of Impulsive Equations

This chapter is based on some ideas of Martynyuk [73] and their development on the basis of heterogeneous Lyapunov matrix functions. To obtain the stability conditions in the form of Theorem 4.3, the estimation technique from the monograph by Martynyuk [69] ascending to Djordjevi`c [25] is applied. In the proof of Theorem 4.4 some results of the paper of Vasundhara Devi and Vatsala [117] are adapted for the family of impulsive equations regularized in terms of the uncertainty parameter (see also Martynyuk et al. [90]).

Chapter 5

Stability of Systems with Aftereffect

This chapter deals with the families of equations with aftereffect and uncertain parameter values. The families of equations with aftereffect are obtained as a result of regularization of equations according to the scheme adopted in the book. For the obtained families of equations, the solution existence conditions are established, the estimates of the distance between extreme sets of solutions are found, and the stability conditions for the set of stationary solutions on a finite time interval are obtained.

5.1 Introduction In this chapter we consider a set of equations with aftereffect and uncertain parameters. As a result of regularization of the family of equations according to the scheme adopted in the book, a set of equations with aftereffect are obtained, for which the solution existence conditions are established, an estimate of the distance between the extreme solution sets is obtained, and stability conditions for the set of stationary solutions on a finite time interval are found as well as the attenuation conditions for the set of trajectories. In Sect. 5.2 we state the problem of analysis of the set of trajectories for the family of equations with aftereffect. Section 5.3 presents conditions for the existence of solutions for the regularized equations with aftereffect. Section 5.4 deals with the estimate of the distance between the extreme sets of solutions of the family of equations with aftereffect. In Sect. 5.5 we discuss the problem of stability on a finite time interval for the class of equations under consideration. In Sect. 5.6 we establish the estimate of the decay time of the set of trajectories. In Sect. 5.7, comments and bibliography are given.

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_5

123

124

5 Stability of Systems with Aftereffect

5.2 Statement of the Problem and Designations In the subset of convex compact sets Kc (Rn ) of the space Rn , we consider the set of functions C0 = C([−τ, 0], Kc (Rn )), where τ > 0 is a given magnitude. For any sets P , Q ∈ C0 we introduce a metric by the formula D0 [P , Q] = max D[P (s), Q(s)], −τ ≤s≤0

where D[· ] is a Hausdorff metric. For the zero set Q = 0 the metric D0 [P , 0 ] = max P (s). −τ ≤s≤0

Designate I0 = [t0 − τ, t0 + τ ], β > 0 , and X ∈ C(I0 , Kc (Rn )) for any t < t0 , t ∈ Rn . For any t ∈ [τ, τ + β] we define the set Xt ∈ C0 by the correlation Xt (s) = X(t + s), −τ ≤ s ≤ 0. Consider the family of uncertain differential equations with aftereffect DH X(t) = F (t, Xt , α),

(5.1)

Xt0 ∈ P0 ∈ C0 ,

(5.2)

where the mapping F ∈ C(J × C0 × I, Kc (Rn )) and J = [t0 , t + β], α ∈ I ⊂ Rd is the uncertainty parameter. Definition 5.1 The mapping Xt (t) : J → Kc (Rn ) is the solution of the problem (5.1), (5.2) if it is absolutely continuous on J and satisfies Eq. (5.1) for the initial functions (5.2) and any value of the parameter α ∈ J . Compute the boundary mappings Fm (t, Xt ) = co



F (t, Xt , α),

α∈I

FM (t, Xt ) = co



F (t, Xt , α),

α∈I

and consider the families of equations DH Y (t) = Fm (t, Yt ),

Yt0 ∈ Kc (Rn ),

(5.3)

DH V (t) = FM (t, Vt ),

Vt0 ∈ Kc (R ),

(5.4)

DH W (t) = Fβ (t, Wt ),

Wt0 ∈ Kc (Rn ),

(5.5)

n

where Fβ (t, ·) = Fm (t, ·)β + (1 − β)FM (t, ·), β ∈ [0, 1].

5.3 Existence of the Set of Solutions

125

The family of equations (5.5) is regularized with respect to uncertainty parameter α ∈ I for the initial family of equations (5.1). It is of interest to study the dynamic properties of the set trajectories of the family of equations (5.3)–(5.5) obtained in terms of the family of equations (5.1).

5.3 Existence of the Set of Solutions For the initial problem (5.5) the following result holds true. Theorem 5.1 For the family of equations (5.5), let a constant K(β), β ∈ [0, 1], exist so that D[Fβ (t, P ), Fβ (t, Q)] ≤ K(β)D0 [P , Q] for all t ∈ J, and all (P , Q) ∈ Q0 . Then the initial problem (5.5) possesses at least one solution W (t) on I0 for any β ∈ [0, 1]. Proof We shall consider a set of trajectories W (t) ∈ C(I0 , Kc ⊂ Rn ) such that W (t) = P (t) for all t0 − τ ≤ t ≤ t0 and W (t) ∈ C(J, Kc ⊂ Rn ) with the initial value W (0) = P (0) if P (t) ∈ Kc (Rn ) for −τ ≤ t ≤ 0. On the set of functions C(I0 , Kc (Rn )) , define the metric D1 [P , Q] =

max

t0 −τ ≤t ≤t0 +β

D[P (t), Q(t)e−λt ],

where λ > 0 and the operator

T W (t) =

⎧ ⎪ ⎨P (t)

for all t0 − τ ≤ t ≤ t0 , t ⎪ ⎩T W (t) = P (0) + Fβ (s, W (s)) ds for all t > t0 , t0

for any β ∈ [0, 1]. For the values −τ ≤ s ≤ 0 we have D[T W (t0 + s), T P (t0 + s)] = D[P (t0 + s), P (t0 + s)] = 0.

126

5 Stability of Systems with Aftereffect

Compute D[T W (t), T P (t)] for t > t0 . In view of conditions of Theorem 5.1 and the properties of the Hausdorff metric we get

t

D[T W (t), T P (t)] = D P (0) +

Fβ (s, W (s)) ds, P (0) t0





t

+

Fβ (s, P (s)) ds t0

 =D



t

t

Fβ (s, W (s)) ds, t0

Fβ (s, P (s)) ds

(5.6)

t0

t ≤

D[Fβ (s, W (s)), Fβ (s, P (s))] ds t0

t ≤ K(β)

D0 [W (s), P (s)] ds t0

for all β ∈ [0, 1]. Having accomplished simple transformations in the inequality (5.6), one can easily obtain the estimate exp(−λt)D[T W (t), T P (t)] ≤

K(β) D1 [W (t), P (t)] λ

(5.7)

for all β ∈ [0, 1] and t > t0 . Choosing the value λ = 2K(β) for all β ∈ [0, 1] , we find from estimate (5.7) that D1 [T W (t), T P (t)] ≤

1 D1 [W (t), P (t)] 2

(5.8)

for all t > t0 . Inequality (5.8) implies that the operator T on the set of functions C(I0 , Kc (Rn )) is contractive. In this case there exists a fixed “point” W ∈ C(I0 , Kc (Rn )). Therefore, the set is a unique solution of the initial problem (5.5). The theorem is proved.

5.4 Funnel for the Set of Trajectories Consider the families of equations (5.3) and (5.4) and estimate the funnel for the set of solutions of the regularized equation (5.5).

5.4 Funnel for the Set of Trajectories

127

Theorem 5.2 Assume that the following conditions are satisfied: (1) The mappings Fm ∈ C(R+ × C0 , Kc (Rn )) and FM ∈ C(R+ × C0 , Kc (Rn )). (2) There exists a function g(t, w) nondecreasing in w and such that D[Fm (t, Yt ), FM (t, Vt )] ≤ g(t, D0 [Yt , Vt ]) for all t ∈ R+ and Yt0 , Vt0 ∈ C0 . (3) For all t ≥ t0 there exists a maximum solution r(t, t0 , w0 ) of the initial problem dw = g(t, w) dt w(t0 ) = w0 ≥ 0. Then for all t ≥ t0 the estimate D[Y (t), V (t)] ≤ r(t, t0 , w0 ) is valid whenever D[Yt0 , Vt0 ] ≤ w0 . Proof Let Y (t) and V (t) be the sets of trajectories of Eqs. (5.3) and (5.4) under the initial conditions Yt0 and Vt0 , respectively. For sufficiently small h > 0 we calculate the differences Y (t + h) − Y (t) and V (t + h) − V (t) for all t ∈ R+ and designate m(t) = D[Y (t), V (t)]. In view of the properties of the Hausdorf metric, for the difference m(t + h) − m(t) we get the estimate m(t + h) − m(t) = D[Y (t + h), V (t + h)] − D[Y (t), V (t)] ≤ D[Y (t + h), Y (t) + hFm (t, Yt )] + D[Y (t) + hFm (t, Yt ), V (t) + hFM (t, Vt )] + D[V (t) + hFM (t, Vt ), V (t + h)] − D[Y (t), V (t)] ≤ D[Y (t + h), Y (t) + hFm (t, Yt )] + D[V (t) + hFM (t, Vt ), V (t + h)] + hD[Fm (t, Yt ), FM (t, Vt )]. The fact that [m(t + h) − m(t)]h−1 ≤ D[(Y (t + h) − Y (t))h−1 , Fm (t, Yt )] + D[FM (t, Vt ), (V (t + h) − V (t))h−1 ] + D[Fm (t, Yt ), FM (t, Vt )]

128

5 Stability of Systems with Aftereffect

as h → 0+ implies the estimate D + m(t) = [m(t + h) − m(t)]h−1 ≤ D[Fm (t, Yt ), FM (t, Vt )] ≤ g(t, D0 [Yt , Vt ]) = g(t, mt0 ), where mt0 = D0 [Yt0 , Vt0 ]. From condition (3) of Theorem 5.2 and Theorem 1.5.2 from the monograph by Lakshmikantham et al. [51], it follows that for all t ≥ t0 the estimate D[Y (t), V (t)] ≤ r(t, t0 , w0 )

(5.9)

holds true whenever D[Yt0 , Vt0 ] ≤ w0 . Inequality (5.9) establishes the estimate of the distance between the extreme sets of the trajectories of Eq. (5.1) between which the set of solutions to the family of equations (5.5) is located.

5.5 Finite-Time Stability We shall consider a regularized family of equations (5.5) and establish sufficient stability conditions for the set of trajectories on a finite interval. Further we shall need the following definition. Definition 5.2 For the given estimates of the values (λ, A, t0 , T ), λ < A , and t0 < T the set of trajectories of the family of equations (5.5) is stable on a finite interval if the condition D0 [P , 0 ] < λ yields the estimate D[W (t), 0 ] < A for all t ∈ [t0 , t0 + T ] and all β ∈ [0, 1]. Further, in the space Kc (Rn ) we consider the domains S(a) = {W ∈ Kc (Rn ) : D[W, 0 ] < a},

(b) = {P ∈ C0 : D0 [P , 0 ] < b}. The family of equations (5.5) is considered under the following hypotheses: H1 . There exist a functional V ∈ C(R+ × W, R+ ) and a constant L1 > 0 such that (a) |V (t, W1 ) − V (t, W2 )| ≤ L1 D[W1 , W2 ]; (b) b(D[W, 0 ]) ≤ V (t, W ) ≤ a(D[W, 0 ]), where a and b ∈ K are of the Hahn class K.

5.5 Finite-Time Stability

129

H2 . There exists a function g ∈ C(R+ × R2 , R) such that D + V (t, W ) = lim sup{[V (t + h, W + Fβ (t, Wt )) − V (t, W )]h−1 : h → 0+ } ≤ g(t, V (t, W )) for all (t, W ) ∈ R+ × S(A) and any value of β ∈ [0, 1]; H3 . There exists a maximum solution of the comparison equation dw = g(t, w), dt

w(t0 ) = w0 ≥ 0,

(5.10)

for all t > 0. We shall present a result on (λ, A, t0 , T )-stability of the set of trajectories. Theorem 5.3 Let the conditions of hypotheses H1 –H3 be satisfied and, moreover, let the maximum solution of Eq. (5.10) with the initial condition w0 < a(t0 , λ) satisfy the estimate w(t, t0 , w0 ) < b(A) for all t ∈ [t0 , t0 + T ]. Then the set of trajectories Wt of the family of equations (5.5) is stable on a finite interval. Proof Let 0 < λ < A and a value 0 < T < ∞ be given. Let us show that if D0 [P (t), 0 ] < λ, then D0 [W (t), 0 ] < A for all t ∈ [t0 , t0 + T ] and for any β ∈ [0, 1]. If this is not the case, then there are (t2 > t1 ) ∈ [t0 , t0 + T ] such that D[W (t1 ), 0 ] = λ and D[W (t2 ), 0 ] = A and λ ≤ D[W (t), 0 ] ≤ A for t1 ≤ t ≤ t2 . Hypotheses H1 –H3 imply that V (t, Wt ) ≤ r(t, t, t0 , w0 ) for all t1 ≤ t ≤ t2 , where r(t, ·) is a maximum solution of Eq. (5.10). Since r(t, t0 , w0 ) is stable on a finite interval, there is a function b of the Hahn class K such that r(t, t0 , w0 ) < b(A) for all t1 ≤ t ≤ t2 . Conditions (5.11) imply b(D[W (t), 0 ]) ≤ V (t, W ) ≤ r(t, t0 , w0 ) < b(A) for all t1 ≤ t ≤ t2 . Hence D[W (t2 ), 0 ] < A.

(5.11)

130

5 Stability of Systems with Aftereffect

The contradiction obtained completes the proof of Theorem 5.3. Remark 5.1 If T = ∞, then the conditions of Theorem 5.3 are sufficient for the practical stability of the set trajectories of the family of equations (5.5). Definition 5.3 For the given estimates of the values C1 , C2 , t0 , T , 0 < C1 < C2 , t0 < T , and the functional V (t, w) , the set of trajectories of the family of equations (5.5) is stable on a finite interval, provided that the condition V (t0 , ) ≤ C1 implies the estimate V (t, t ) ≤ C2 for all t ∈ [t0 , t0 + T ). The following result is valid. Theorem 5.4 Let the following conditions be satisfied: (1) Condition (a) of hypothesis H1 is fulfilled. (2) There exist a positive integrable function c(t) and a constant k > 1 such that D + V (t, W ) ≤ c(t)V k (t, W ) for all (t, W, P ) ∈ R+ × S(A) × (A) and any value of β ∈ [0, 1]. (3) For all t ∈ [t0 , t0 + T ) the following inequality holds true: t N(t) = 1 − (k − 1)V

k−1

c(s)ds > 0.

(t0 , ) t0

(4) The following estimate takes place: (N(t))− k−1 < 1

C2 C1

for all t ∈ [t0 , t0 + T ).

Then the set of trajectories W (t) of the family of equations (5.5) is stable on a finite interval. Proof Condition (2) of Theorem 5.4 for all t ≥ t0 yields V (t, W (t)) ≤ V (t0 , Wt0 ) t +

(5.12)

c(s)V k−1 (s, W (s))V (s, W (s)) ds. t0

From inequality (5.12) we get

t V (t, W (t)) ≤ V (t0 , Wt0 ) exp

 c(s)V k−1 (s, W (s)) ds

t0

5.5 Finite-Time Stability

131

for all t ≥ t0 . Since k − 1 > 0, we have V

k−1

(t, W (t)) ≤ V

−1



t

(t0 , Wt0 ) exp (k − 1)

 c(s)V k−1 (s, W (s)) ds .

(5.13)

t0

Multiplying both parts of inequality (5.13) by −(k −1)c(t) we arrive at the estimate −(k − 1)c(t)V k−1 (t, W (t)) ≥ −(k − 1)V k−1 (t0 , Wt0 )c(t)

 t × exp (k − 1) c(s)V k−1 W (s, W (s)) ds t0

which provides the inequality −(k − 1)c(t)V

k−1

 t k−1 (t, W (t)) exp (k − 1) c(s)V (s, W (s)) ds t0

(5.14)

≥ −(k − 1)V k−1 (t0 , Wt0 )c(t). We find from (5.14) that  

α d exp − (k − 1) c(s)V k−1 (s, W (s))ds dt t0

≥ −(k − 1)V k−1 (t0 , Wt0 )c(t). Integrating this inequality between t0 and t we get the estimate 

t exp − (k − 1) c(s)V k−1 (s, W (s)) ds t0

t ≥ 1 − (k − 1)V

k−1

(t0 , Wt0 )

c(s)ds t0

which implies

 t k−1 exp (k − 1) c(s)V (s, W (s)) ds ≤ (N(t))−1 t0

(5.15)

132

5 Stability of Systems with Aftereffect

for all t ∈ [t0 , t0 + T ). Further, in view of (5.15), inequality (5.13) becomes V k−1 (t, W (t))V k−1 (t0 , Wt0 ) ≤ (N(t))−1 , and taking into account that V k−1 (t0 , Wt0 ) > 0 , we obtain the estimate 1

V (t, W (t)) ≤ V (t0 , Wt0 )(N(t))− k−1 for all t ∈ [t0 , t0 + T ). Let the family of trajectories W (t) go out of the set V (t0 , Wt0 ) ≤ C1 and at some time t1 ∈ [t0 , t0 + T ) reach the boundary of the set V (t1 , Wt1 ) = C2 . Under condition (4) of Theorem 5.4 we have V (t1 , W (t1 )) ≤ V (t0 , Wt0 )(N(t)) ≤ C1 (N(t))− k−1 < C2 . 1

The contradiction obtained completes the proof of Theorem 5.4, i.e., there is no t1 ∈ [t0 , t0 + T ) for which V (t1 , W (t1 )) = C2 . Theorem 5.4 is proved.

5.6 Damping Time for the Set Trajectories Further we shall need the following definition. Definition 5.4 The set of trajectories W (t) of the family of equations (5.5) is called damping if the following conditions are satisfied: (1) W (t) is stable in the sense of Lyapunov. (2) There exists a δ > 0 such that for small Wt0 ∈ Kc (Rn ) : D[Wt0 , 0 ] < δ , there is 0 ≤ T (Wt0 ) < +∞ such that D[W (t), 0 ] = 0 for all t ≥ T (Wt0 ).  The functional T0 (Wt0 ) = inf T (Wt0 ) ≥ 0 : D[W (t), 0 ] = 0 for all t ≥ T (Wt0 ) estimates the damping time of the set of trajectories W (t). Let us show that the following result is valid. Theorem 5.5 For the family of equations (5.5), assume that the following conditions are satisfied: (1) There exists a functional V (t, W ) such that b(D[Wt0 , 0 ]) ≤ V (t, W ), where b is of the Hahn class K.

5.6 Damping Time for the Set Trajectories

  (2) D + V (t, W )

(5.5)

133

≤ −g(V (t, W )), where g(u) is a nondecreasing function and

for u > 0 the inequality ε

du < +∞ g(u)

0

is fulfilled for any ε > 0. Then the set of trajectories W (t) of the family of equations (5.5) is damping with the damping time 

V (t0 ,Wt0 )

T0 (Wt0 ) ≤

du . g(u)

0

Proof Let V (t, W ) : R+ × S(a) → R+ be a functional satisfying conditions (1)– (2) of Theorem 5.5. We shall prove that the set of trajectories W (t) with the initial state Wt0 is damping with the estimate 0 ≤ T0 (Wt0 ) ≤ ∞. For condition (2) of Theorem 5.5 being satisfied, consider the comparison equation du = −g(u), dt

u(t0 ) = u0 ≥ 0,

(5.16)

for which the solution u(t, t0 , u0 ) estimates the variation of the functional V (t, W (t)) by the inequality V (t, Wt ) ≤ u(t, t0 , u0 )

for all t ≥ 0,

(5.17)

where u(t, ·) is a maximum solution of the comparison equation (5.16). It follows from (5.16), (5.17) that 

V (t0 ,Wt0 )

D[W (t), 0 ] = 0 for t ≥

du g(u)

0

when all t ≥ t0 and D[W (t), 0 ] < δ. Hence it follows that 

V (t0 ,Wt0 )

T0 (Wt0 ) ≤

du < +∞ g(u)

0

is an estimate of the damping time of the set of trajectories W (t) of the family of equations (5.5). This completes the proof of Theorem 5.5.

134

5 Stability of Systems with Aftereffect

5.7 Notes and References One of the first researchers who described a real system with aftereffect in the study of problems of the theory of elasticity was Boltzmann [14]. When studying the stabilization of motion of a ship on the course by an automatic control system, Minorsky [95] showed the importance of taking into account the aftereffect in the feedback mechanism. Mathematical statement of many problems for a system with aftereffect together with the original results is given in the monographs by Burton [16], Hale [32], Lakshmikantham and Rama Mohana Rao [44], Yoshizawa [122], and others. Krasovskii [40] generalized the direct Lyapunov method for the systems with aftereffect on the basis of the functionals introduced by him instead of the Lyapunov functions. A new method for stability analysis of systems with aftereffect was developed under the guidance of Professor N.V. Azbelev (see Azbelev and Simonov [9] and the bibliography therein). This method does not incorporate the range of ideas of the direct Lyapunov method and relies on a new form of the Cauchy matrix for linear functional equation and the corresponding estimates in an appropriate Banach space. Modern classification of the equations with aftereffect is given in the dissertation paper by Schoen [107]. Important studies of this class of systems were carried out by El’sgol’ts [26], Razumikhin [104], Milman and Myshkis [94], Stamova [110], Lakshmikantham et al. [48], and others. In the paper by Lakshmikantham and Martynyuk [43] the results of development of the direct Lyapunov method for systems with aftereffect based on the Lyapunov functionals are presented. In addition, it is shown that the combination of the Lyapunov function and the functional simplifies the procedure of applying the direct Lyapunov method for systems with aftereffect. In Martynyuk’s monograph [69] the theory of stability of systems with aftereffect is developed on the basis of matrix-valued Lyapunov functionals. Their application in the study of the stability of large-scale systems with aftereffect is presented. For the set of equations with aftereffect, some results are obtained by Vasundhara Devi and Vatsala [118]. The main results of this chapter are new.

Chapter 6

Impulsive Systems with Aftereffect

In the first part of the chapter a family of differential equations with aftereffect under impulsive perturbations is considered. For such equations, some results of set trajectories analysis based on the matrix-valued function defined on the product of spaces are given. In the second part of the chapter, for the first time, uncertain sets of equations under impulse perturbations are investigated. Estimates of the distance between extremal sets of trajectories are derived for the systems under consideration. In addition, conditions for the global existence of the sets of solutions regularized with respect to the uncertainty parameter are proved.

6.1 Introduction This chapter presents the results of analysis of the set trajectories of the families of equations with delay and impulsive perturbations. The analysis is based on a theorem of the principle of comparison in terms of the auxiliary matrix function defined on the product of spaces. Sufficient conditions for the stability of the set of stationary solutions are established on the basis of scalar comparison equation. As an application, we consider a system of ordinary differential equations with impulsive perturbations. Sufficient conditions of stability are established in the case when both continuous and discrete components of the system may have unstable zero solution. In Sect. 6.2 we present preliminary information on the delay systems under impulsive perturbation. Section 6.3 deals with a theorem of the comparison principle with the matrixvalued Lyapunov function. Some simple stability criteria for the stability of the set stationary solutions are given in Sect. 6.4. Section 6.5 addresses a funnel problem for the set of solutions to the uncertain impulse system with aftereffect.

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_6

135

136

6 Impulsive Systems with Aftereffect

In Sect. 6.6 a general theorem on stability of the set of uncertain equations with aftereffect is established and an example is given. Finally, Sect. 6.7 presents some comments and bibliography to this chapter.

6.2 Preliminaries Let Rn be the n-dimensional Euclidean space with norm  · , and let R+ = [0, ∞). We denote by Kc (Rn ) the space of all nonempty compact convex subsets of Rn . For a given τ > 0 consider the set C0 = C([−τ, 0], Kc (Rn )). For any two sets A, B ∈ C0 introduce the metric D0 [A, B] = max D[A(s), B(s)], −τ ≤s≤0

where D[A, B] is the Hausdorff distancebetween the sets A and B, defined as D[A, B]= max sup d(y, A), sup d(x, B) , where d(y, A)= inf{y−a : a∈A}. y∈B

x∈A

For some value a > τ > 0 consider the set-valued function X ∈ C(J, Kc (Rn )) on the interval J = [t0 − τ, t0 + a], where t0 ∈ R+ . Denote by Xt the translation of X onto the interval [t − τ, t], where t ∈ [t0 , t0 + a]. For this set the translation Xt ∈ C0 is defined by Xt (s) = X(t + s) for −τ ≤ s ≤ 0. For the multivalued function X : J → Kc (Rn ) consider the generalized derivative in the sense of Hukuhara DH X(t). Let {tk } be a sequence of points defined on R+ such that 0 ≤ t0 < t1 < . . . < tk < . . . and lim tk = ∞. k→∞

We consider the initial value problem for the families of differential equations with aftereffect under impulsive perturbations of the type DH X = F (t, Xt ), Xt + = Ik (Xtk ), k

t = tk , t = tk ,

(6.1)

Xt0 = 0 ∈ C0 , where F ∈ (R+ × C0 , Kc (Rn )), Ik : C0 → C0 for all k = 0, 1, 2, . . .. A motion or a physical process is correctly described by the family of equations (6.1) if and only if the piecewise continuous sets of functions X(t0 , 0 )(t), defined on the interval [t0 , ∞), are continuous from the left on each of the intervals (tk , tk+1 ] and

X(t0 , 0 )(t) =

⎧ ⎪

0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨X0 (t0 , 0 )(t),

t0 − τ ≤ t ≤ t0 , t0 ≤ t ≤ t1 ,

X1 (t1 , 1 )(t), t1 < t ≤ t2 , ⎪ ⎪ ⎪ ⎪ .............. ⎪ ⎪ ⎪ ⎩X (t , )(t), t < t ≤ t , k k k k k+1

6.3 Comparison Principle

137

is a solution of the family of differential equations with aftereffect DH X = F (t, Xt ), Xt + = k k

for all k = 0, 1, 2, . . . ,

where k is the set of initial functions at the moment tk+ . Suppose that the set of systems of impulse equations (6.1) satisfies the following conditions: H1 . There exists a stationary solution 0 ∈ Kc (Rn ), i.e., F (t, 0 ) = 0 and Ik (0 ) = 0 for all k = 0, 1, 2, . . .. H2 . For the sets (0 , 0 ) ∈ Kc (Rn ) there exists a Hukuhara difference 0 −0 = W0 ∈ Kc (Rn ). H3 . The set of equations (6.1) for t ≥ t0 has a unique solution X(t, t0 , 0 −0 ) = X(t, t0 , W0 ). Definition 6.1 The stationary solution X(t0 , W0 )(t) = 0 of the family of impulsive equations (6.1) is said to be: (a) stable if for every t0 ∈ R+ and every ε > 0 there exists δ = δ(t0 , ε) > 0 such that D0 [W0 , 0 ] < δ for all t ≥ t0 implies D[X(t), 0 ] < ε; (b) uniformly stable if the number δ in (a) does not depend on t0 ; (c) asymptotically stable if it is stable and for every ε > 0 and t0 ∈ R+ there exist δ0 = δ0 (t0 ) and T = T (t0 , ε) > 0 such that D0 [W0 , 0 ] < δ0 implies D[X(t), 0 ] < ε for all t ≥ t0 + T (t0 , ε); (d) uniformly asymptotically stable if it is uniformly stable and the numbers δ0 and T in (c) are independent on t0 ∈ R+ .

6.3 Comparison Principle Together with the set of impulsive equations with aftereffect (6.1) we will consider the matrix-valued function U (t, X, ) = [Uij (t, X, )],

i, j = 1, 2,

(6.2)

with the elements U11 : R+ × Kc (Rn ) → R+ , U22 : R+ × C0 → R+ , and Uij : R+ × Kc (Rn ) × C0 → R for i = j . Based on the matrix function (6.2) we will construct a bi-graded “function” (functional) V (t, X, , η) = ηT U (t, X, )η,

η ∈ R2+ ,

(6.3)

138

6 Impulsive Systems with Aftereffect

and suppose that V : R+ × Kc (Rn ) × C0 × R2+ → R+ satisfies the following conditions. The function V (t, X, , η) belongs to the class V0 , and the following conditions are satisfied: A1 . The function V (t, X, , η) is continuous on (tk−1 , tk ] × Kc (Rn ) × C0 and for any value of X ∈ Kc (Rn ) and ∈ C0 , k = 1, 2, . . ., there exists the limit lim V (t, X, , η) = V (tk+ , X, , η);

t →tk+

A2 . The function V ∈ (tk−1 , tk ] × Kc (Rn ) × C0 × R2+ satisfies the estimate |V (t, X, , η) − V (t, Y, , η)| ≤ LD[X, Y ],

L = const > 0.

On the product of spaces (tk−1 , tk ] × Kc (Rn ) × C0 , we define the total derivative of the function V (t, X, , η) with respect to the set of equations (6.1) as follows: D + V (t, X, , η) = lim sup {[V (t + h, X(t + h) + hF (t, Xt ), Xt +h (t, ), η)  − V (t, X(t), , η)] h−1 : h → 0+ . Remark 6.1 The use of the functional V (t, X, , η), defined on a product of spaces, simplifies the procedure of estimation of its total derivative with respect to the set of equations under consideration. Example 6.1 The simplest matrix-valued function defined on the product of spaces Kc (Rn ) × C0 is the function U (X, ) which maps the product Kc (Rn ) × C0 onto R2×2 U (X, ) = D[∗, 0 ]D T [∗, 0 ], where D[∗, 0 ] = (D1 [X, 0 ], D2 [ , 0 ]). Here D1 : Kc (Rn ) D2 : C0 → R+ . If D1 [X, 0 ] × D2 [ , 0] = 0 , then

→

U (X, ) = diag(D12 [X, 0 ], D22 [ , 0 ]), i.e., U (X, ) is a diagonal matrix-valued function on the product of spaces.

R+ ,

6.3 Comparison Principle

139

Together with the set of equations of problem (6.1) we will consider the impulsive scalar differential equation dw = g(t, w), dt

t = tk ,

w(tk+ ) = Jk (w(tk )),

(6.4)

t = tk ,

w(t0 ) = w0 , where g ∈ P C(R2+ , R), Jk : R+ → R for each k = 1, 2, . . . . We assume that there exists a maximal solution r(t, t0 , w0 ) of Eq. (6.4) for all t ≥ t0 . We will prove the following comparison theorem. Theorem 6.1 Assume that: (1) for the set of equations (6.1) there exist a matrix-valued function (6.2) and a scalar function V : R+ × Kc (Rn ) × C0 × R2+ → R+ , V ∈ V0 , such that for t = tk (a) D + V (t, X, , η) ≤ g(t, V (t, X, , η)), where g : (tk−1 , tk ] × R+ → R, g(t, w) is nondecreasing in w, and for every w ∈ R+ there exists the limit lim g(t, z) = g(tk+ , w); (t,z)→(tk+,w)

(b) for any t = tk  V tk+ , X(t0 , 0 )(tk+ ), Xt + (t0 , 0 ), η k   ≤ Jk V tk , X(t0 , 0 )(tk ), Xtk (t0 , 0 ), η , where Jk : R+ → R, Jk (w) is nondecreasing in w. (2) for t ≥ t0 there exists the maximal solution r(t, t0 , w0 ) of the scalar equation (6.4). Then for any solution X(t0 , 0 )(t) of the set of equations (6.1) for the function (6.3) the following estimate: V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) ≤ r(t, t0 , w0 )

for t ≥ t0 ,

is valid whenever V (t0 , X(t0 , 0 )(t0 ), 0 , η) ≤ w0 . Proof Let the solution X(t0 , 0 )(t) of the set of equations (6.1) be defined for all t ≥ t0 . We set m(t) = V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η)

140

6 Impulsive Systems with Aftereffect

and suppose that m(t0 ) ≤ w0 , where w0 is the initial value of the solution w(t, t0 , w0 ) of the problem (6.4). For t ∈ (tk−1 , tk ], k = 1, 2, . . ., calculate the difference m(t + h) − m(t) as h → 0. Taking the condition A2 into account, we obtain m(t + h) − m(t) = V (t + h, X(t0 , 0 )(t + h), Xt +h (t0 , 0 ), η) − V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) = V (t + h, X(t0 , 0 )(t + h), Xt +h (t0 , 0 ), η) − V (t + h, X(t0 , 0 )(t) + hF (t, Xt ), Xt +h (t0 , 0 ), η) + V (t + h, X(t0 , 0 )(t) + hF (t, Xt ), Xt +h (t0 , 0 ), η) − V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) ≤ LD[X(t0 , 0 )(t) + hF (t, Xt ), X(t0 , 0 )(t + h)] + V (t + h, X(t0 , 0 )(t) + hF (t, Xt ), Xt +h (t0 , 0 ), η) − V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η). Hence, we have   D + m(t) = lim sup [m(t + h) − m(t)] h−1 : h → 0+ ≤ D + V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) −1

+ L lim sup{D[X(t0 , 0 )(t + h), X(t0 , 0 )(t) + hF (t, Xt )]h

(6.5) :

+

h → 0 }. Taking into account the properties of the Hausdorff metric and the fact that X(t0 , 0 )(t) is a solution of the set of equations (6.1), from the inequality (6.5) we get lim sup{D[X(t0 , 0 )(t + h), X(t0 , 0 )(t) + hF (t, Xt )]h−1 : h → 0+ } = 0, so D + m(t) ≤ D + V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) ≤ g(t, V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η)) = g(t, m(t))

for t = tk ,

k = 1, 2, . . . .

(6.6)

6.4 Stability of the Set of Stationary Solutions

141

For t = tk from the condition 1(b) of Theorem 6.1, we obtain m(tk+ ) = V (tk+ , X(t0 , 0 )(tk+ ), Xt + (t0 , 0 ), η) k

≤ Jk (V (tk , X(t0 , 0 )(tk ), Xtk (t0 , 0 ), η)) = Jk (m(tk )),

(6.7)

k = 1, 2, . . . .

From (6.6), (6.7) according to Theorem 4.9.1 from Lakshmikantham et al. [51] it follows that m(t) ≤ r(t, t0 , w0 )

for all t ≥ t0 .

This proves Theorem 6.1.

6.4 Stability of the Set of Stationary Solutions Next, we use the following open neighborhoods of the set of stationary solutions   S(H ) = X ∈ Kc (Rn ) : D[X, 0 ] < H , Q(H1 ) = { ∈ C0 : D0 [ , 0 ] < H1 } ,

H = const > 0;

H1 = const > 0,

H1 ≤ H.

The following theorem holds. Theorem 6.2 Assume that for the set of equations (6.1) a matrix-valued function V (t, X, , η) exists on the product of spaces Kc (Rn ) × C0 such that (1) condition 1(a) of Theorem 6.1 is satisfied for all (t, X, ) ∈ R+ × S(H ) × Q(H1 ) with the function g(t, u), g(t, 0) ≡ 0 for u = 0; (2) there exists H1 > 0 such that Xtk ∈ Q(H1 ) implies Jk (Xtk ) ∈ Q(H ), and condition 1(b) of Theorem 6.1 is satisfied with Jk (u), Jk (0) ≡ 0 for u = 0, and k = 1, 2, . . .; (3) for any η ∈ R2+ there exist constant symmetric (2 × 2)-matrices A = A(η) and B = B(η) and vector comparison functions a, b ∈ K (K is the Hahn class of functions) such that T

b (D[X, 0 ])Ab(D[X, 0 ]) ≤ V (t, X, , η) ≤ a T (D0 [ 0 , 0 ])Ba(D0 [ 0 , 0 ]) for all (t, X, ) ∈ R+ × S(H ) × Q(H1 ). Then, if the matrices A and B are positive definite, the stability of the trivial solution of comparison equation (6.4) implies the corresponding stability type of the set of stationary solutions 0 of the family of equations (6.1).

142

6 Impulsive Systems with Aftereffect

Proof First, we will transform the condition (3) of Theorem 6.2 to the following form: λm (A)b(D[X, 0]) ≤ V (t, X, , η) ≤ λM (B)a(D0 [ 0 , 0 ]) for all (t, X, ) ∈ R+ × S(H ) × Q(H1 ), where λm (A) > 0 and λM (B) > 0 are, respectively, the minimum and the maximum T eigenvalue of matrices A and B, b(r) ≤ b (r)b(r) and a(r) ≥ a T (r)a(r), a, b ∈ K, K is the Hahn class of functions. Suppose that the zero solution of the impulsive comparison equation (6.4) is stable. Let t0 ∈ R+ and 0 < ε < min(H, H1 ) be given. For the given λm (A)b(ε) > 0 and t0 ∈ R+ there exists δ1 = δ1 (t0 , ε) > 0 such that w0 < δ1 implies r(t, t0 , w0 ) < λm (A)b(ε) for all t ≥ t0 . Let w0 = λM (B)a(D0 [ 0 , 0 ]). We choose δ = δ(t0 , ε) > 0 so that λM (B)a(δ) < δ1 . We will show that if D0 [ 0 , 0 ] < δ, then D[X(t0 , 0 )(t), 0 ] < ε for all t ≥ t0 . If this is not true, then a solution X(t) = X(t0 , 0 )(t) of the set of equations (6.1) with D0 [ 0 , 0 ] < δ and a moment t ∗ : tk < t ∗ < tk+1 for some k exist, so that D[X(t0 , 0 )(t ∗ ), 0 ] ≥ ε and D[X(t0 , 0 )(t), 0 ] < ε for t0 < t < tk . Since 0 < ε < H1 , from condition (2) of Theorem 6.2, we have D[X(t0 , 0 )(t), 0 ] < ε for t = tk and D0 [Xt + (t0 , 0 ), 0 ] = D0 [Ik (Xtk (t0 , 0 )), 0 ] < H . Therefore, k

there exists a moment t 0 , tk < t 0 < t ∗ , such that ε ≤ D[X(t0 , 0 )(t 0 ), 0 ] < H. The application of Theorem 6.1 for t ∈ [t0 , t 0 ] leads to V (t, X(t0 , 0 )(t), Xt (t0 , 0 ), η) ≤ r (t, t0 , λM (B)a(D0 [ 0 , 0 ])) ,

(6.8)

where r(t, t0 , w0 ) is the maximal solution of the comparison equation (6.4). From the inequality (6.8) we have λm (A)b(ε) ≤ λm (A)b(D0[Xt 0 (t0 , 0 ), 0 ]) ≤ V (t, X(t0 , 0 )(t 0 ), Xt 0 (t0 , 0 ), η)   ≤ r t 0 , t0 , λM (B)a(D0 [ 0 , 0 ])     < r t 0 , t0 , λM (B)a(δ) < r t 0 , t0 , δ1 < λm (A)b(ε), which contradicts the existence of a t ∗ > t0 , tk < t ∗ ≤ tk+1 , such that ε ≤ D[X(t0 , 0 )(t ∗ ), 0 ]. The contradiction obtained shows that the set of stationary solutions 0 of the family of impulsive equations (6.1) is stable. The proofs for the other stability types are similar.

6.5 Funnel for the Set of Uncertain Equations

143

Corollary 6.1 If the elements Uij : R+ × Kc (Rn ) × C0 → R of the matrix (6.2) for i = j are zeros, and for the function V (t, X, , η) = ηT diag U (t, X, )η all conditions of Theorem 6.2 are satisfied, then its conclusions remain true.

6.5 Funnel for the Set of Uncertain Equations In this section we consider a family of equations with uncertain parameters of the form DH U = F (t, Ut , α),

t = tk ,

U (tk+ ) = Ik (Utk , α),

t = tk ,

U (t0 ) = 0 ∈ Kc (Rn ),

(6.9) (6.10)

where F ∈ P C(R+ × C0 × S), Kc (Rn ), C0 = C([−τ, 0], Kc (Rn )) and {tk } is the sequence 0 ≤ t1 < t2 < . . . < tk < such that lim tk = ∞. k→∞

We define the boundary values of the mapping F (t, Ut , α): Fm (t, Vt ) = co



F (t, Vt , α),

α∈I

FM (t, Vt ) = co



F (t, Vt , α).

α∈I

Ikm (Vtk ) = co



Ik (Xtk

α∈I

IkM (Vtk ) = co



Ik (Xtk

α∈I

and consider the following families of equations: DH W = Fm (t, Wt ), W (tk+ )

= Ikm (Wt ),

W (t0 ) = 0 ∈ Kc (Rn ),

t = tk , t = tk , k = 0, 1, 2, . . .

(6.11) (6.12)

144

6 Impulsive Systems with Aftereffect

and DH Z = FM (t, Zt ),

t = tk ,

Z(tk+ ) = IkM (Zt ), Z(t0 ) = Z0 ∈ Kc (Rn ),

t = tk , k = 0, 1, 2, . . .

(6.13) (6.14)

We will establish an estimate of the distance between solutions of the families of equations (6.11) and (6.13) under the initial conditions (6.12), (6.14). The following theorem holds. Theorem 6.3 Assume that: (1) the maps Fm , FM ∈ P C(R+ × C0 , Kc (Rn )) and there exist local solutions of the families of equations (6.11), (6.13); (2) there exists a nondecreasing in w function g ∗ (t, w) such that D[Fm (t, ), FM (t, )] ≤ g ∗ (t, D0 [ , ]) for t ∈ R+ , t = tk , ,  ∈ C0 ; (3) there exists a nondecreasing in w function Jk∗ (w) such that D0 [Ikm (Wt ), IkM (Zt )] ≤ Jk∗ (D0 [Wt , Zt ]) for t = tk and k = 0, 1, 2, . . .; (4) there exists a maximal solution rM (t) = rM (t, t0 , ω0 ) of the scalar initial value problem dω = g ∗ (t, w), dt w(tk+ )

=

Jk∗ (wtk ),

w(t0 ) = w0 ,

t = tk ,

(6.15)

t = tk ,

k = 0, 1, 2, . . .

(6.16)

on the interval [t0 , ∞). Then, for the distance between the solutions W (t) = W (t0 , 0 )(t) and Z(t) = Z(t0 , Z0 )(t) of the families of equations (6.11) and (6.13) the following estimate: D[W (t), Z(t)] ≤ rM (t)

for all t ≥ t0

is valid whenever D0 [0 , Z0 ] ≤ w0 . Proof Let the conditions of Theorem 6.3 be satisfied on T0 = [t0 , t1 ]. According to Theorem 5.5.2 from Lakshmikantham et al. [49] we obtain the estimate D[W (t), Z(t)] ≤ rM (t)

for all

from which it follows that D[W (t1 ), Z(t1 )] ≤ rM (t1 ).

t ∈ T0 ,

6.6 Regularization of the Set of Equations with Aftereffect

145

From condition (2) of Theorem 6.3 we have that for t = t1+ D0 [Wt + , Zt + ] = D0 [I1m (Wt1 ), I1M (Zt1 )] 1

1

≤ J1∗ (D0 [Wt1 , Zt1 ]) ≤ J1∗ (rM (t1 )) ≡ rM (t1+ ). Hence, D0 [Wt1 , Zt1 ] ≤ rM (t1+ ).

(6.17)

Further we consider, on the interval T1 = (t1 , t2 ], the estimate (6.17) and the conditions of Theorem 6.3, which lead to the estimate D[W (t), Z(t)] ≤ rM (t, t0 , w0 )

for all

t ∈ T1 .

Continuing this process on the intervals (tk−1 , tk ], we arrive at the conclusions of Theorem 6.3.

6.6 Regularization of the Set of Equations with Aftereffect In this section, we will regularize the family of equations (6.9) and (6.10) with respect to the parameter α ∈ S and obtain a family of equations of the type DH V = Fβ (t, Vt ), Vt + = Ikβ (Vtk ), k

t = tk , t = tk ,

Vt0 = V0 ∈ Kc (Rn ),

(6.18) (6.19)

where β ∈ [0, 1] and Fβ (t, Vt ) = Fm (t, Vt )β + FM (t, Vt )(1 − β); Ikβ (Vtk ) = Ikm (Vtk )β + IkM (Vtk )(1 − β). It is assumed that for any β ∈ [0, 1] the mappings Fβ and Ikβ take their values in Kc (Rn ). For the regularized family of equations (6.18) we will prove the following theorem. Theorem 6.4 Assume that for the regularized family of equations (6.18) the following conditions are satisfied: (1) for any β ∈ [0, 1] the map Fβ ∈ P C(R+ ×C0 , Kc (Rn )) satisfies conditions for the existence of a local solution of the initial value problem (6.18) and (6.19);

146

6 Impulsive Systems with Aftereffect

(2) there exists a function g ∈ P C(R2+ , R), g(t, w) is nondecreasing in w for any t ∈ R+ and such that D[Fβ (t, ), 0 ] ≤ g(t, D0 [ , 0 ])

for t = tk

and β ∈ [0, 1]; (3) there exists a function Jk (w), Jk (w) is nondecreasing in w for any k = 0, 1, 2, . . . and such that D[Ikβ (Vtk ), 0 ] ≤ Jk (D0 [Vtk , 0 ])

for t = tk

and β ∈ [0, 1]; (4) the maximal solution rM (t, t0 , w0 ) of the scalar comparison equation dw = g(t, w), dt

t = tk ,

w(tk+ ) = Jk (wtk ),

t = tk ,

w(t0 ) = w0 ,

k = 0, 1, 2, . . .

(6.20) (6.21)

exists on [t0 , ∞). Then the solution V (t0 , V0 )(t) of the family of equations (6.18) is continuable on [t0 , ∞). Proof Let β = β1 ∈ [0, 1], T0 = [t0 , t1 ] and Fβ1 ∈ P C([t0 , t1 ] × C0 , Kc (Rn )). We consider the initial value problem of the family of equations with aftereffect DH V = Fβ1 (t, Vt ),

(6.22)

Vt0 = 0 ∈ Kc (R )

(6.23)

n

on the interval T0 . It follows from the conditions of Theorem 6.4 that all conditions of Theorem 5.5.1 from Lakshmikantham et al. [49] are satisfied, and therefore, the solution of the initial value problem (6.22) and (6.23) exists on [t0 , t1 ]. Further, we set β = β2 ∈ [0, 1], t = t1 , Vt1 = V (t0 , 0 )(t1 ), and Vt + = 1 I1β2 (Vt1 ). Let 1 = Vt + and T1 = (t1 , t2 ]. For the family of equations with 1 aftereffect DH V = Fβ2 (t, Vt ),

t ∈ T1 ,

Vt + = 1 ∈ Kc (Rn ) 1

(6.24) (6.25)

we use the same arguments as for the initial value problem (6.22) and (6.23) and prove the existence of the solution V (t1 , 1 )(t) for all t ∈ T1 . Continuing this

6.6 Regularization of the Set of Equations with Aftereffect

147

process for β = β3 ∈ [0, 1] and t = t2 , etc., we come to the conclusion about the existence of solution of the initial value problem for the family of equations (6.24) and (6.25) on the interval T2 = (t2 , t3 ]. Repeating these arguments for β = βk ∈ [0, 1] and t = tk we can prove the existence of the solution of the corresponding initial value problem on the intervals of the type of (tk−1 , tk ] and for k → ∞ on the interval [t0 , ∞). Thus, the proof of Theorem 6.4 is complete. For the family of regularized equations (6.18), we make the following assumptions: A3 . For any β ∈ [0, 1] the map Fβ (t, 0 ) = 0. A4 . For each k = 0, 1, 2 . . . the map Ikβ (0 ) = 0, for any β ∈ [0, 1]. We will formulate the stability conditions for the set of stationary solutions of the family of equations (6.18) in the form of the following theorem. Theorem 6.5 Assume that for the family of equations (6.18) conditions A3 and A4 are satisfied and a function L : R+ × S(H ) × Q(H1 ) → R+ , L ∈ V0 exists, so that (1) for any β ∈ [0, 1] D + L(t, V , ) ≤ g(t, L(t, V , )), t = tk , g : R2+ → R, g(t, 0) = 0 and g(t, w) is nondecreasing in w; (2) there exists a H1 > 0 such that Vtk ∈ Q(H1 ) implies Jkβ (Vtk ) ∈ Q(H ) for every k = 0, 1, 2 . . . and L(tk+ , V (t0 , 0 )(tk+ ), Vt + (t0 , 0 )) K

≤ Jk (L(tk , V (t0 , 0 )(tk ), Vtk (t0 , 0 )))

for t = tk ,

Vtk ∈ S(H1 ), where Jk : R+ → R+ , Jk is nondecreasing, Jk (0) = 0 for all k = 0, 1, 2 . . .; (3) there exist functions a, b ∈ K such that b(D0 [V , 0 ]) ≤ L(t, V , ) ≤ a(D0 [, 0 ]). Then the stability properties of the solution w = 0 of the problem (6.20) and (6.21) imply the corresponding properties of the set of states 0 of the family of equations (6.18). The proof of Theorem 6.5 is similar to the proof of Theorem 6.2. Example 6.2 We consider the family of equations (6.18) under the following assumptions: A5 . There exist a positive continuous function λ(t) and a value β = β ∗ ∈ [0, 1] such that D[Fβ ∗ (t, V (t)), 0 ] ≤ λ(t)D[V (t), 0 ].

148

6 Impulsive Systems with Aftereffect

A6 . There exist constants γi ≥ 0 such that  D[ Iiβ ∗ (Vti ), 0 ] ≤ γi D[Vti , 0 ]. t0 0 or into the contractive strip for all t ∈ [t0 , t0 + a).

156

7 Dynamics of Systems with Causal Operator

We return back to the regularized family of equations (7.5) and present the following result. Theorem 7.2 Let Q(β) ∈ C(E × [0, 1], Kc (Rn )) be a causal mapping for which there exists a function g ∗ ∈ C(J × R+ , R) such that D[(Q(β)Y )(t), 0 ] ≤ g ∗ (t, D0 [Y, 0 ](t)) for all t ∈ [t0 , t0 + a) and any β ∈ [0, 1]. If the equation dw = g ∗ (t, w), dt

w(t0 ) = w0 ≥ 0,

possesses a maximum solution on [t0 , t0 + a) and D[Y0 , 0 ] ≤ w0 , then for the family of solutions Y (t) = Y (t, t0 , Y0 ) the estimate D[Y (t), 0 ] ≤ rM (t, t0 , w0 ) is valid for all t ∈ [t0 , t0 + a) and any β ∈ [0, 1]. The proof of this theorem is similar to that of Theorem 7.1 and therefore is omitted here.

7.5 Global Existence of Solutions Further we shall consider the family of equations (7.5). We denote the set B = B(X0 , b) = {X ∈ E : D[X, X0 ] ≤ b} and prove the following result. Theorem 7.3 For the family of equations (7.5) assume that the following conditions are satisfied: (1) for any β ∈ [0, 1] the mapping Qβ ∈ C(B, E) and there exist constants Mβ > 0 such that D0 [(Qβ X), 0 ](t) ≤ Mβ for all X ∈ B and β ∈ [0, 1]; (2) there exist a function  g (t, w), which is nondecreasing with respect to w for every t ∈ [t0 , t0 + a), and a constant M2 > 0 such that  g (t, w) ≤ M2 on J × [0, 2b],  g (t, 0) = 0 for all t ∈ [t0 , t0 + a), and D[(Qβ X), (Qβ Y )](t) ≤  g (t, D0 [X, Y ](t)) on the set B for all β ∈ [0, 1];

7.5 Global Existence of Solutions

157

(3) the scalar equation dw = g (t, w), dt

w(t0 ) = 0,

(7.11)

has a unique solution w(t) = 0 for all t ∈ [t0 , t0 + a). Then the successive approximations t Xn+1 (t) = X0 +

n = 0, 1, 2, . . . ,

(Qβ Xn )(s) ds,

(7.12)

t0

exist on J0 = [t0 , t0 + ), where  = min{a, b/M}, M = max{M1 , M2 }, M1 = max Mβ , β ∈ [0, 1], and converge uniformly to the solution X(t) of the family of β

equations (7.5). Proof It follows from the correlations (7.12) that



t

D[Xn+1 (t), X0 ] = D X0 +

(Qβ Xn )(s) ds, X0 t0

t

 t (Qβ Xn )(s) ds, 0 ≤ D[(Qβ Xn )(s), 0 ] ds

t0

t0

=D t

D0 [(Qβ X), 0 ](t) ds ≤ Mβ (t − t0 ) ≤ M1 (t − t0 ).

≤ t0

For problem (7.11) we have w0 (t) = M1 (t − t0 ) ≤ M(t − t0 ), t wn+1 (t) =

 g (s, wn (s)) ds,

t ∈ J0 ,

n = 0, 1, 2, . . . ,

t0

and for n = 1 the correlation t w1 (t) =

 g (s, w0 (s)) ds ≤ M2 (t − t0 ) ≤ M(t − t0 ) = w0 (t) t0

is true.

(7.13)

158

7 Dynamics of Systems with Causal Operator

Let for some k > 1 and t ∈ J0 the inequality wk (t) ≤ wk−1 (t) hold. Then, by virtue of condition (2) of Theorem 7.3, we have the estimate t wk+1 (t) =

t  g (s, wk (s)) ds ≤

t0

 g (s, wk−1 (s)) ds = wk (t), t0

which implies that the sequence {wk (t)} is monotone decreasing. In view of the fact that dwk = g (t, wk−1 (t)) ≤ M2 , dt

t ∈ J0 ,

and according to the Arzela–Ascoli theorem, it follows that the sequence {wk (t)} is monotone and lim wk (t) = w(t)

k→∞

uniformly in t ∈ J0 . Since w(t) is a solution of problem (7.11), by virtue of condition (3) it follows that w(t) ≡ 0 on J0 . Further, it follows from (7.13) that for any t ∈ J0 and t0 ≤ s ≤ t the correlation



s

D[X1 (s), X0 ] = D X0 +

(Qβ X0 )(τ ) dτ, X0 t0

s



s

(Qβ X0 )(τ ) dτ, 0 ≤

=D t0

  D (Qβ X0 )(τ ), 0 dτ

t0

    ≤ D0 (Qβ X0 ), 0 (s)(s − t0 ) ≤ D0 (Qβ X0 ), 0 (t)(t − t0 ) ≤ Mβ (t − t0 ) ≤ M1 (t − t0 ) ≤ M(t − t0 ) = w0 (t) is valid for any value β ∈ [0, 1]. Hence it follows that D0 [X1 , X0 ](t) ≤ w0 (t)

for all

t ∈ J0 .

It is easy to show that for any t ∈ J0 and t0 ≤ s ≤ t the estimate D0 [Xn+1 (t), Xn (t)] ≤ wn (t),

n = 0, 1, 2, . . . ,

(7.14)

7.5 Global Existence of Solutions

159

is valid and the sequence {Xn (t)} is a Cauchy sequence for the family of causal equations (7.5). Let us show this assertion to be true. Let m ≥ n and ν(t) = D[Xn (t), Xm (t)]. Taking (7.13) into account, we obtain D + ν(t) ≤ D[DH Xn (t), DH Xm (t)](t)   = D (Qβ Xn−1 )(t), (Qβ Xm−1 )(t)     ≤ D (Qβ Xn−1 )(t), (Qβ Xn )(t) + D (Qβ Xn )(t), (Qβ Xm )(t)   + D (Qβ Xm )(t), (Qβ Xm−1 )(t) g (t, D0 [Xn , Xm ](t)) ≤ g (t, D0 [Xn−1 , Xn ](t)) +  + g (t, D0 [Xm−1 , Xm ](t)) ≤ g (t, wn−1 (t)) +  g (t, ν(t)) +  g (t, wn−1 (t)) = g (t, ν(t)) + 2 g (t, wn−1 (t)) . (7.15) Applying Lemma 7.1 to inequality (7.15) and the comparison equation drn = g (t, rn ) + 2 g (t, wn−1 (t)) , dt

rn (t0 ) = 0,

we get, for any n = 0, 1, 2, . . ., the estimate ν(t) ≤ rn (t)

for all t ∈ J0 .

Since for n → ∞ the expression 2 g (t, wn−1 (t)) → 0 uniformly in t ∈ J0 , we obtain by Lemma 7.1 that rn (t) → 0 as n → ∞ uniformly with respect to t ∈ J0 . Consequently, according to (7.14), the sequence {Xn (t)} converges to X(t) on J0 and X(t) is a solution of the family of equations (7.5) for any β ∈ [0, 1]. We shall indicate conditions for the global existence of a family of solutions for Eq. (7.5). Theorem 7.4 Let Qβ ∈ C(E × [0, 1], E) be a causal mapping and there exist a function gˆ ∈ C(R2+ , R) (g(t, ˆ w) is nondecreasing with respect to w) such that D[(Qβ X)(t), 0 ] ≤ g(t, ˆ D0 [X, 0 ](t)) for all β ∈ [0, 1] and t ∈ R+ . Suppose that there exists a maximum solution of the comparison equation dw = g(t, ˆ w), dt on the interval [t0 , ∞).

w(t0 ) = w0 ≥ 0,

(7.16)

160

7 Dynamics of Systems with Causal Operator

If for (t0 , X0 ) ∈ R+ × Kc (Rn ) there exists a local solution of equation (7.5), then the maximal interval of existence for any solution X(t) to the family of equations (7.5) is [t0 , ∞) whenever D[X0 , 0 ] ≤ w0 for any value of β ∈ [0, 1]. Proof Let X(t) = X(t, t0 , X0 ) be any solution of the family of equations (7.5) with the initial conditions D[X0 , 0 ] ≤ w0 existing on the interval [t0 , σ ), t0 < σ < +∞. Assume that σ cannot be increased. Consider the function m(t) = D[X(t), 0 ], having noted that, by the hypothesis of Theorem 7.4, D[X0 , 0 ] ≤ w0 . It is easy to see that   ˆ D0 [X, 0 ](t)) D + m(t) ≤ D[DH X(t), 0 ] ≤ D (Qβ X)(t), 0 ≤ g(t, (7.17) for any β ∈ [0, 1] and all t ∈ [t0 , ∞). Applying Lemma 7.1 to inequality (7.17) and Eq. (7.16), we get the estimate m(t) ≤ rM (t),

t0 ≤ t < σ.

Further, for any t0 < t1 < t2 < σ it is easy to obtain

t1 D[X(t1 ), X(t2 )] = D

(Qβ X)(s) ds, t0

t2 ≤



t2 (Qβ X)(s) ds t1





t2

D (Qβ X)(s), 0 ds ≤ t1

g(s, ˆ D0 [X, 0 ] (s)) ds t1

for any β ∈ [0, 1]. In view of the monotonicity of function g(t, ˆ w), the last estimate implies t2 D[X(t1 ), X(t2 )] ≤

g(s, ˆ r(s)) ds = r(t2 ) − r(t1 ). t1

From the fact that lim r(t, t0 , w0 ) exists, it follows that {Xn (t)} is a Cauchy t →σ −

sequence and lim X(t, t0 , w0 ) = Xσ exists. In this case we shall consider the problem

t →σ −

DH X(t) = (Qβ X)(t),

X(σ ) = Xσ .

(7.18)

The solution X(t) = X(t, σ, Xσ ) can be extended beyond the value σ . This contradicts the assumption made above and proves Theorem 7.4 provided that D[X0 , 0 ] ≤ w0 , i.e., the solution X(t) = X(t, t0 , X0 ) exists for all t ∈ [t0 , ∞) and any value β ∈ [0, 1].

7.6 Comparison Principle

161

7.6 Comparison Principle The principle of comparison, developed in the theory of equations, can also be applied in the theory of causal equations. We shall formulate the main theorem of this principle in terms of the matrix-valued Lyapunov functions for the regularized family of equation (7.5). Definition 7.2 Let Qβ ∈ C(E, E) for all β ∈ [0, 1]. The regularized operator Qβ is called the robust causal operator if, for all t0 ≤ s ≤ t < ∞, the condition X(s) = V (s) implies (Qβ X)(s) = (Qβ V )(s) for all t0 ≤ s ≤ t < ∞ and for any value β ∈ [0, 1]. Together with the families of equations (7.3)–(7.5) we shall consider the matrixvalued function S(t, X) = [Sij (t, X)],

i, j = 1, 2,

(7.19)

whose elements sij are associated with Eqs. (7.3)–(7.5) as follows: S11 (t, X) ∈ C(R+ × Kc (Rn ), R+ ) is associated with the family of equations (7.3); S22 (t, X) ∈ C(R+ × Kc (Rn ), R+ ) is associated with the family of equations (7.4); and S12 (t, X) = S21 (t, X) ∈ C(R+ × Kc (Rn ), R) is associated with the family of equations (7.5). For the given vector ψ ∈ R2+ , ψ > 0, we construct the function V (t, X, ψ) = ψ T S(t, X)ψ

(7.20)

and assume that V ∈ C(R+ × Kc (Rn ) × R2+ , R+ ) and V (t, X, ψ) is locally Lipschitzian in X, i.e., there exists a constant K > 0 such that |V (t, X, ψ) − V (t, Y, ψ)| ≤ KD[X, Y ] for all X, Y ∈ Kc (Rn ), t ∈ R+ , and ψ ∈ R2+ . For function (7.20), we define the expression D− V (t, X, ψ) = lim inf{[V (t + h, X(t) + h(Qβ X)(t)) − V (t, X, ψ)]h−1 : h → 0− }

(7.21)

for any value β ∈ [0, 1]. If functions (7.20) and (7.21) solve the problem on stability of the state 0 ∈ Kc (Rn ) of the family of equations (7.5), then the function (7.20) is called the Lyapunov function for the causal equation (7.5).

162

7 Dynamics of Systems with Causal Operator

Remark 7.1 Alongside the derivative D− V (t, X, ψ), with function (7.20) the derivative D + V (t, X, ψ) can be applied. Theorem 7.5 Assume that for the families of equations (7.3)–(7.5) the function (7.20) is constructed, being locally Lipschitzian in X, and there exists a function g(t, w) ∈ C(R+ × R+ , R) such that: (1) D− V (t, X, ψ) ≤ g(t, V (t, X, ψ)) for all t ∈ [t0 , ∞) and X ∈ B, ψ ∈ R2+ and any value β ∈ [0, 1]; (2) there exists a maximal solution rM (t, t0 , w0 ) of the comparison equation dw = g(t, w), dt

w(t0 ) = w0 ≥ 0,

(7.22)

for all t ∈ [t0 , ∞). Then, along any family of solutions X(t) = X(t, t0 , X0 ) ∈ B of Eq. (7.5) under the initial conditions V (t0 , X0 , ψ) ≤ w0 , the estimate V (t, X(t), ψ) ≤ rM (t, t0 , w0 )

(7.23)

is valid for all t ∈ [t0 , ∞) and any value β ∈ [0, 1]. Proof Let B = {X ∈ Kc (Rn ) : D[X, X0 ] ≤ b} and X(t, t0 , X0 ) ∈ B for all t ∈ [t0 , t1 ). For the function m(t) = V (t, X(t), ψ) and arbitrarily small h > 0 we have m(t + h) − m(t) = V (t + h, X(t + h), ψ) − V (t, X(t), ψ) = V (t + h, X(t + h), ψ) − V (t + h, X(t) + h(Qβ X)(t), ψ) + V (t + h, X(t) + h(Qβ X)(t), ψ) − V (t, X(t), ψ)

(7.24)

≤ KD[X(t + h), X(t) + h(Qβ X)(t)] + V (t + h, X(t) + h(Qβ X)(t), ψ) − V (t, X(t), ψ) for all values β ∈ [0, 1]. According to (7.24), as h → 0− , estimate (7.23) yields D− m(t) = D− V (t, X, ψ) ≤ g(t, V (t, X, ψ)) = g(t, m(t)). Applying Theorem 1.5.2 from the monograph by Lakshmikantham et al. [51] to this inequality, we obtain the required estimate. Theorem 7.5 is proved.

7.7 Stability Analysis The principle of comparison for the family of equations with causal operator makes it possible to investigate the stability of stationary solution to the family of equations (7.5) in terms of the generalized direct Lyapunov method. Further we shall

7.7 Stability Analysis

163

assume that the solution Y (t) = Y (t, t0 , Y0 ) of the set of equations (7.5) is correctly defined for all α ∈ J and all t ≥ t0 . Consider the initial problem DH Y (t) = (Qβ Y )(t),

Y (t0 ) = Y0 ∈ Kc (Rn ),

(7.25)

where (Qβ Y )(t) : R+ × E → Kc (Rn ). If Qβ 0 = 0 for all β ∈ [0, 1], then Y (t) = 0 is a set of stationary solutions of the family of equations (7.25). Since t → diam(X(t)) is a nondecreasing function for t ≥ t0 , the application of the norm X(t) ∼ diam(X(t)) for the family of equations (7.25) in the stability problem seems to be impossible. Further, for Eq. (7.25) the following assumptions are introduced: H1 . For all β ∈ [0, 1], the family of equations (7.25) has a stationary solution 0 , i.e., (Qβ 0 )(t) = 0 . H2 . There exists a set W0 ∈ Kc (Rn ) such that for the sets (V0 , Y0 ) ∈ Kc (Rn ) there exists a Hukuhara difference Y0 − V0 = W0 . H3 . The family of equations (7.25) possesses a unique solution Y (t) = Y (t, t0 , Y0 − V0 ) = X(t, W0 ) for the initial values (t0 , W0 ) ∈ R+ × Kc (Rn ). Definition 7.3 The set 0 of stationary solutions of the family of equations (7.25) is robustly stable if for any t0 ∈ R+ and ε > 0 there exists a δ = δ(t0 , ε) > 0 such that D[Y (t), 0 ] < ε for all t ≥ t0 and any β ∈ [0, 1], whenever D[W0 , 0 ] < δ. Other types of stability of the stationary set 0 are defined in the same way as in Definition 7.3. Further, we establish sufficient conditions for the stability and uniform stability of the stationary solution 0 ∈ Kc (Rn ). Theorem 7.6 For the family of equations (7.25) assume that the conditions of assumptions H1 –H3 are satisfied and, in addition: (1) there exist a matrix-valued function S(t, Y ) and a vector θ ∈ R2+ such that the function V (t, Y, θ ) = θ T S(t, Y )θ satisfies the Lipschitz condition |V (t, Y, θ ) − V (t, Y , θ )| ≤ LD[Y, Y ], where L > 0 and (Y, Y ) ∈ B = {Y ∈ Kc (Rn ) : D[Y, 0 ] ≤ ρ0 } for all t ≥ t0 and V (t, 0 , θ ) = 0; (2) there exist a comparison vector function ψ 1 , ψ 1 is of the class K, and a constant symmetric positive definite (2 × 2)-matrix A1 such that T

ψ 1 (D[Y, 0 ])A1 ψ 1 (D[Y, 0 ]) ≤ V (t, Y, θ ) for all (t, Y ) ∈ R+ × B;

164

7 Dynamics of Systems with Causal Operator

(3) there exists a function g ∈ C(R+ × R+ , R), g(t, 0) = 0 for all t ∈ R+ , such that  D− V (t, Y, θ )(7.25) ≤ g(t, V (t, Y, θ )) for all (t, Y ) ∈ R+ × B and all β ∈ [0, 1]. Then the stability of zero solution to the equation dw = g(t, w), dt

w(t0 ) = w0 ≥ 0,

(7.26)

implies the robust stability of the stationary set of states 0 of the family of equations (7.25). Proof We transform condition (2) of Theorem 7.6 to the form λm (A1 )ψ1 (D[Y, 0 ]) ≤ V (t, Y, θ ),

(7.27)

where λm (A1 ) is a minimal eigenvalue of the matrix A1 and ψ1 is of the class K such that T

ψ1 (D[Y, 0 ]) ≤ ψ 1 (D[Y, 0 ])ψ 1 (D[Y, 0 ]) for all (t, Y ) ∈ R+ × B. Suppose ε > 0 and t0 ∈ R+ are given. Since the function V (t, Y, θ ) is positive definite, estimate (7.27) is satisfied for all (t, Y ) ∈ R+ × B. Let the zero solution of Eq. (7.26) be stable. In this case, for the given ε > 0 and t0 ∈ R+ , λm (A1 )ψ1 (ε) > 0, there exists a δ(t0 , ε) > 0 such that if w0 < δ, then w(t, t0 , w0 ) ≤ λm (A1 )ψ1 (ε)

for all

t ≥ t0 ,

(7.28)

where w(t, t0 , w0 ) is any solution to Eq. (7.25). Suppose that w0 = V (t0 , W0 , θ ) < δ. Then, since V (t, Y (t), θ ) is a continuous function and V (t, 0 , θ ) = 0, there exists a δ1 = δ1 (t0 , ε) > 0 such that the inequalities D[W0 , 0 ] ≤ δ1 and V (t, W0 , θ ) ≤ δ are fulfilled simultaneously. We shall show that if D[W0 , 0 ] ≤ δ1 , then D[Y (t), 0 ] < ε for all t ≥ t0 and any β ∈ [0, 1]. Suppose this is not the case. Then, for the set of solutions Y (t) = Y (t, t0 , W0 ) of the family of equations (7.25) there exists a t1 > t0 such that D[Y (t1 ), 0 ] = ε and D[Y (t), 0 ] < ε for all t0 < t < t1 < +∞ and all β ∈ [0, 1]. According to estimate (7.28), we have V (t1 , Y (t1 ), θ ) ≥ λm (A1 )ψ1 (ε). Therefore, for all t ∈ [t0 , t1 ] and β ∈ [0, 1] the inclusion Y (t) ∈ B is true. Consequently, under condition (3) of Theorem 7.6, for w0 = V (t0 , W0 , θ ) we obtain

7.7 Stability Analysis

165

the estimate V (t, Y (t), θ ) ≤ r(t, t0 , w0 )

for all

t ∈ [t0 , t1 ].

(7.29)

Getting together estimates (7.27)–(7.29) we arrive at λm (A1 )ψ1 (ε) ≤ V (t1 , Y (t1 ), θ ) ≤ r(t1 , t0 , w0 ) < λm (A1 )ψ1 (ε)

for some

t0 < t1 < ∞.

The contradiction obtained shows that D[Y (t), 0 ] < ε for all t ≥ t0 and any β ∈ [0, 1]. This completes the proof of Theorem 7.6. Consider next the conditions for the robust asymptotic stability of the families of equations (7.25). The following result holds. Theorem 7.7 Suppose that there exist functions V (t, Y, θ ) and g(t, w) given in Theorem 7.6 and, in addition: (1) conditions (1) and (2) of Theorem 7.6 are satisfied; (2) condition (3) of Theorem 7.6 holds true for all (t, Y ) ∈ R+ × B and all β ∈ [0, 1]; (3) the zero solution of the comparison equation (7.22) is asymptotically stable. Then the set of stationary states 0 of the family of equations (7.25) is robustly asymptotically stable. Proof Let 0 < ε < ρ0 and t0 ∈ R+ be given. Next, we find ε1 = λm (A1 )ψ1 (ε) and note that the stability of the zero solution of (7.22) implies that for ε1 > 0 there exists δ = δ(t0 , ε1 ) > 0 such that if w0 < δ, then w(t, t0 , w0 ) < ε1

for all t ≥ t0 .

(7.30)

Estimate (7.30) and the hypotheses of Theorem 7.7 imply that for w0 = V (t0 , W0 , θ ) the state w = 0 of Eq. (7.22) is stable, which yields the robust stability of the stationary solution 0 of the family of equations (7.25). Let Y (t, t0 , W0 ) be any solution of the family of equations  (7.25) with the initial conditions D[W0 , 0 ] ≤ δ0 , where δ0 = δ t0 , 12 ρ0 . Since the state w = 0 of Eq. (7.22) is stable, we have D[Y (t), 0 ] < 12 ρ0 for all t ≥ t0 . It follows from (7.27) that λm (A1 )ψ1 (D[Y (t), 0 ]) ≤ V (t, Y (t), θ ) ≤ r(t, t0 , w0 )

(7.31)

for all t ≥ t0 for any solution Y (t) = Y (t, t0 , W0 ) such that D[W0 , 0 ] ≤ δ0 . Suppose that the set 0 is not robustly asymptotically stable. Then, there exists a sequence {tk }, tk > t0 + T , tk → ∞ as k → ∞, such that D[Y (tk ), 0 ] > ε for some solution Y (t), for which D[W0 , 0 ] ≤ δ0 .

166

7 Dynamics of Systems with Causal Operator

Inequalities (7.30), (7.31) and condition (3) of Theorem 7.7 imply that the set of stationary states 0 is robustly asymptotically stable for the family of equations (7.25). We shall establish conditions for the uniform robust stability of the stationary solution 0 to the family of equations (7.25). Theorem 7.8 For the family of equations (7.25) let conditions of assumptions H1 –H3 be satisfied and there exists a function V (t, Y, θ ) satisfying the following conditions: (1) there exist comparison vector functions ψ 1 and ψ 2 of the Hahn class K, and symmetric positive definite (2 × 2)-matrices A1 and A2 such that T

ψ 1 (D[Y, 0 ])A1 ψ 1 (D[Y, 0 ]) ≤ V (t, Y, θ ) T

≤ ψ 2 (D[Y, 0 ])A2 ψ 2 (D[Y, 0 ]) for all (t, Y ) ∈ R+ × B; (2) there exist a (2 × 2)-matrix A3 (β) and a comparison vector function ψ 3 of the Hahn class K such that  T D− V (t, Y (t), θ )(7.25) ≤ ψ 3 (D[Y, 0 ])A3 (β)ψ 3 (D[Y, 0 ]) for all (t, Y ) ∈ R+ × B and any β ∈ [0, 1]; (3) there exists a constant symmetric (2 × 2)-matrix A3 such that A3 (β) ≤ A3 for all β ∈ [0, 1].

(7.32)

1 2

 T A3 (β)+

Then, if the matrix A3 is negative definite, then the set of stationary states 0 is robustly uniformly stable for the family of equations (7.25). Proof Condition (1) of Theorem 7.8 can be rewritten as λm (A1 )ψ1 (D[Y, 0 ]) ≤ V (t, Y, θ ) ≤ λM (A2 )ψ2 (D[Y, 0 ]),

(7.33)

where the functions ψ1 and ψ2 are of Hahn class K so that T

ψ1 (D[Y, 0 ]) ≤ ψ 1 (D[Y, 0 ])ψ 1 (D[Y, 0 ]), T

ψ2 (D[Y, 0 ]) ≥ ψ 2 (D[Y, 0 ])ψ 2 (D[Y, 0 ]), λM (A2 ) is a maximum eigenvalue of the matrix A2 . Let ε ∈ (0, ρ) and t0 ∈ R+ be given. We choose the value δ = δ(ε) > 0 so that λM (A2 )ψ2 (δ) < λm (A1 )ψ1 (ε).

(7.34)

7.8 Hyers–Ulam–Rassias Stability of the Set of Causal Equations

167

We shall show that, if D[W0 , 0 ] ≤ δ, then D[Y (t), 0 ] < ε for all t ≥ t0 and any value β ∈ [0, 1]. Let this be not the case. Then, there exist a solution Y (t) with the initial conditions D[W0 , 0 ] ≤ δ and an instant t2 > t0 such that D[Y (t2 ), 0 ] = ε and D[Y (t), 0 ] < ε for all t ∈ [t0 , t2 ). From the inequality (7.33) it follows that, for t = t2 , the following estimate holds true: V (t2 , Y (t2 ), θ ) ≥ λm (A1 )ψ1 (ε).

(7.35)

Note that, since 0 < ε < ρ, we have Y (t) ∈ B for all t ≥ t0 and any β ∈ [0, 1]. According to the chosen w0 = V (t0 , W0 , θ ), under the condition  D− V (t, Y (t), θ )(7.25) ≤ 0 for all t ≥ t0 and Y (t) ∈ B, we get the estimate V (t, Y (t), θ ) ≤ V (t0 , W0 , θ ),

t ∈ [t0 , t2 ).

(7.36)

In view of inequalities (7.34)–(7.36), we get λm (A1 )ψ1 (ε) ≤ V (t2 , Y (t2 ), θ ) ≤ λM (A2 )ψ2 (D[W0 , 0 ]) ≤ λM (A2 )ψ2 (δ) < λm (A1 )ψ1 (ε). The contradiction obtained proves the uniform robust stability of the stationary set 0 for the family of equations (7.31).

7.8 Hyers–Ulam–Rassias Stability of the Set of Causal Equations In this section we present a result on Hyers–Ulam–Rassias stability for the family of causal equations. Consider a family of causal equations in the form DH U (t) = F (t, U (t), (QU )(t)),

U (t0 ) = U0 ∈ Kc (Rn ),

(7.37)

where U ∈ Kc (Rn ), F ∈ C(R+ × Kc (Rn ) × E, Kc (Rn )) and E = C(R+ , Kc (Rn )) with the norm D0 [U, ] = sup D[U (t, )]. t ∈R+

We shall give the definition of H.U.R. stability of the family of equations (7.37). Definition 7.4 The family of causal equations (7.37) is Hyers–Ulam–Rassiasstable with respect to the set of functions ∗ (t) ∈ Kc (Rn ) if there exists a constant C ∗ > 0 such that for each Y (t) ∈ Kc (Rn ), for which DH Y (t) − F (t, Y (t), (QY )(t)) ⊆ ∗ (t),

t ∈ R+ ,

168

7 Dynamics of Systems with Causal Operator

a solution U (t) ∈ Kc (Rn ) to the family of equations (7.37) is found so that D[Y (t), U (t)] ≤ C ∗ D[ ∗ (t), 0 ]

for all t ∈ R+ .

For the family of causal equations (7.37) the following result is valid. Theorem 7.9 Assume that: (1) F ∈ C(R+ × Kc (Rn ) × E, Kc (Rn )), where Q is the causal operator acting on the space E; (2) for the given set of functions ∗ (t) ∈ Kc (Rn ) for each Y (t) ∈ Kc (Rn ) the inclusion DH Y (t) − F (t, Y (t), (QY )(t)) ⊆ ∗ (t) is satisfied for all t ∈ R+ ; (3) there exist a function μ(t) ∈ L1 (R+ , R+ ) and a constant δ > 0 such that F satisfies the generalized Lipschitz condition D[F (t, U, (QU )(t)), F (t, Y, (QY )(t))] ≤ μ(t)D[U (t), Y (t)] + D[(QU )(t), (QY )(t)], where D[(QU )(t), (QY )(t)] ≤ δD[U (t), Y (t)] for all

(t, U, Y ) ∈ R+ × Kc (Rn ) × Kc (Rn );

(4) there exists a constant γ > 0 such that t

D[ ∗ (s), 0 ]ds ≤ γ D[ ∗ (t), 0 ] for all t ∈ R+ .

0

Then the family of causal equations (7.37) is H.U.R.-stable with respect to the set of functions ∗ (t) ∈ Kc (Rn ). Proof Let Y (t) ∈ Kc (Rn ) be a solution of the inclusion from condition (2) of Theorem 7.9. From the family of equations (7.37) for U0 = Y0 ∈ Kc (Rn ) we have t U (t) = Y0 +

F (s, U (s), (QU )(s))ds 0

(7.38)

7.8 Hyers–Ulam–Rassias Stability of the Set of Causal Equations

169

and t Y (t) − Y0 −

t F (s, Y (s), (QY )(s))ds ⊆

0

∗ (s)ds

(7.39)

0

for all t ∈ R+ . Correlations (7.38) and (7.39) yield

t D[U (t), Y (t)] = D Y0 + F (s, U (s), (QU )(s))ds, 0

t Y0 +

t F (s, Y (s), (QY )(s))ds +

0

∗



(s)ds 0

t ≤D

F (s, U (s), (QU )(s))ds, 0

t +D



t F (s, Y (s), (QU )(s))ds 0



∗ (s)ds, 0 ≤ γ D[ ∗ (t), 0 ]

0

t +

(μ(s) + δ)D[U (s), Y (s)]ds for all t ≥ 0. 0

The inequality t

∗

D[U (t), Y (t)] ≤ γ D[ (t), 0 ] +

(μ(s) + δ)D[U (s), Y (s)]ds 0

implies that t

∗

D[U (t), Y (t)] ≤ γ D[ (t), 0 ] +

γ D[ ∗ (s), 0 ]

0

t × (μ(s) + δ) exp s

(μ(u) + δ)du ds

170

7 Dynamics of Systems with Causal Operator

 t

 t ≤ γ D[ (t), 0 ] 1 − exp (μ(u) + δ)du ds ∗

s

0



t

t  = γ D[ ∗ (t), 0 ] 1 − exp (μ(u) + δ)du s

0

t

= γ D[ (t), 0 ] exp δ (μ(u)ds) ∗

0

≤ γ exp δ

∞

μ(u)du D[ ∗ (t), 0 ] ≤ C ∗ D[ ∗ (t), 0 ],

0

where C ∗

∞

= γ exp δ μ(u)du . 0

Thus, we arrive at the estimate D[U (t), Y (t)] ≤ C ∗ D[ ∗ (t), 0 ] for all t ≥ 0, which is required in Definition 7.4. This completes the proof of Theorem 7.9.

7.9 Notes and References The concept of causality is a mainstay of scientific thinking (see Birkhoff and Lewis [12]), it is critical, for instance, in the study of uniqueness of solution to the initial problem for the system of ordinary differential equations. The term “causal operator” is adopted from the engineering applications and the general theory of operators. The concept of causality and the causal equations are the tool of unified description of real processes modeled by ordinary differential equations, integro-differential equations, equations with finite and/or infinite memory, Volterra integral equations, and the equations of other types. The general theory of equations with causal operator is presented in the monographs by Corduneanu [19] and Lakshmikantham et al. [50]. This theory covers many classes of equations that have been studied and applied in various fields of science and technology (see Karakostas [37], Burton [16], Hale [32], etc.). At present, the concept of causality is being developed in many directions (see, for example, Mooij et al. [96], Lupulescu [58] and the bibliography therein).

7.9 Notes and References

171

Definition 7.1 is formulated with allowance for the results of the monographs by Corduneanu [19] and Lakshmikantham et al. [50]. Lemma 7.1 is due to Lakshmikantham et al. [51]. In [71] Martynyuk proposed to use the matrix Lyapunov functions for the analysis of robust stability of the set equations with causal operator. In the monograph by Martynyuk and Martynyuk-Chernienko [83] this approach has been developed to some extent in combination with the idea of regularization of the uncertain equations with causal operator. This chapter is based on the main ideas and approaches due to Martynyuk [71], and Martynyuk and Martynyuk-Chernienko [83] as well as their developments in context with the matrix Lyapunov functions. The results of Sect. 7.8 are new. The analysis of Hyers–Ulam–Rassias stability of equations with causal operator is advantageous in mathematical biology, economics, etc. (see Rus [106], Corduneanu et al. [20] and the bibliography therein).

Chapter 8

Finite-Time Stability of Standard Systems Sets

For standard form nonlinear equations with generalized derivative, estimates of deviation of a set of exact solutions from the averaged ones are established and the deviation of a set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integrations and stability over finite interval are considered as applications.

8.1 Introduction For standard form nonlinear equations with generalized derivative, estimates of deviation of a set of exact solutions from the averaged ones are established and the deviation of a set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integrations and stability over finite interval are considered as applications. The chapter is arranged as follows: In Sect. 8.2 the problem for a set of standard systems of differential equations with generalized derivative is stated. In Sect. 8.3 an estimate of distance between the set of solutions to the initial and the averaged system is found. In Sect. 8.4 the application of Theorem 8.1 to the analysis of set of solutions to quasi-linear equations is considered. In Sect. 8.5 an estimate of deviation of the set of solutions to the affine system from zero is given. In Sect. 8.6 concluding remarks are presented and some problems for further investigations are discussed.

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3_8

173

174

8 Finite-Time Stability of Standard Systems Sets

8.2 Statement of the Problem The set of systems of differential equations DH X = μF (t, X), X(t0 ) = X0 ∈ Kc (Rn ),

(8.1)

where F ∈ C(R × Kc (Rn ), Kc (Rn )), and μ ∈ (0, 1] is a small parameter, is called a generalized standard system. Together with Eq. (8.1) we shall consider a partially averaged differential equation DH Y = μG(t, Y ),

Y (t0 ) = Y0 ∈ Kc (Rn ),

(8.2)

for which 1 D lim T →∞ T

T

T

G(s, Y )ds = 0,

F (s, X)ds, 0

 (8.3)

0

for X, Y ∈ D ∗ ⊂ Kc (Rn ). We assume on the families of equations (8.1) and (8.2) as follows. H1 . There exists a function M(t, ·) > 0, integrable on J , for all t ∈ J such that μD[G(t, X), G(t, Y )] ≤ M(t, μ)D[X, Y ] for all 0 < μ < μ1 ∈ (0, 1]; H2 . There exist a function f (t, ·) > 0, integrable on J , lim f (t, μ) = 0 as t → ∞, and p > 1 such that μD[F (t, X), G(t, Y )] ≤ f (t, μ)D p [X, Y ] for all (X, Y ) ∈ D ∗ ⊆ Kc (Rn ) and 0 < μ < μ2 ∈ (0, 1]. This chapter is aimed to obtain the estimate of deviation between solutions to the family of equations (8.1) and (8.2) and deviation of solutions to averaged equations (8.2) from the equilibrium state 0 ∈ Kc (Rn ).

8.3 On Distance Between the Sets of Solutions We shall estimate deviations between the sets of solutions to the families of equations (8.1) and (8.2). Let us show that the following result holds true. Theorem 8.1 In the domain Q = {(t, X) : t ≥ t0 ≥ 0, X ∈ D ∗ ⊆ Kc (Rn )} let the following conditions be satisfied:

8.3 On Distance Between the Sets of Solutions

175

(1) the solution X(t) of the initial problem for the family of equations (8.1) exists for all t ≥ t0 and 0 < μ < μ1 , μ1 ∈ (0, 1]; (2) the solution Y (t) of the family of equations (8.2) with the initial condition Y0 ∈ D ∗ ⊂ D exists for all t ≥ t0 and 0 < μ < μ2 , μ2 ∈ (0, 1]; (3) the limit (8.3) exists uniformly with respect to X ∈ D; (4) the conditions of hypotheses H1 and H2 are satisfied; (5) for all t ∈ J and 0 < μ < μ0 the inequality t 1 − (p − 1)D

p−1

[X0 , Y0 ]

f (s, μ) 0



s × exp 2(p − 1) M(τ, μ) dτ ds > 0 0

is valid. Then the deviation between the sets of solutions to equations (8.1) and (8.2) is estimated as t D[X(t), Y (t)] ≤ D[X0 , Y0 ] exp

M(s, μ)ds

0

× 1 − (p − 1)D p−1 [X0 , Y0 ]

(8.4)

t

1

− p−1 s f (s, μ) exp 2(p − 1) M(τ, μ) dτ ds

0

0

×

for all t ∈ J and 0 < μ < μ0 , μ0 = min(μ1 , μ2 ). Proof We represent the families of equations (8.1) and (8.2) in the equivalent form t X(t) = X0 + μ

F (s, X(s)) ds,

X0 ∈ D ⊂ Kc (Rn ),

0

and t Y (t) = Y0 + μ

G(s, Y (s)) ds, 0

Y0 ∈ D ∗ ⊂ D,

176

8 Finite-Time Stability of Standard Systems Sets

and assume that D[X0 , Y0 ] = 0 for all X0 and Y0 in the domain of their values. It is easy to get the following estimates:

t

D[X(t),Y (t)] = D X0 + μ

F (s, X(s)) ds, Y0 + μ 0

G(s, Y (s)) ds 0

t = D[X0 , Y0 ] + μD

F (s, X(s)) ds,

G(s, Y (s)) ds 0

t ≤ D[X0 , Y0 ] + μD



t

0



t F (s, X(s)) ds,

0

t + μD



t

G(s, Y (s)) ds 0



t G(s, X(s)) ds,

0

G(s, Y (s)) ds 0

t ≤ D[X0 , Y0 ] + μ

D[F (s, X(s)), G(s, Y (s))] ds 0

t +μ

D[G(s, X(s)), G(s, Y (s))]ds.

(8.5)

0

In view of D[F (t, X), G(t, X)] ≤ D[F (t, X), G(t, Y )] + D[G(t, Y ), G(t, X)], under hypotheses H1 and H2 we get from estimate (8.5) that t D[X(t), Y (t)] ≤ D[X0 , Y0 ] + 2

M(s, μ)D[X(t), Y (t)] ds 0

(8.6)

t +

f (s, μ)D p [X(s), Y (s)] ds, 0

where 0 < μ < μ0 , μ0 = min(μ1 , μ2 ). We designate D[X(t), Y (t)] = m(t) and represent inequality (8.6) as t m(t) ≤ m(t0 ) + 2

t M(s, μ)m(s) ds +

0

f (s, μ)mp (s) ds, 0

(8.7)

8.3 On Distance Between the Sets of Solutions

177

where 0 < μ < μ0 , μ0 = min(μ1 , μ2 ), and t ∈ J . Inequality (8.7) is rewritten in the pseudo-linear form t m(t) ≤ m(t0 ) +

(2M(s, μ) ds + f (s, μ)mp−1 (s))m(s) ds 0

for all t ∈ J . Applying to this inequality the summand estimation technique from Louartassi et al. [56] and Martynyuk [77] we get t

m(t) ≤ m(t0 ) exp 2 M(s, μ) ds 0

× 1 − (p − 1)mp−1 (t0 )

(8.8)

t

1

− p−1 s f (s, μ) exp 2(p − 1) M(τ, μ) dτ ds

0

0

×

under condition (5) of Theorem 8.1. Estimate (8.8) yields the assertion of Theorem 8.1. Estimate (8.4) allows one to establish sufficient conditions for the presence of (A, λ)-estimate of approximate integration of the family of equations (8.1) in the sense of the definition below. Definition 8.1 The set of solutions Y (t) of the family of differential equations (8.2) satisfies (A, λ)-estimate of approximate integration of the family of equations (8.1) if, given the values λ and A (0 < λ < A), the condition D[X0 , Y0 ] < λ implies that D[X(t), Y (t)] < A for 0 < μ < μ0 on the common existence interval of solutions X(t) and Y (t). Corollary 8.1 Let all conditions of Theorem 8.1 be satisfied and for given values λ and A the inequality  t  exp 2 M(s, μ)ds

A 0     1 < λ t s p−1 1 − (p − 1)λp−1 f (s, μ) exp 2(p − 1) M(τ, μ)dτ ds 0

0

hold true for all 0 < μ < μ0 and all t ∈ J . Then for the set of solutions X(t) of the family of equations (8.1) the (A, λ)estimate of approximate integration takes place.

178

8 Finite-Time Stability of Standard Systems Sets

The assertion of Corollary 8.1 follows immediately from the estimate (8.4) and Definition 8.1. Further we consider the set of the quasi-linear equation X(0) = X0 ⊂ D ∗ ,

DH X = A(t)X + μF (t, X),

(8.9)

where A(t) is a bounded operator on R+ , F (t, X) is a mapping containing X in power higher than 2. The solution of problem (8.9) is the mapping X(t) = X(t, t0 , X0 ) satisfying the family of equations (8.9) almost everywhere on J . Together with the family of equations (8.9) we consider a family of averaged equations DH Y = A(t)Y + μG(t, Y ),

Y (t0 ) = Y0 ⊂ D ∗ ,

(8.10)

where A = lim

T →∞

1 T

T A(s) ds

(8.11)

0

and G(t, Y ) satisfies relation (8.3). We assume on the family of equations (8.10) as follows. H3 . There exists an integrable function b(t) > 0 for all t ∈ J such that A(t) − A ≤ b(t). We find the estimate of deviation of the set of solutions to the averaged equation (8.10) from the solutions to the initial equations (8.9). Theorem 8.2 In the domain Q = {(t, X) : t ≥ 0 and X ∈ D ⊂ Kc (Rn )} and let the following conditions be satisfied: (1) there exists the limit (8.11); (2) the conditions of hypotheses H1 –H3 are satisfied; (3) for Y0 ∈ D ∗ the solution of averaged equations (8.10) is defined for all t ≥ 0 and 0 < μ < μ0 ; (4) for all t ∈ J and 0 < μ < μ0 the inequality 2(p − 1)mp−1 (t0 ) t × 0

is true.



s M(s, μ) exp (p − 1) (b(τ ) + f (τ, μ))dτ ds < 1 0

8.3 On Distance Between the Sets of Solutions

179

Then the deviation between the sets of solutions to Eqs. (8.9) and (8.10) is estimated as t D[X(t), Y (t)] ≤ D[X0 , Y0 ] exp

(b(s) + 2M(s, μ)) ds

0

t × 1 − 2(p − 1)mp−1 (0) M(s, μ)

(8.12)

0 1

− p−1 s × exp (p − 1) (b(τ ) + f (τ, μ)) dτ ds

0

for all t ∈ J and 0 < μ < μ0 . Proof The relations t X(t) = X0 +

(A(s)X(s) + μF (s, X(s))) ds, 0

(8.13)

t Y (t) = Y0 +

(A(s)Y (s) + μG(s, Y (s))) ds 0

for D[X0 , Y0 ] = 0 yield

t

D[X(t), Y (t)] = D X0 +

(A(s)X(s) + μF (s, X(s))) ds, 0



t Y0 +

(A(s)Y (s) + μG(s, Y (s))) ds 0

t ≤ D[X0 , Y0 ] + D

A(s)X(s) ds, 0

t + μD



t A(s)Y (s) ds 0



t F (s, X(s)) ds,

0

G(s, Y (s)) ds 0

t ≤ D[X0 , Y0 ] +

(A(s) − A)D[X(s), Y (s)] ds 0

180

8 Finite-Time Stability of Standard Systems Sets

t +μ

D[F (s, X(s)), G(s, X(s))] ds 0

t +μ

D[F (s, X(s)), G(s, Y (s))] ds. 0

Hence, according to hypotheses H1 –H3 it follows that t D[X(t), Y (t)] ≤ D[X0 , Y0 ] +

(b(s) + 2M(s, μ))D[X(s), Y (s)] ds 0

(8.14)

t +

f (s, μ)D p [X(s), Y (s)] ds 0

for all t > t0 and 0 < μ < μ0 . Inequality (8.14) is rewritten as t m(t) ≤ m(t0 ) +

[(b(s) + 2M(s, μ))m(s) + f (s, μ)mp (s)] ds 0

and further t m(t) ≤ m(t0 ) +

[(b(s) + 2M(s, μ)) + f (s, μ)mp−1 (s)]m(s) ds.

(8.15)

0

As in the analysis of inequality (8.7), we get from estimate (8.15) the inequality t m(t) ≤ m(t0 ) exp

(b(s) + 2M(s, μ))ds

0

t × 1 − (p − 1)(0) f (s, μ) 0 1

− p−1 s × exp (p − 1) (b(τ ) + 2M(τ, μ))dτ ds

0

(8.16)

8.4 Finite-Time Stability for the Set of Averaged Equations

181

provided that condition (4) of Theorem 8.2 is satisfied for all t ∈ J and 0 < μ < μ0 . In view of the designation m(t) = D[X(t), Y (t)], for all t ∈ J estimate (8.16) completes the proof of Theorem 8.2. Corollary 8.2 Let all conditions of Theorem 8.2 be satisfied and for the given estimates of the values λ and A the inequality

t (b(s) + 2M(s, μ))ds

exp 0

t × 1 − 2(p − 1)mp−1 (0) M(s, μ) 0

 × exp (p − 1)

s

 −1 A (b(τ ) + f (τ, μ))dτ ds < λ

0

holds for all t > t0 and 0 < μ < μ0 . Then for the set of solutions X(t) of the family of equations (8.9) the (A, λ)-estimate of approximate integration takes place. The assertion of Corollary 8.2 follows from estimate (8.12) and Definition 8.1.

8.4 Finite-Time Stability for the Set of Averaged Equations Further we shall consider a family of averaged equations (8.9). Assume that the following conditions are satisfied. H4 . There exists a constant a > 0 such that A < a,

a = const > 0.

H5 . There exists a function N(∗, t) > 0, which is integrable on J , such that μD[G(t, Y ), 0 ] ≤ N(μ, t)D p [Y, 0 ] for all t ∈ J and 0 < μ < μ0 in the domain of values Y ⊂ D ∗ . We shall show that the following result is valid. Theorem 8.3 In the domain Q = {(t, Y ) : t ≥ 0, Y ∈ D ∗ ⊂ Kc (Rn )} let the following conditions be satisfied. (1) there exists a solution Y (t) = Y (t, t0 , Y0 ) of the averaged equation (8.10) for all t ≥ 0 and Y ∗ ∈ D ∗ ; (2) the conditions of hypotheses H4 and H5 are satisfied.

182

8 Finite-Time Stability of Standard Systems Sets

Then the deviation of the set of solutions Y (t) from the equilibrium state is estimated as t D[Y (t), 0 ] ≤ n(0) exp

A ds 0

1



− p−1 t s × 1 − (p − 1)np−1 (0) N(μ, s) exp (p − 1) Adτ ds

0

(8.17)

0

for all t ∈ J and 0 < μ < μ0 provided that t (p − 1)n

p−1

(t0 )



s N(μ, s) exp (p − 1) Adτ ds < 1

0

0

for all t ∈ J and 0 < μ < μ0 . Proof For correlation (8.13) we have t D[Y (t), 0 ] ≤ D[Y0 , 0 ] +

AD[Y (s), 0 ] ds 0

t +μ

D[G(s, Y (s)), 0 ] ds ≤ D[Y0 , 0 ] 0

t +

t AD[Y (s), 0 ] ds +

0

N(μ, s)D p [Y (s), 0 ] ds. 0

Hence, for the function n(t) = D[Y (t), 0 ] estimating the deviation of the set of solutions to the averaged equations from zero in Kc (Rn ), we have the inequality t n(t) ≤ n(0) +

t An(s) ds +

0

N(μ, s)np (s) ds 0

for all t ∈ J and 0 < μ < μ0 . Applying to this inequality the technique used for the proof of Theorem 8.1 we arrive at estimate (8.17). This proves Theorem 8.3. Estimate (8.17) allows one to establish conditions for finite-time stability of the set of solutions to equations (8.10).

8.5 Boundedness of the Set of Solutions of Standard Affine Systems

183

Definition 8.2 For given estimates of the values λ, A, J the set of solutions to the averaged equations (8.10) is finite-time stable if D[Y (t), 0 ] < A for all t ∈ J , whenever D[Y0 , 0 ] < λ and 0 < μ < μ0 . Corollary 8.3 Let all conditions of Theorem 8.2 be satisfied and for given estimates of the values λ, A and J the inequality exp

+

 t

Ads



t

A   , 1 < λ s p−1 N(μ, s) exp (p − 1) Adτ ds

0

0

0

1 − (p − 1)λp−1

holds true for all t ∈ J and 0 < μ < μ0 . Then the set of solutions Y (t) of the family of equations (8.10) is finite-time stable. The assertion of Corollary 8.3 follows from estimate (8.17) and Definition 8.2.

8.5 Boundedness of the Set of Solutions of Standard Affine Systems Consider a family of affine systems of the form DH X(t) = μ(f (t, X) + g(t, X)U (t)),

(8.18)

X(t0 ) = X0 ∈ D ⊂ Kc (R ),

(8.19)

n

where f (t, X) : R+ × D → Kc (Rn ), g(t, X) is an n × n-matrix, g(t, X) : R+ × D ∗ → Kc (Rn ), U (t) ∈ W ⊂ Kc (Rn ) is a control. Together with the family of equations (8.18) we consider the averaged equations DH Y (t) = μ(f (t, Y ) + g(t, Y )V (t)), Y (t0 ) = Y0 ∈ D ⊂ Kc (Rn ),

(8.20)

where 1 D lim T →∞ T

T

T f (s, X(s))ds,

0

1 D T →∞ T

T

lim

T

for all (U, V ) ∈ W ⊂ Kc (Rn ).

(8.21)

 g(s, Y (s))V (s)ds = 0;

(8.22)

0

g(s, X(s))U (s)ds, 0

 f (s, Y (s))ds = 0;

0

184

8 Finite-Time Stability of Standard Systems Sets

We shall estimate the deviation of solutions to the family of equations (8.20) from the state 0 ∈ Kc (Rn ). Assume as follows: H6 . There exists a function f1 (t, μ) > 0, integrable on J and such that D[f (t, Y ), 0 ] ≤ f1 (t, μ)D[Y, 0 ] for all Y ∈ D and 0 < μ < μ1 . H7 . There exists a function f2 (t, μ) > 0, integrable on J and such that D[g(t, Y )V (t), 0 ] ≤ f2 (t, μ)D 2 [Y, 0 ] for all Y ∈ D ∗ , V (t) ∈ W and 0 < μ < μ1 . Theorem 8.4 In the domain Q = {(t, Y ) : t ≥ 0, Y ∈ D ∗ ⊂ Kc (Rn )} for Eqs. (8.18) and (8.20) let (1) there exist limits (8.21) and (8.22); (2) the conditions of hypotheses H6 and H7 be satisfied; (3) for all t ≥ 0 and 0 < μ < μ0 the inequality t

t 1 − D[Y0 , 0 ]

f2 (s, μ) exp 0

f1 (τ, μ)dτ ds > 0

(8.23)

0

hold true. Then the deviation of the set of solutions to the family of equations (8.20) from zero is estimated as D[Y0 , 0 ] exp

 t

 f1 (s, μ)ds

0

D[Y (t), 0 ] ≤

t

 t

0

0

1 − D[Y0 , 0 ] f2 (s, μ) exp

 f1 (τ, μ)dτ ds

(8.24)

for all t ≥ 0, Y ∈ D ∗ and 0 < μ < μ0 . Proof Let the limiting relations (8.21) and (8.22) be satisfied. From Eq. (8.20) we have t Y (t) = Y0 + μ 0

 f (s, Y (s)) + g(s, Y (s))V (s) ds

8.5 Boundedness of the Set of Solutions of Standard Affine Systems

185

and further t D[Y (t), Y0 ] ≤ D[Y0 , 0 ] + μ

D[f (s, Y (s)), 0 ] ds 0

(8.25)

t +μ

D[g(s, Y (s))V (s), 0 ] ds. 0

Under the conditions of hypotheses H6 and H7 we find from inequality (8.25) that t D[Y (t), 0 ] ≤ D[Y0 , 0 ] + μ

f (s, Y (s))D[Y (s), 0 ] 0

(8.26)

+ f2 (s, μ)D 2 [Y (s), 0 ] ds. Designate n(t) = D[Y (t), 0 ] and from (8.26) we get t n(t) ≤ n(t0 ) + μ

(f1 (s, μ) + f2 (s, μ)n(s))n(s) ds.

(8.27)

0

Applying Gronwall-Bellman lemma to inequality (8.27) we arrive at t n(t) ≤ n(t0 ) exp

(f1 (s, μ) + f2 (s, μ))n(s) ds .

0

Hence t

t −n(t) exp

(−f2 (s, μ))n(s) ds ≥ n(t0 ) exp 0

f1 (s, μ) ds .

0

Multiplying both sides of this inequality by f2 (t, μ) > 0 we obtain d dt



exp



t −

f2 (s, μ)n(s)ds 0

t ≥ −n(t0 )f2 (t, μ) exp 0

f1 (s, μ)ds .

186

8 Finite-Time Stability of Standard Systems Sets

Integrating this inequality between t0 and t we get −1

(n(t))

t exp

f1 (s, μ)ds

0

t

t ≥ 1 − n(t0 )

f2 (s, μ) exp 0

f1 (τ, μ)dτ ds.

0

Hence follows the estimate of deviation of the family of solutions to equation (8.20) from the equilibrium state in the (8.24) form under condition (8.23). Theorem 8.4 is proved. Estimate (8.24) allows one to establish boundedness conditions for the set of solutions to the averaged affine system (8.20). Definition 8.3 The set of solutions to equations (8.20) is bounded if for any ε > 0 and t0 ∈ R+ there exists a δ(t0 , ε) > 0 such that D[Y (t), 0 ] < ε for all t ≥ t0 and 0 < μ < μ0 , whenever D[Y0 , 0 ] < δ. If δ does not depend on t0 , the boundedness of the set of solutions Y (t) is uniform with respect to t0 . Corollary 8.4 Let all conditions of Theorem 8.4 be satisfied and for any ε > 0 there exist a δ(ε) > 0 such that exp

 t

 f1 (s, μ)ds

0

t

 t

0

0

1 − δ(ε) f2 (s, μ) exp

ε  < δ(ε) f1 (τ, μ)dτ ds

for all t ≥ 0 and 0 < μ < μ0 . Then the set of solutions of equations (8.20) is uniformly bounded. The assertion of Corollary 8.4 follows from estimate (8.24) and Definition 8.3.

8.6 Notes and References A key element of the approach is the use of nonlinear integral inequalities in the problems of qualitative analysis of the set of trajectories of generalized standard systems. The resulting estimates of the deviation of the set of trajectories from

8.6 Notes and References

187

the equilibrium state, and the estimate of the distance between the sets of initial and averaged solutions to systems of equations are applicable in many problems of mechanics and applied mathematics in which the processes models are the system of equations (8.9) and (8.18). The averaging technique developed in the framework of nonlinear mechanics is a powerful tool widely used for the analysis of nonlinear systems found in various applied investigations (see Krylov and Bogoliubov [41], Bogoliubov and Mitropolskii [13], Grebenikov et al. [29], etc.). In the book by Martynyuk [63] the averaging techniques are combined with the comparison method and Lyapunov’s direct method using scalar, vector or matrix Lyapunov functions (for details, see Müller [97]). On the basis of the estimates obtained, the distances between the solutions of the initial and averaged families of equations are estimated and the conditions for finite-time stability are established as well as the (λ, A)-estimates of approximate integration. These conditions are a generalization of known results obtained for systems of ordinary differential equations (see Martynyuk [61, 81], Lakshmikantham et al. [47]). The averaging scheme is used here for the set equations, described in the monograph by Plotnikov and Skripnik [102]. The results of the chapter are new (see also Martynyuk [80]).

Postface

The qualitative and analytic theories of multivalued differential equations are developed in the wake of two trends: the one is the internal development of the theory of differential equations and the other is the attempts to describe realworld processes and phenomena giving rise to multivalued differential equations. By now there is still no methodology developed for the description of realworld phenomena by means of multivalued differential equations at the level of engineering applications. At the same time, the general theory of such equations has advanced essentially. The ground for appearing of a multivalued right-hand side in the differential equations are the uncertainties of different nature in the description of parameters of the system in question. In this regard three problems arise. Problem 1 How to carry out qualitative analysis of the properties of solutions to the families of equations with uncertain parameter values? In this book, a regularization procedure for the right-hand side of the family of equations with respect to the uncertainty parameter is proposed to solve the problem. The resulting regularized family of equations enables one to apply some developed approaches in the framework of the general theory of multivalued differential equations. Namely, the techniques applied are: a comparison principle in the integral and differential forms, a direct Lyapunov method based on matrix-valued functions, and other. Generalization of the direct Lyapunov method based on the auxiliary matrix-valued function developed for many types of equations turned out to be efficient as well for the family of regularized equations. Problem 2 How to effectively construct a Lyapunov function for regularized equations? In view of the fact that for the families of equations (exact or uncertain) there exists no partial differential equation similar to that considered in the investigation of systems of ordinary differential equations, the “distance” type functions are taken as © Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3

189

190

Postface

the matrix function components. This approach allows for upper and lower bounds for the multivalued right-hand side and for the regularized family of equations. The idea is used in the book to construct the Lyapunov function for some types of the families of equations for which the Hausdorff metric is defined. However, in general, this problem remains open. Problem 3 How to estimate the effect of small perturbations on the dynamical properties of solutions to regularized equations? In fact, a regularized family of equations is an approximation of the family of initial uncertain equations. With this in mind, in the book a method of estimating the deviation of the set of solutions to the family of equations from the equilibrium state is proposed basing on the pseudo-linear representation of nonlinear integral inequalities occurred when the dynamics of the regularized family of equations and small perturbations are taken into account. This technique works effectively in the problems under consideration and possesses a considerable potential for further development. This book provides answers to the questions posed, and even so, in view of the high rate development of the qualitative analysis methods for the systems of equations describing real processes in engineering and technology, the following should not be left unmentioned. The opportunities in the engineering sciences are changing at such a quick pace, the writing of any book cannot keep up with!

Bibliography

1. Akhmetov, M.U., Zafer, A.: Stability of zero solution of impulsive differential equations by the Lyapunov second method. J. Math. Anal. Appl. 248, 69–82 (2000) 2. Aleksandrov, A.Yu., Platonov, A.V.: Aggregation and stability analysis of nonlinear complex systems. J. Math. Anal. Appl. 342(2), 989–1002 (2008) 3. Aleksandrov, A.Yu., Platonov, A.V.: Conditions of ultimate boundedness of solutions for a class of nonlinear systems. Nonlinear Dyn. Syst. Theory 8(2), 109–122 (2008) 4. Aleksandrov, A.Yu., Zhabko, A.P.: Preservation of stability under discretization of systems of ordinary differential equations. Sib. Math. J. 51(3), 383–395 (2010) 5. Aleksandrov, A.Yu., Martynyuk, A.A., Platonov, A.V.: Analysis of a set of nonlinear dynamics trajectories: stability of difference equations. J. Math. Anal. Appl. 421(1), 105–117 (2015) 6. Aleksandrov, A.Yu., Martynyuk, A.A., Platonov, A.V.: Stability analysis of nonlinear switched difference systems via the comparison method. Pan Am. Math. J. 25(2), 71–88 (2015) 7. Aleksandrov, A.Yu., Martynyuk, A.A., Platonov, A.V.: Dwell time stability analysis for nonlinear switched difference systems. Nonlinear Dyn. Syst. Theory 16(3), 221–234 (2016) 8. Aubin, J.-P.: Mutational equations in metric spaces. Set-Valued Anal. 1, 3–46 (1993) 9. Azbelev, N.V., Simonov, P.M.: Stability of Differential Equations with Aftereffect. Taylor and Francis, London (2002) 10. Bainov, D.D., Simeonov, P.S.: Systems with Impulsive Effect: Stability, Theory and Applications. Ellis Horwood, Chichester (1989) 11. Belen’ky, I.M.: Some Questions of the Theory of Trajectories. Abstract of the dis. Doct. Fiz.mat. Sciences, Leningrad (1964) 12. Birkhoff, G.D., Lewis, D.C.Jr.: Stability in causal systems. Philos. Sci. 2(3), 304–333 (1935) 13. Bogoliubov, N.N., Mitropolskii, Yu.A.: Asymptotic Methods in Theory of Nonlinear Oscillations. Izd-vo Akademii nauk Ukr. SSR, Moscow (1963) 14. Boltzmann, L.: Zur Theorie der elastischen Nachwirkungen. In: Wissenschaftliche Abhandlung von Ludwig Boltzmann, I. Band (1865–1874), pp. 616–644. Verlag Johann Ambrosius Barth, Leipzig (1909) 15. Bridgland, T.F.: Trajectory integrals of set valued functions. Pac. J. Math. 33(1), 43–68 (1970) 16. Burton, T.A.: Stability and Periodic Solutions for Ordinary and Functional Differential Equations. Academic, New York (1985) 17. Conrad, K.: Zorn’s lemma and some applications. Expository Papers (2016) 18. Corduneanu, C.: Application of differential inequalities to stability theory. Analele Stiintifice, Univ. Iasi VI, 47–58 (1960)

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3

191

192

Bibliography

19. Corduneanu, C.: Functional Equations with Causal Operators. Taylor and Francis, New York (2003) 20. Corduneanu, C., Li, Y., Mehran, M.: Functional Differential Equations: Advances and Applications. Wiley, New York (2016) 21. Darboux, G.: Lecons sur la theorie generate des surfaces, vol. 2. Gautier Villars et Fils, Paris (1889) 22. de Blasi, F.S., Iervolino, F.: Equazioni differentiali con soluzioni a valore compatto convesso. Bollettino della Unione Matematica Italiana 2(4–5), 491–501 (1969) 23. Deimling, K.: Ordinary Differential Equations in Banach Spaces. Springer, Berlin (1977) 24. Deimling, K.: Multivalued Differential Equations. Walter de Gruyter, Berlin (1992) 25. Djordjevi`c, M.Z.: Zur Stabilitat Nichtlinearer Gekoppelter Systeme mit der Matrix-LjapunovMethode. Diss. ETH 7690, Zurich (1984) 26. El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day, San Francisco (1966) 27. Fillipov, A.F.: Classical solutions of differential equations with multivalued hand side. Vestn. Mosc. gos. un-ta (3), 16–26 (1967) 28. Gajic, Z.: Quantitative methods in nonlinear dynamics: novel approaches to Liapunov’s matrix functions by A. A. Martynyuk, New York: Marcel Dekker, 2002. Int. J. Robust Nonlinear Control 16, 465–466 (2006) 29. Grebenikov, E.A., Mitropolsky, Yu.A., Ryabov, Y.A.: Asymptotic Methods on Resonance Analytical Dynamics. Chapmen and Hall/CRC, Boca Raton (2004) 30. Haddad, W.M., Chellaboina, V.S., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006) 31. Hahn, W.: Stability of Motion. Springer, Berlin (1995) 32. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977) 33. Hertz, G.: Principles of Mechanics, Set out in a New Connection. Trans. from German. Izd-vo Akademii nauk Ukr. SSR, Moscow (1959) 34. Hukuhara, M.: Sur l’application semicontinue dont la valeur est un past convex. Functional Ekvac. 10, 48–66 (1967) 35. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U. S. A. 27, 222–224 (1941) 36. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhauser Verlag, Boston (1998) 37. Karakostas, G.: Uniform asymptotic stability of causal operator equations. Integr. Equ. 5, 59–71 (1983) 38. Kats, I.Ya., Martynyuk, A.A.: Stability and Stabilization of Nonlinear Systems with Random Structure. Stability and Control: Theory, Methods and Applications, vol. 18. Taylor and Francis, London (2002) 39. Kikuchi, N.: On some fundamental theorem of contingent equations in connections with the control problem. Publ. Res. Inst. Math. Sci. A 3, 177–201 (1967) 40. Krasovskii, N.N.: Stability on Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963) 41. Krylov, N.M., Bogoliubov, N.N.: Introduction to Nonlinear Mechanics. Izd-vo Akademii nauk Ukr. SSR, Kiev (1937) 42. Ladde, G.S, Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985) 43. Lakshmikantham, V., Martynyuk, A.A.: Development of the direct Lyapunov’s method for systems with delay. Int. Appl. Mech. 29(2), 3–14 (1993) 44. Lakshmikantham, V., Rama Mohana Rao, M.: Theory of Integro-Differential Equations. Gordon and Breach Science Publishers, Amsterdam (1995) 45. Lakshmikantham, V., Vatsala, A.S.: Set differential equations and monotone flows. Nonlinear Dyn. Syst. Theory 3(2), 33–43 (2003) 46. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulse Differential Equations. World Scientific, Singapore (1989)

Bibliography

193

47. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990) 48. Lakshmikantham, V., Leela, S., Sivasundaram, S.: Lyapunov functions on product spaces and stability theory of delay differential equations. J. Math. Anal. Appl. 154(2), 391–402 (1991) 49. Lakshmikantham, V., Bhaskar, T.G., Vasundhara Devi, J.: Theory of Set Differential Equations in Metric Spaces. Cambridge Scientific Publishers, Cambridge (2006) 50. Lakshmikantham, V., Leela, S., Drici, Z., McRae, F.A.: Theory of Causal Differential Equations. Atlantic Press/World Scientific, Amsterdam (2009) 51. Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems, 2nd edn. Springer, Basel (2015) 52. LaSalle, J.P.: The Stability and Control of Discrete Process. Springer, New York (1986) 53. Lyapunov, A.M.: The General Problem of the Stability of Motion. GITTL, Moscow (1950) 54. Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003) 55. Liberzon, D., Morse, A.S.: Basic problems in stability and design of switched systems. IEEE Contr. Syst. Mag. 19(15), 59–70 (1999) 56. Louartassi, Y., El Mazoudi, El.H., El Alami, N.: A new generalization of lemma Gronwall– Bellman. Appl. Math. Sci. 6(13), 621–628 (2012) 57. Lukyanova, T.A., Martynyuk, A.A.: Robust stability: three approaches for discrete-time systems. Nonlinear Dyn. Syst. Theory 2(1), 45–55 (2002) 58. Lupulescu, V.: Functional differential equations with causal operators. Int. J. Nonlinear Sci. 11(4), 499–505 (2011) 59. Marchaud, A.: Sur les champs de demi-cones et equations differentieles du premier order. Bull. Soc. Math. France (62), 1–38 (1934) 60. Marchaud, A.: Sur les champs de demi-cones convexes. Bull. Sci. Math. LXII(2), 229–240 (1938) 61. Martynyuk, A.A.: Practical Stability of Motion. Naukova Dumka, Kiev (1983) 62. Martynyuk, A.A.: Qualitative analysis of nonlinear systems by the method of matrix Lyapunov functions. Rocky Mt. J. Math. 25(1), 397–415 (1995) 63. Martynyuk, A.A.: Stability Analysis: Nonlinear Mechanics Equations. Gordon and Breach Publishers, Philadelphia (1995) 64. Martynyuk, A.A.: Stability by Liapunov’s Matrix Function Method with Applications. Marcel Dekker, New York (1998) 65. Martynyuk, A.A.: Stability analysis of discrete-time systems. Int. Appl. Mech. 36(7), 3–35 (2000) 66. Martynyuk, A.A.: Qualitative Methods in Nonlinear Dynamics. Novel Approaches to Liapunov’s Matrix Function. Marcel Dekker, New York (2002) 67. Martynyuk, A.A.: Advances in Stability Theory at the End of the 20th Century. Stability and Control: Theory, Methods and Applications, vol. 13. Taylor and Francis, London (2003) 68. Martynyuk, A.A.: On the theory of Lyapunov’s direct method. Dokl. Math. 73(4), 376–379 (2006) 69. Martynyuk, A.A.: Stability of Motion. The Role of Multicomponent Liapunov’s Functions. Cambridge Scientific Publishers, Cambridge (2007) 70. Martynyuk, A.A.: Stability of a set of trajectories of nonlinear dynamics. Dokl. Math. 75(3), 385–389 (2007) 71. Martynyuk, A.A.: Comparison principle for a set equations with robust causal operator. Dokl. Acad. Nauk 427(6), 750–753 (2009) 72. Martynyuk, A.A.: Asymptotic stability criterion for nonlinear monotonic systems and its applications (Review). Int. Appl. Mech. 47(5), 475–534 (2011) 73. Martynyuk, A.A.: On stability of the set impulsive equations. Dokl. Acad. Nauk 436(5), 593– 596 (2011) 74. Martynyuk, A.A.: Invariance of solutions of set regularized equations. Dokl. NAS Ukr. 17, 3–7 (2012) 75. Martynyuk, A.A.: On the stabilization of a system with aftereffect by pulsed perturbations. Dokl. NAS Ukr. 9, 62–65 (2012)

194

Bibliography

76. Martynyuk, A.A.: On the stability of trajectories of the set of difference equations. Dokl. NAS Ukr. 5, 65–69 (2014) 77. Martynyuk, A.A.: Novel bounds for solutions of nonlinear differential equations. Appl. Math. 6, 182–194 (2015) 78. Martynyuk, A.A.: On a method for estimating solutions of quasilinear systems. Dokl. NAS Ukr. 2, 19–23 (2015) 79. Martynyuk, A.A.: Elements of the theory of stability of hybrid systems (Review). Int. Appl. Mech. 51(3), 243–302 (2015) 80. Martynyuk, A.A.: Analysis of a set of trajectories of generalized standard systems: averaging technique. Nonlinear Dyn. Syst. Theory 17(1), 29–41 (2017) 81. Martynyuk, A.A.: Deviation of the set of trajectories from the state of equilibrium. Dokl. NAS Ukr. 10, 10–16 (2017) 82. Martynyuk, A.A., Obolenskij, A.Yu.: Stability of solutions of autonomous Wazewskij systems. Differ. Equ. 16(8), 1392–1407 (1980) 83. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Uncertain Dynamical Systems: Stability and Motion Control. CRC Press Taylor and Francis Group, Boca Raton (2012) 84. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Analysis of set trajectories of nonlinear dynamics: the estimation of solutions and the comparison principle. Differ. Equ. 48(10), 1395–1403 (2012) 85. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Analysis of the set of trajectories nonlinear dynamics: stability and boundedness of motions. Differ. Equ. 49(1), 20–31 (2013) 86. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Analysis of the set of trajectories of nonlinear dynamics: impulsive systems with aftereffect. Differ. Equ. 65(8), 47–53 (2012). https://doi.org/10.1134/S03740641140400 87. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: On robust stability of impulsive systems with aftereffect. Dokl. NAS Ukr. 8, 47–53 (2012) 88. Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Existence, uniqueness and an estimate of the solutions of the set of equations of the perturbed motion. Ukr. Mat. J. 65(2), 273–295 (2013) 89. Martynyuk, A.A., Miladzhanov, V.G.: Stability Analysis of Nonlinear Systems Under Structural Perturbations. Cambridge Scientific Publishers, Cambridge (2014) 90. Martynyuk, A.A., Stamova, I.M.: Stability analysis of set trajectories for families of impulsive equations. Appl. Anal. (2017). https://doi.org/10.1080/00036811.2017.1403589 91. Martynyuk, A.A., Chernetskaya, L.N., Martynyuk-Chernienko, Yu.A.: Dynamic analysis of the trajectories on product of convex compacts. Dokl. NAS Ukr. 9, 44–50 (2016) 92. McRae, F.A., Vasundhara Devi, J.: Impulsive set differential equations with delay. Appl. Anal. 84(4), 329–341 (2005) 93. Michael, E.A.: Continuous selections, I. Ann. Math. 63(2), 361–381 (1956) 94. Milman, V.D., Myshkis, A.D.: On motion stability with shocks. Sibirsk. Mat. Zh. I(2), 233– 237 (1960) 95. Minorsky, N.: Self-excited oscillations in dynamical systems possessing retarded actions. J. Appl. Mech. 9(1), 65–71 (1942) 96. Mooij, J.M., Janzing, D., Scholkopf, B.: From ordinary differential equations to structural causal models: the deterministic case. In: Proceedings of the 29th Annual Conference on Uncertainty in Artificial Intelligence (UAI-13), pp. 440–448 (2013) 97. Müller, W.: Stability Analysis: Nonlinear Mechanics Equations by Martynyuk A.A. Stability and Control: Theory, Methods and Applications, vol. 2. Gordon and Breach Science Publishers, Philadelphia (1995). Zbl. Math. 0840.34003 98. N’Doye, I.: Generalisation du Lemme de Gronwall—Bellman pour la Stabilisation des Systemes Fractionnaires. Nancy-Universite, PhD These (2011) 99. Ovsyannikov, D.A., Egorov, N.V.: Mathematical Modeling of Formation Systems Electron and Ion Beams. Ed. St. Petersburg University, St. Petersburg (1998) 100. Pandit, S.G., Deo, S.G.: Differential Systems Involving Impulses. Lecture Notes in Mathematics, vol. 954. Springer, Berlin (1982)

Bibliography

195

101. Pinto, B.L., De Blasi, A.J., Ievorlino, F.: Uniqueness and existence theorems for differential equations with convex-valued solutions. Boll. Unione Mat. Italy 3, 47–54 (1970) 102. Plotnikov, A.V., Skripnik, N.V.: Differential Equations with “Clear” and Fuzzy Multi-Valued Right-Hand Side. Asymptotic Methods. Astroprint, Odessa (2009) 103. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) 104. Razumikhin, B.S.: On the stability of systems with a delay. J. Appl. Math. Mech. 22, 215–227 (1958) 105. Ricci, G., Levi-Civita, T.: Methodes de calcul absolu et leurs applications. Math. Ann. 54, 125–201 (1901) 106. Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26(1), 103–107 (2010) 107. Schoen, G.M.: Stability and stabilization of time-delay systems. Zurich, Diss. ETH No. 11166 (1995) 108. Shen, J., Luo, Z., Liu, X.: Impulsive stabilization of functional differential equations via Liapunov functionals. J. Math. Anal. Appl. 240, 1–5 (1999) 109. Shurz, H.: Stability by Lyapunov’s Matrix Function Method with Applications by Martynyuk A.A. Pure and Applied Mathematics, vol. 214. Marcel Dekker, New York (1998). Zentralblatt MATH 907, 166–167 110. Stamova, I.M.: Stability Analysis of Impulsive Functional Differential Equations. Walter De Gruyter Inc., New York (2009) 111. Stamova, I.M., Stamov, G.: Applied Impulsive Mathematical Models. CMS Books in Mathematics. Springer, Cham (2016) 112. Stefanini, L.: A generalization of Hukuhara difference. In: Dubois, D., et al. (eds.) Soft Methods for Hand Var. and Imprecision. ASC, vol. 48, pp. 203–210. Springer, Berlin (2008) 113. Synge, J.L.: Classical Dynamics. Springer, Berlin (1960) 114. Szarski, J.: Differential Inequalities. PWN, Warsaw (1965) 115. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publ., New York (1960) 116. Vassilyev, S.N., Kosov, A.A., Malikov, A.I.: Stability analysis of nonlinear switched systems via reduction method. In: Proceedings of the 18th IFAC World Congress, Milano, vol. 18, Part 1, pp. 5718–5723 (2011) 117. Vasundhara Devi, J., Vatsala, A.S.: Monotone iterative technique for impulsive and set differential equations. Nonlinear Stud. 11(4), 639–658 (2004) 118. Vasundhara Devi, J., Vatsala, A.S.: A study of set differential equations with delay, dynamics of continuous, discrete and impulsive systems. Math. Anal. A 11, 287–300 (2004) 119. Walter, W.: Differential and Integral Inequalities. Springer, Berlin (1970) 120. Wang, Q.: Stability and Boundedness of Impulsive Systems with Time Delay. University of Waterloo, Waterloo (2007) 121. Wazewski, T.: Systemes de comande et equations au contingent. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9, 865–867 (1961) 122. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Mathematics Society Japan, Tokyo (1966) 123. Zaremba, S.K.: Sur les equations au paratingent. Bull. Sci. Math. bfLX(2), 139–160 (1936) 124. Zubov, V.I.: Mathematical Methods of Investigations of Systems Automatical Control. Sudpromgiz, Leningrad (1959)

Index

(A, λ)-estimate of approximate integration, 177 ε-approximate solution to the set of equations, 28 ε-neighborhood of the set, 3

Cartesian product, 33 Comparison equation, 65 system, 64 Complete separable metric space, 4 Cone in Kc (Rn ), 17, 51 Coupled pair of lower and upper solutions of type I, 20 of type II, 20 of type III, 21

equations regularized, 7, 153 equations stable on a finite interval, 130 equations H.U.R. stable, 79 impulsive differential equations, 104 perturbed motion equations, 6 Family of impulsive equations, 137 asymptotically stable, 137 stable, 137 uniformly asymptotically stable, 137 uniformly stable, 137 Function differentiable in the Hukuhara sense, 5 scalar, 54 support, 4 vector, 55, 74

Generalized standard system, 174 Gronwall-Bellman lemma, 38 Diagonal matrix-valued function, 138 Diameter of the set, 5 Differential inclusion, 44 Distance between the nonempty closed subsets, 3 from the point x to the set, 3

Euler solution for the family of equations, 30

Family of affine systems, 183 averaged equations, 178 causal equations, 152, 167 differential equations with aftereffect, 136

Hausdorff metric, 3 Hausdorff separation of sets, 3 Hukuhara difference, 89 Hyers-Ulam-Rassias stability, 167

Impulsive perturbations, 136 Integral of the multivalued function, 5 Interconnected subsystems, 91

Limiting mappings, 6 Lipschitz multivalued mapping, 4

© Springer Nature Switzerland AG 2019 A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, https://doi.org/10.1007/978-3-030-07644-3

197

198 Lyapunov function, 88 first difference, 88 heterogeneous, 106 heterogeneous vector, 106 strengthened, 73

Matrix-valued function, 53, 74 Maximal solution, 17 of the family of equations, 51 Measurable selector, 4 Minimal solution, 17 Multivalued mapping, 4, 8, 30

Natural lower and upper solutions, 20 Nonlinear integral inequalities pseudo-linear representation, 36

Pair (, Fβ ) strictly invariant, 35 weakly invariant, 33 Pair of upper and lower solutions, 20, 115 of type I, 115 of type II, 116 Partial ordering in the metric space, 17

Index Set of discrete-time equations, 87 Set of solutions X(t), 70 equi-bounded, 70 equi-ultimate bounded, 71 quasi-equiultimate bounded, 71 quasi-uniformly ultimate bounded, 71 uniformly bounded, 70 uniformly ultimate bounded, 71 Set of solutions bounded, 186 Set of stationary solutions, 108 asymptotically stable, 108 attractive, 108 robustly stable, 163 stable, 108 Set of system states, 6 Solution to problem (1.2) ((1.3)), 8 Stability on a finite interval, 128 Stationary solution, 89 asymptotically stable, 89 attractive, 89 stable, 89 Stationary solution 0 , 57 asymptotically stable, 57 attractive, 57 stable, 57 strictly stable, 57 Total derivative by virtue of equation, 54

Robust causal operator, 152, 161

Selector of the multivalued mapping, 4 Set of damping trajectories, 132

Uncertain differential equations with aftereffect, 124 Uncertainty parameter of the mapping, 6