Stability Theory for Dynamic Equations on Time Scales [1st ed.] 331942212X, 978-3-319-42212-1, 978-3-319-42213-8

Table of contents :
Front Matter....Pages i-xi
Elements of Time Scales Analysis....Pages 1-23
Method of Dynamic Integral Inequalities....Pages 25-84
Lyapunov Theory for Dynamic Equations....Pages 85-144
Comparison Method....Pages 145-183
Applications....Pages 185-214
Back Matter....Pages 215-223

Citation preview

Systems & Control: Foundations & Applications

Anatoly A. Martynyuk

Stability Theory for Dynamic Equations on Time Scales

Systems & Control: Foundations & Applications Series Editor Tamer Ba¸sar, University of Illinois at Urbana-Champaign, Urbana, IL, USA Editorial Board Karl Johan Åström, Lund University of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing, China Bill Helton, University of California, San Diego, CA, USA Alberto Isidori, Sapienza University of Rome, Rome, Italy Miroslav Krstic, University of California, San Diego, CA, USA H. Vincent Poor, Princeton University, Princeton, NJ, USA Mete Soner, ETH Zürich, Zürich, Switzerland; Swiss Finance Institute, Zürich, Switzerland Roberto Tempo, CNR-IEIIT, Politecnico di Torino, Italy

More information about this series at http://www.springer.com/series/4895

Anatoly A. Martynyuk

Stability Theory for Dynamic Equations on Time Scales

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

ISSN 2324-9749 ISSN 2324-9757 (electronic) Systems & Control: Foundations & Applications ISBN 978-3-319-42212-1 ISBN 978-3-319-42213-8 (eBook) DOI 10.1007/978-3-319-42213-8 Library of Congress Control Number: 2016948874 Mathematics Subject Classification (2010): 34D10, 34D20, 39A11, 45G10, 70K20, 93D05, 93D20, 93D30 © Springer International Publishing Switzerland 2016 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. Printed on acid-free paper This book is published under the trade name Birkhäuser The registered company is Springer International Publishing AG Switzerland (www.birkhauser-science.com)

Contents

1

Elements of Time Scales Analysis . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Description of a Time Scale . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Delta Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Matrix Exponential Functions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Variation of Constants Formula . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Nabla Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Diamond-Alpha Derivative . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 Comments and Bibliography . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 4 6 8 11 13 15 19 22

2 Method of Dynamic Integral Inequalities . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Dynamic Integral Inequalities . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 Gronwall inequalities .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Some nonlinear inequalities .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Stability of Linear Dynamic Equations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Nonautonomous systems . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Time-invariant system . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Elements of Floquet theory.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Stability of Nonlinear Dynamic Equations . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Estimations of solutions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Theorems on stability .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Stability of quasilinear equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Exponential stability . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 Scalar quasilinear equation .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Preservation of Stability Under Perturbations . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Linear systems under parametric perturbations . . . . . . . . . . . . . . . 2.4.2 Quasilinear dynamic equations.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Comments and Bibliography . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 25 25 31 37 37 44 48 52 52 57 61 73 75 78 78 80 84 v

vi

Contents

3 Lyapunov Theory for Dynamic Equations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Auxiliary Functions for Dynamic Equations . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Scalar functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Vector functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Matrix-valued functions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Theorems of Stability and Instability . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 General systems of dynamic equations . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Stability of linear systems . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Existence and Construction of Lyapunov Functions . . . . . . . . . . . . . . . . . . 3.4.1 Converse theorem .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Solution of dynamic Lyapunov equation . .. . . . . . . . . . . . . . . . . . . . 3.4.3 Lyapunov function for linear periodic system . . . . . . . . . . . . . . . . . 3.5 Stability Under Structural Perturbations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Description of structural perturbations for dynamic equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Periodic linear system . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Polydynamics on Time Scales . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.2 Analysis of polydynamics .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.3 Conditions for stability and instability . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Comments and Bibliography . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85 85 86 88 88 89 90 92 92 105 114 114 116 118 125

4 Comparison Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Theorems of the Comparison Method .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Stability Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Stability of Conditionally Invariant Sets . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Stability with Respect to Two Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Stability of a Dynamic Graph . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Description of a dynamic graph .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.2 Problem of stability . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.3 Evolution of a dynamic graph .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.4 Matrix-valued functions and their applications . . . . . . . . . . . . . . . 4.5.5 A variant of comparison principle . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Comments and Bibliography . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

145 145 146 150 159 168 174 174 176 178 179 180 182

5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Stability of Neuron Network . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Stability of a Complex-Valued Neuron Network . .. . . . . . . . . . . . . . . . . . . . 5.3 Volterra Model on Time Scale . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 Generalization of Volterra model . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

185 185 185 193 199 200 202

125 128 136 137 138 139 143

Contents

5.4 Stability of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Statement of the problem .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 Stability under structural perturbations . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Comments and Bibliography . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

vii

206 206 206 214

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 215 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221

Preface

During the last decades, many areas of modern applied mathematics, such as the automatic control theory, the study of aircraft and spacecraft dynamics, and others, have been faced with the problems coming to the analysis of behavior of solutions to time continuous-discrete linear and/or nonlinear perturbed motion equations. As far back as 1937, in his book Men of Mathematics, E.T. Bell wrote: A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. This is exactly what the mathematical analysis on time scale does. Equations which adequately describe processes in continuous-discrete systems are called the “dynamic equations” (not to be confused with classical dynamic systems in the sense of Nemytskii-Stepanov). These equations, as a subject of intensive research, appeared 20 years ago due to the development of mathematical analysis based on the generalized concept of a derivative and an integral on a time scale representing some closed subset of real numbers. Dynamic equations have made it possible to present time continuousdiscrete processes as a single whole, keeping at the same time the parity of the contribution of both the continuous and the discrete components of the system into the total dynamics of the process and/or equation under consideration. One of the most important problems for dynamic equations is the problem of stability of their solutions and/or their equilibrium state. It is clear that the techniques of stability analysis of solutions to dynamic equations must take account of the specific character of these equations brought about by new definitions of the derivative of the state vector of a system and the corresponding integrals. This monograph is the first of the kind in the world literature and is focused on three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequalities and fundamental matrix of solutions to linear approximation of dynamic equations. The second approach implies a generalization of the direct Lyapunov method for equations on time scales. To this end, scalar, vector, and matrix-valued auxiliary functions are used.

ix

x

Preface

The third approach involves the application of auxiliary functions (scalar, vector, or matrix-valued ones) in combination with differential dynamic inequalities. It appears to be an adaptation of the comparison method which is well developed for time continuous and time-discrete systems. The book consists of the preface, five chapters, and a list of references. Note that the latter contains only those monographs and papers which were referred to in this study; therefore, the list is not a comprehensive bibliography. Chapter 1 provides the main concepts, definitions, and theorems (without proofs) of the mathematical analysis on time scales. This knowledge is necessary for understanding of the results of Chapters 2–5 of this book. This chapter has been kindly read over and amended by Professor M. Bohner. In Chapter 2, the reader will find some results for dynamic integral inequalities, starting from inequalities of Gronwall-Bellman type. Application of these inequalities in the problems of stability of solutions to linear, quasilinear, and nonlinear dynamic equations makes the basic content of this chapter. Whereas the method of integral inequalities for continuous systems is adequately developed (see Martynyuk and Gutowsky [1], Martynyuk, Lakshmikantham, and Leela [1]), only the first results have been obtained for dynamic equations via this method. Chapter 3 presents a generalization of the direct Lyapunov method for dynamic equations. Here the main theorems of the direct Lyapunov method are given, including theorems on exponential stability and polystability on time scales. Moreover, for the construction of appropriate Lyapunov functions, matrix-valued auxiliary functions are used. A special section of the chapter is dedicated to the converse of the theorem on exponential stability and solution of Lyapunov equations for linear periodic systems on time scales. Chapter 4 gives an account of results of the development of the comparison method on a time scale. The theory and applications of this method for continuous, discrete, and some other classes of systems are presented in several monographs (see Matrosov [1]; Lakshmikantham, Leela, and Martynyuk [1]; Šiljak [1]). For dynamic equations, the comparison method is presented in the context with an auxiliary matrix-valued function and dynamic comparison equations. On the basis of general theorems of the comparison principle, some problems of stability of invariant sets, the stability with respect to two measures, and the dynamic graph stability are discussed. In Chapter 5, some general results of the previous chapters are applied to dynamic equations having real physical significance. In particular, the problems of stability of neural networks and stability of oscillations under structural perturbations on time scales are addressed. Research in the area of stability theory of dynamic equations on time scales is already established. This research has potential applications in such areas as theoretical and applied mechanics, neurodynamics, mathematical biology, finance, and others.

Acknowledgments

Writing this book would not be possible without the aid of many people to whom I would like to express my sincere gratitude. First of all, I would like to thank Professors M. Bohner, L. Erbe, V. Lakshmikantham, A. Peterson, Q. Sheng, and A. Vatsala for their active support on my work during the preparation of the Russian variant of this book. I am grateful to the collaborators of the Department of Processes Stability of the S.P. Timoshenko Institute of Mechanics NAS of Ukraine S.V. Babenko, L.N. Chernetskaya, T.A. Luk’yanova, S.N. Rasshyvalova, and V.I. Slyn’ko for their careful reading of some chapters of the book and their remarks and discussion of the text. Finally, I would like to thank Birkhäuser for their interest to this project and helping me in the preparation of the final version of the book. Kiev, Ukraine March 2016

A.A. Martynyuk

xi

Chapter 1

ELEMENTS OF TIME SCALES ANALYSIS

1.0 Introduction In this chapter we give without proof some well-known results from mathematical analysis on time scales. For further reading we refer to the book by Bohner and Peterson [1]. The chapter is organized as follows. Section 1.2 deals with the description of a time scale and some examples of the most commonly used time scales. In Section 1.3, a derivative of the function defined on time scales is introduced, and the relation of this derivative with ordinary derivative and ordinary forward difference operator is shown. In Section 1.4 some results related to the integration on time scales are given. Section 1.5 presents a definition of the exponential function on time scales together with its main properties. In Section 1.6, a matrix exponential function on time scales is introduced and some of its properties are described. In Section 1.7, a formula of variation of constants on time scales for scalar and vector dynamic equations is discussed. Section 1.8 provides a description of some results of the mathematical analysis for the nabla derivative and nabla integrals on time scales. In the final Section 1.9, the diamond-alpha derivative and its properties are considered.

1.1 Description of a Time Scale Definition 1.1.1 (Time scale) A time scale T is an arbitrary nonempty closed subset of the set of real numbers R. © Springer International Publishing Switzerland 2016 A.A. Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42213-8_1

1

2

Elements of Time Scales Analysis

Example 1.1.1 (Some time scales) The most commonly used time scales are the real numbers T D R for continuous calculus, the integer numbers T D Z for discrete calculus, and the quantum numbers T D qN0 D fqn W n 2 N0 g;

where q > 1;

for quantum calculus. Other examples of time scales contain the natural numbers N; the nonnegative natural numbers N0 ; the set hZ D fhkW k 2 Zg;

where h > 0I

the Cantor set; the set Pa;b D

1 [

Œk.a C b/; k.a C b/ C a;

where a; b > 0I

kD0

the set Np D fnp W n 2 Ng;

where p > 0I

the set of harmonic numbers (

n X 1 kD1

k

) W n2N I

and the set qZ D fqk W k 2 Zg [ f0g;

where q > 0:

Definition 1.1.2 (Jump operators) For any t 2 T, the forward jump operator is defined by .t/ D inffs 2 TW s > tg; while the backward jump operator is given by .t/ D supfs 2 TW s < tg: In this definition, it is assumed that inf ¿ D sup T (i.e., .t/ D t if T contains the largest element t) and sup ¿ D inf T (i.e., .t/ D t if T contains the smallest element t). Using the operators W T ! T and W T ! T, the points t on the time scale T are classified as follows: If .t/ D t, then the point t 2 T is said to be right-dense, while if .t/ > t, then the point t 2 T is said to be right-scattered. If

Elements of Time Scales Analysis

3

Table 1.1 Some time scales and their characteristics

T  .t/ .t/ .t/

R t t 0

Z tC1 t1 1

cZ tCc tc c

qN qt t=q .q  1/t

2N 2t t=2 t

N20 p 2 tC1 2 p t1 p 2 tC1

.t/ D t, then the point t 2 T is called left-dense, while if .t/ < t, then the point t 2 T is called left-scattered. Definition 1.1.3 (Kappa operator) Along with the set T, the set T is defined as follows: if T contains the left-scattered maximum M; then T D TnfMg. Otherwise, we put T D T. Then, ( 

T D

T n ..sup T/; sup T;

if sup T < 1;

T;

if sup T D 1:

(1.1.1)

Definition 1.1.4 (Graininess) The distance from an arbitrary element t 2 T to the closest element on the right is called the graininess of the time scale T and is determined by the formula .t/ D .t/  t:

(1.1.2)

Some examples of time scales and their characteristics (forward jump, backward jump, and graininess) are shown in Table 1.1. Example 1.1.2 For the scale Pa;b obtain

.t/ D

.t/ D

.t/ D

8 ˆ ˆ 0; we define the strip 

  Zh D z 2 CW  < Im.z/  h h



and the set  1 Ch D z 2 CW z ¤  : h 

Moreover, let Z0 D C0 D C be a set of complex numbers. Then for h  0, we define the cylinder transformation h W Ch ! Zh by the formula 8 < 1 Log.1 C zh/; if h > 0; h D h :z; if h D 0: where Log is the principal logarithm. Now we introduce the exponential function on time scales. Definition 1.4.3 (Exponential function) For p 2 R and s 2 T, the exponential function ep .; s/ is defined by !

Zt ep .t; s/ D exp

. / . p. // ;

t 2 T;

s

where is a cylindric transformation introduced in Definition 1.4.2. The following properties of the exponential function are well known (see Bohner and Peterson [1, Section 2.2]). Theorem 1.4.2 Let p; q 2 R and t; r; s 2 T. Then (i) e0 .t; s/  1 and ep .t; t/  1I 1 D ep .s; t/I (ii) ep .t; s/ D ep .s; t/

10

Elements of Time Scales Analysis

(iii) ep .t; s/ep .s; r/ D ep .t; r/I (iv) ep .t; s/eq .t; s/ D ep˚q .t; s/I ep .t; s/ D epq .t; s/I (v) eq .t; s/ (vi) ep ..t/; s/ D .1 C .t/p.t//ep .t; s/I ep .t; s/ I (vii) ep .t; .s// D 1 C .s/p.s/ (viii) .ep .; s// D pep .; s/I (ix) .ep .t; // D .p/ep .t; /I (x) ep .t; s/ > 0 if p 2 RC We now give some examples of exponential function on some time scales. Example 1.4.1 Let T D R. If p 2 R, then ep .t; s/ D e

Rt s

p. /d

:

If p.t/  ˛, then ep .t; s/ D e˛.ts/ : Example 1.4.2 Let T D Z. If p 2 R, then t1 Y

ep .t; s/ D

.1 C p. //:

Ds

If p.t/  ˛, then ep .t; s/ D .1 C ˛/ts : Example 1.4.3 Let T D hZ with h > 0. If p 2 R, then t

ep .t; s/ D

h 1 Y

.1 C hp. //:

D hs

If p.t/  ˛, then ep .t; s/ D .1 C h˛/

ts h

:

Example 1.4.4 Define the time scale of the harmonic numbers by T D fHn W n 2 N0 g;

where H0 D 0

and Hn D

n X 1 kD1

k

;

n 2 N:

Elements of Time Scales Analysis

11

If ˛  0 is some constant, then nC˛ e˛ .Hn ; 0/ D n

! D

.n C ˛/  : : :  .1 C ˛/ : nŠ

Example 1.4.5 Let T D qN0 with q > 1. Let p 2 R. The initial value problem y D p.t/y;

y.1/ D 1

is equivalent to the recursive problem y D .1 C .q  1/tp.t//y;

y.1/ D 1;

and its solution is the function ep .t; 1/ D

Y

.1 C .q  1/sp.s//:

s2T\.0;t/

Some other examples of exponential functions on time scales can be found in the monograph by Bohner and Peterson [1, Section 2.3].

1.5 Matrix Exponential Functions Definition 1.5.1 Let A be an m  n-matrix-valued function on a time scale T. The function A is called rd-continuous on T if all of its entries, aij W T ! R, 1  i  m, 1  j  n, are rd-continuous on T. The class of all such functions will be denoted by Crd D Crd .T/ D Crd .T; Rnn /: If all entries aij W T ! R are differentiable, then A is called differentiable and the m  n-matrix-valued function consisting of the entries a ij is denoted by A . Moreover,  we denote by A the m  n-matrix-valued function consisting of the entries aij . The following results are well known (see Bohner and Peterson [1, Section 5.1]). Theorem 1.5.1 Let A and B be differentiable n  n-matrix-valued functions on T. Then the following relations hold: (i) (ii) (iii) (iv)

A D A C A ; .A C B/ D A C B ; .cA/ D cA , where c 2 R; .AB/ D A B C AB D A B C A B;

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(v) .A1 / D .A /1 A A1 D A1 A .A /1 provided that AA is invertible; (vi) .AB1 / D .A  AB1 B /.B /1 D .A  .AB1 / B /B1 provided that BB is invertible. Definition 1.5.2 (Regressivity) An rd-continuous n  n-matrix-valued function is called regressive if I C .t/A.t/

is invertible for all t 2 T ;

where I denotes the n n-identity matrix. The class of all such matrices A is denoted by R D R.T/ D R.T; Rnn /: For two functions A; B 2 R, we define the circle plus by .A ˚ B/.t/ D A C B C .t/A.t/B.t/ Theorem 1.5.2 (Regressive group) The pair .R.T; Rnn /; ˚/ is a group. The inverse element of A 2 R with respect to ˚ is .A/.t/ D ŒI C .t/A.t/1 A.t/ D A.t/ŒI C .t/A.t/1 ;

t 2 T :

Thus, if A; B 2 R, then A ˚ B 2 R. In order to define the matrix exponential function on time scales, we make use of the following fundamental result. Theorem 1.5.3 If A 2 R and t0 2 T, then the initial value problem Y  D A.t/Y;

Y.t0 / D I;

(1.5.1)

has a unique solution. Definition 1.5.3 (Matrix exponential function) The unique solution of the initial value problem (1.5.1) on T is called the matrix exponential function, and it is denoted by eA .; t0 /. Example 1.5.1 If T D R and A is a constant n  n-matrix such that I C hA is invertible, then eA .t; s/ D .I C hA/

ts h

:

The following properties of the matrix exponential function are well known (see Bohner and Peterson [1, Section 5.1]).

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13

Theorem 1.5.4 Let A; B 2 R and t; r; s 2 T. Then: (i) (ii) (iii) (iv) (v) (vi)

e0 .t; s/ D I and eA .t; t/ D I;  eA .t; s/ D e1 A .s; t/ D eA .s; t/; eA .t; s/eA .s; r/ D eA .t; r/; eA .t; s/eB .t; s/ D eA˚B .t; s/ provided that eA .t; s/ and B commute; eA ..t/; s/ D .I C .t/A.t//eA .t; s/; .eA .; s// D AeA .; s/.

Here . / denotes the conjugate transpose of A.

1.6 Variation of Constants Formula Definition 1.6.1 (Regressivity) A first-order homogeneous equation x D p.t/x

(1.6.1)

is called regressive provided p 2 R.T; R/. A first-order nonhomogeneous equation x D p.t/x C f .t/;

(1.6.2)

is called regressive provided p 2 R.T; R/ and f 2 Crd .T; R/. The following two results are well known (see Bohner and Peterson [1, Theorem 2.62 and Theorem 2.77]). Theorem 1.6.1 Suppose (1.6.1) is regressive. Let t0 2 T and x0 2 R. Then the unique solution of the initial value problem x D p.t/x;

x.t0 / D x0 ;

(1.6.3)

is given by x.t/ D ep .t; t0 /x0

for all t 2 T:

Theorem 1.6.2 (Variation of constants) Suppose (1.6.2) is regressive. Let t0 2 T and x0 2 R. Then the unique solution of the initial value problem x D p.t/x C f .t/;

x.t0 / D x0

(1.6.4)

ep .t; . //f . / :

(1.6.5)

is given by Zt x.t/ D ep .t; t0 /x0 C t0

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Elements of Time Scales Analysis

Proof Indeed, the equation (1.6.2) can be rewritten as  x .t/ D p.t/ x..t//  .t/x C f .t/: Hence .1 C .t//p.t/ x .t/  p.t/x..t// D f .t/; and since p 2 R, we obtain x .t/ C .p/.t/x..t// D

f .t/ : 1 C .t/p.t/

(1.6.6)

We multiply the equation (1.6.6) by the “integrating factor” ep .t; t0 / to obtain .xep .; t0 // .t/ D ep .t; t0 /

f .t/ : 1 C .t/p.t/

(1.6.7)

Equation (1.6.7) can now be integrated Zt x.t/ep .t; t0 /  x.t0 /ep .t0 ; t0 / D

ep . ; t0 / t0

f . /  : 1 C . /p. /

(1.6.8)

Multiplying (1.6.8) by ep .t; t0 / yields Zt x.t/ D x0 ep .t; t0 / C

ep .t; / t0

f . /  : 1 C . /p. /

(1.6.9)

The fact that ep .t; / D ep .t; . // 1 C . /p. / and the relation (1.6.9) imply the formula (1.6.5). Definition 1.6.2 (Regressivity) and f W T ! Rn , then the system

If A.t/ is an n  n-matrix-valued function on T x D A.t/x C f .t/;

is called regressive provided A 2 R and f 2 Crd . The following result is due to Bohner and Peterson [1, Theorem 5.24].

(1.6.10)

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15

Theorem 1.6.3 (Variation of constants) Suppose (1.6.10) is regressive. Let t0 2 T and x0 2 Rn . Then the unique solution of the initial value problem x D A.t/x C f .t/;

x.t0 / D x0

(1.6.11)

eA .t; . //f . /  :

(1.6.12)

is given by Zt x.t/ D eA .t; t0 /x0 C t0

The general initial value problem on time scales is formulated as follows. Definition 1.6.3 (Initial value problem) Suppose f W T  Rn ! Rn . Let t0 2 T and x0 2 Rn be given. The system of dynamic equations x .t/ D f .t; x.t//

(1.6.13)

x.t0 / D x0

(1.6.14)

with the initial conditions

is called an initial value problem. The function x.t/W T ! Rn with (1.6.14) satisfying the system of dynamic equation (1.6.13) at all t 2 T is called the solution of the initial value problem (1.6.13)–(1.6.14).

1.7 Nabla Derivative Definition 1.7.1 (Kappa operator) For the time scale T, we define the set T D T n fmg if T contains the right-scattered minimum m. Otherwise, we put T D T. Now we consider a function f W T ! R and define its r-derivative at the point t 2 T . Definition 1.7.2 (Nabla derivative) The function f W T ! R is said to be nabla differentiable (or r-differentiable) at the point t 2 T if there exists ˛ 2 R such that for any " > 0, there exists a neighborhood U of t so that the inequality jŒ f ..t//  f .s/  ˛Œ.t/  sj  "j.t/  sj holds for all s 2 U. In this case, we denote f r .t/ D ˛. If the function f W T ! R is nabla differentiable at any t 2 T , then f is called r-differentiable. Example 1.7.1 If T D R, then f r D f 0 , which is an ordinary derivative of the function f W R ! R. If T D Z, then f r D rf , which is an ordinary backward

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difference of the function f W Z ! R defined by rf .t/ D f .t/  f .t  1/: Definition 1.7.3 (Backward graininess) scale T is defined by the formula

The backward graininess of the time

.t/ D t  .t/: For the r-derivative of the function f , the following three results are known (see Bohner and Peterson [1], Theorem 8.39, Theorem 8.41, and Theorem 8.50). Theorem 1.7.1 Assume that f W T ! R and let t 2 T . Then the following statements hold: (i) if f is nabla differentiable at the point t, then f is continuous at t; (ii) if f is continuous at t and t is left-scattered, then f is r-differentiable at t with f r .t/ D

f .t/  f ..t// I .t/

(iii) if t is left-dense, then f is r-differentiable at t if and only if the limit lim s!t

f .t/  f .s/ ts

exists as a finite number. In this case f r .t/ D lim s!t

f .t/  f .s/ I ts

(iv) if f is r-differentiable at the point t, then f ..t// D f .t/  .t/f r .t/: Theorem 1.7.2 Assume that f ; gW T ! R are r-differentiable at the point t 2 T . Then the following statements hold: (i) the sum f C g is r-differentiable at t and . f C g/r .t/ D f r .t/ C gr .t/I (ii) for any constant c, the scalar multiple cf is r-differentiable at t and .cf /r .t/ D cf r .t/I

Elements of Time Scales Analysis

17

(iii) the product fg is r-differentiable at t and . fg/r .t/ D f r .t/g.t/ C f ..t//gr .t/ D f .t/gr .t/ C f r .t/g..t//I (iv) if f .t/f ..t// ¤ 0, then 1=f is r-differentiable at t and  r f r .t/ 1 I .t/ D  f f .t/f ..t// (v) if g.t/g..t// ¤ 0, then the quotient f =g is r-differentiable at t and  r f f r .t/g.t/  f .t/gr .t/ : .t/ D g g.t/g..t// Theorem 1.7.3 Let t0 2 T. Assume f , f  , f r are continuous. If Zt g.t/ D

Zt f .t; s/ s

t0

and h.t/ D

f .t; s/ rs; t0

then 

Zt

g .t/ D

f  .; s/.t/ s C f ..t/; t/;

t0 r

Zt

g .t/ D

f r .; s/.t/ s C f ..t/; .t//;

t0 

Zt

h .t/ D

f  .; s/.t/ s C f ..t/; .t//;

t0 r

Zt

h .t/ D

f r .; s/.t/ s C f ..t/; t/:

t0

Definition 1.7.4 (Ld-continuity) The function f W T ! R is called ld-continuous if it is continuous at all left-dense points of T and if right-hand limits exist as a finite number at right-dense points of T. The set of all ld-continuous functions is denoted by Cld .

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Elements of Time Scales Analysis

Definition 1.7.5 (Antiderivative) satisfying

Let f W T ! R. A function FW T ! R

F r .t/ D f .t/

for all t 2 T

is called a r-antiderivative of f . Theorem 1.7.4 (Existence of antiderivative) sesses a r-antiderivative.

Any ld-continuous function pos-

Definition 1.7.6 (Cauchy nabla integral) Let FW T ! R be a r-antiderivative of f W T ! R. The Cauchy r-integral of f is then defined by Zb f .t/ rt D F.b/  F.a/;

where a; b 2 T:

a

Definition 1.7.7 (Regressivity) The function pW T ! R is called -regressive if 1  .t/p.t/ ¤ 0

for all t 2 T :

The class of all -regressive functions is defined by R D fpW T ! RW p is ld-continuous and -regressiveg: The general time scales initial value problem for nabla equations is formulated as follows. Definition 1.7.8 (Initial value problem) Suppose f W T  Rn ! R. Let t0 2 T and x0 2 Rn be given. The system of dynamic equations xr .t/ D f .t; x.t//

(1.7.1)

x.t0 / D x0

(1.7.2)

with the initial conditions

is called an initial value problem. The function xW T ! Rn with (1.7.2) satisfying the system of dynamic equation (1.7.1) at all t 2 T is called the solution of the initial problem (1.7.1)–(1.7.2). The following theorem is given in Bohner and Peterson [2, Theorem 3.13]. Theorem 1.7.5 Suppose p 2 Cld . Let t0 2 T. Then eO p .t; t0 / is the unique solution of yr .t/ D p.t/y;

y.t0 / D 1;

(1.7.3)

Elements of Time Scales Analysis

19

where eO p .t; t0 / is given by the formula Zt eO p .t; s/ D exp

! O. / . p. //r

s

and Oh W Ch ! Zh is the -cylinder transformation defined by Oh .z/ D  1h Log .1  zh/ for h > 0 and O0 .z/ D z.

1.8 Diamond-Alpha Derivative Along with the -derivative and the r-derivative of a function f W T ! R, we now consider its so-called diamond-alpha derivative. Throughout this section we let ˛ 2 Œ0; 1. Definition 1.8.1 (Diamond-alpha derivative) Let 0  ˛  1. Assume f W T ! R is -differentiable and r-differentiable at the point t 2 T D T \T . Then we say that f is diamond-alpha-differentiable (or ˙˛ -differentiable) at t with ˙˛ -derivative f ˙˛ .t/ D ˛f  .t/ C .1  ˛/f r .t/:

(1.8.1)

The following result is given in Sheng et al. [1, Theorem 2.3]. Theorem 1.8.1 Assume that f ; gW T ! R are ˙˛ -differentiable at t 2 T . Then the following statements hold: (i) the sum f C g is ˙˛ -differentiable at t and . f C g/˙˛ .t/ D f ˙˛ .t/ C g˙˛ .t/I (ii) for any constant c, the scalar multiple cf is ˙˛ -differentiable at t and .cf /˙˛ .t/ D cf ˙˛ .t/I (iii) the product fg is ˙˛ -differentiable at t and . fg/˙˛ .t/ D f ˙˛ .t/g.t/ C ˛f ..t//g .t/ C .1  ˛/f ..t//gr .t/I (iv) if f .t/f ..t//f ..t// ¤ 0, then 1=f is ˙˛ -differentiable at t and  ˙˛  1 1 . f ..t// C f ..t///f ˙˛ .t/ .t/ D  f f .t/f ..t//f ..t//   ˛f ..t//f  .t/  .1  ˛/f ..t//f r .t/ I

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Elements of Time Scales Analysis

(v) if g.t/g..t//g..t// ¤ 0, then the quotient f =g is ˙˛ -differentiable at t and  ˙˛  ˙˛ f 1 .t/ D f .t/g..t//g..t// g g.t/g..t//g..t//

  ˛f ..t//g..t//g .t/  .1  ˛/f ..t//g..t//gr .t/ :

The following observations are contained in Sheng [1, Proof of Theorem 2.1]. Remark 1.8.1 Consider the following sets on the time scale T: A D ft 2 TW t are left-dense and right-scatteredg B D ft 2 TW t are left-scattered and right-denseg C D ft 2 TW t are left-scattered and right-scatteredg D D ft 2 TW t are left-dense and right-denseg: Then the  and r-derivatives of f W T ! R, if any, are given by 8 < f ..t//  f .t/ ; t 2 C [ A; .t/ f  .t/ D : 0 f .t/ otherwiseI 8 < f .t/  f ..t// ; t 2 C [ B; r .t/ f .t/ D : 0 f .t/ otherwise:

(1.8.2)

(1.8.3)

Taking into account the expressions (1.8.2) and (1.8.3), we may obtain the following formula for f ˙˛ .t/ : 8 f ..t//  f .t/ ˆ ˆ ˛ C .1  ˛/f 0 .t/; ˆ ˆ .t/ ˆ ˆ ˆ ˆ 0 is a sufficiently small number. To this end in (2.1.22), we make the change of variable .t/ D v 1m .t/, and by definition of -derivative of a function, obtain  .t/ D

v 1m ..t//  v 1m .t/ ..t//  .t/ D .t/ .t/

.v.t/ C .t/v  .t//1m  v 1m .t/ .t/    1m v v  .t/ 1m .t/ D 1 C .t/ 1 .t/ v.t/ v 1m .t/  .1 C .t/r.t/v 1m .t//1m  1 D .t/    .t/ .t/r.t/ 1m 1 D 1C .t/ .t/ D

 F.t; /;

.t0 / D .1 C "/1m : . .t// .t/ .t/

in the case .t/  0 is equal to  1m 1C .t/r.t/ the limit lim .tC

/ .t/ . Further for the function F.t; /D .t/ 1 ,

Besides, it is assumed that the expression

!0

we find that

1 @F.t; / D @ .t/

1C

 m !  1 C .t/r.t/ 0  m .t/r.t/ 1C

m.t/r.t/

for all t 2 Œt0 ; C1/, i.e., the function F.t; / does not increase on the set .0; C1/. Since .t/ 2 .0; 1/ for all t 2 Œt0 ; C1/ (due to connection with the function v.t/), for the indicated values of t, the chain of inequalities holds true F.t; 0/  F.t; .t//  F.t; 1/ > F.t; 1/:

(2.1.23)

Method of Dynamic Integral Inequalities

37 1m

1 We find that F.t; 1/ D .1C.t/r.t// . It is easy to verify that the function .t/ F.t; / satisfies all conditions of the theorem on existence and uniqueness of solution to Cauchy problem for dynamic equation on time scale. Therefore, the Cauchy problem

 .t/ D F.t; .t//;

.t0 / D .1 C "/1m

possesses the only solution .t/, which can be presented in the integral form .t/ D .1 C "/

1m

Zt C

F.s; .s//s:

(2.1.24)

t0

Further, using formula (2.1.24) and inequalities (2.1.23), we arrive at the estimate .t/ D .1 C "/

1m

Zt C

F.s; .s//s  .1 C "/

1m

Zt C

t0

D .1 C "/1m C

Zt t0

F.s; 1/s t0

(2.1.25)

.1 C .s/r.s//1m  1 s; .s/

which is valid for all t 2 Œt0 ; tQ/. For the values of t from the scale T, the expression in the right-hand part of inequality (2.1.25) is positive by Lemma 2.1.9, and therefore, inequality (2.1.25) is equivalent to the inequality v.t/ D v.tI t0 ; 1 C "/  .1 C "/

1m

Zt C t0

1 ! 1m .1 C .s/r.s//1m  1 s .s/

for all t 2 Œt0 ; tQ/. In view of the comparison principle and the passage to the limit as " ! 0, we get inequality (2.1.21). Lemma 2.1.9 is proved.

2.2 Stability of Linear Dynamic Equations 2.2.1 Nonautonomous systems On a time scale T with graininess .t/, the system of dynamic equations x .t/ D f .t; x.t//;

x.t0 / D x0 ;

(2.2.1)

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Method of Dynamic Integral Inequalities

is considered, where 8 < x..t//  x.t/ ;  .t/ x .t/ D : xP .t/

if t 2 A [ C; at the rest of points;

x 2 Rn , f W T  Rn ! Rn . Let us make the following assumptions on the system (2.2.1). H1 .

H2 .

The vector function F.t/ D f .t; x.t// satisfies the condition F 2 Crd .T/, as soon as x is a differentiable function with its values in N, N Rn is an open connected neighborhood of the state x D 0. The vector function f .t; x/ is componentwise regressive, i.e., eT C .t/f .t; x/ ¤ 0 at all t 2 Œt0 ; 1/; where eT D .1; : : : ; 1/T 2 Rn :

H3 . H4 .

f .t; x/ D 0 at all t 2 Œt0 ; 1/ if and only if x D 0. The graininess function 0 < .t/ 2 M at all t 2 T; where M is a compact set.

Below find the standard definitions of some types of stability of unperturbed motion. Definition 2.2.1 The unperturbed motion of the system (2.2.1) is called (S1 ) (S2 ) (S3 )

stable, if for any " > 0 and t0 2 T, there exists ı D ı."; t0 / > 0 such that the condition kx0 k < ı implies the inequality kx.tI t0 ; x0 /k < " at all t  t0 W uniformly stable, if the value ı in the definition (S1 ) does not depend on t0 ; asymptotically stable, if it is stable and there exists ı0 such that the condition kx0 k < ı0 implies lim kx.tI t0 ; x0 /k D 0. t!1

We first establish stability conditions for solutions of linear homogeneous dynamic systems x .t/ D A.t/x.t/;

x.t0 / D x0 ;

(2.2.2)

where A 2 Crd .T; Rnn / and A.t/ are regressive on T. Recall that for the system (2.2.2), the properties of any solution x.t; t0 ; x0 / are equivalent to the corresponding properties of the zero solution of the system (2.2.2). We denote by ˆA .t; t0 / the solution to Y  .t/ D A.t/Y.t/;

Y.t0 / D In ;

where In is the n  n-identity matrix and A.t/ is time varying. Note that ˆA .t; t0 /  eA .t; t0 / only when A.t/  A is a constant matrix.

Method of Dynamic Integral Inequalities

39

Let us agree to say that the system of dynamic equations (2.2.2) is (uniformly, asymptotically, exponentially) stable if the zero solution of the equations (2.2.2) has a corresponding type of stability. Theorem 2.2.1 The system of dynamic equations (2.2.2) is stable if and only if all its solutions are bounded at all t  t0 2 T. Proof Assume that the system (2.2.2) is stable. Since the zero solution x.t/ D 0 of the system (2.2.2) is stable, for any " > 0 and t0 2 T, one can find ı D ı.t0 ; "/ > 0 such that kx.t; t0 ; x0 /k < ", as soon as kx0 k < ı. Note that kx.t; t0 ; x0 /k D kˆA .t; t0 /x0 k < " at all t  t0 2 T. Let the j-th component of the vector x0 do not exceed ı=2 at all j D 1; 2; : : : ; n. Then the relation kˆA .t; t0 /x0 k D jxj .t/j.ı=2/ < " holds at all t  t0 2 T, where xj .t/ is the j-th column of the matrix ˆA .t; t0 /. As a result, we have the estimate kˆA .t; t0 /k D max jxj .t/j < 1jn

2" ı

and for any solution x.t; t0 ; x0 / of the system (2.2.2) the estimate holds: kx.t; t0 ; x0 /k D kˆA .t; t0 /x0 k
0 such that the exponential stability of the system (2.2.11) implies the exponential stability of the system x D Bx

(S8 )

for any matrix B 2 Rnn such that kB  Ak  "; weakly uniformly exponentially stable if there exists a constant ˛ > 0 such that for any s 2 T, one can find K.s/  1 and the estimate keA .t; /k  K.s/e˛.t / holds at all t   s.

Let us dwell on the study of the existence and robustness properties of the exponential stability on a time scale T. Theorem 2.2.7 Let the time scale T be unbounded from above. If the graininess .t/ of the time scale T is bounded from above, i.e., there exists h > 0 such that .t/  h at all t 2 T, then, and only then, there exists a system of the form (2.2.11) whose zero solution is uniformly exponentially stable on T. Proof Let there exist a matrix A 2 Rnn such that the system x D Ax;

x 2 Rn ;

(2.2.12)

is uniformly exponentially stable, i.e., there exist constants K > 0 and ˛ > 0 such that keA .t; s/k  K exp.˛.t  s//

at all t  s:

(2.2.13)

Show that the matrix A ¤ 0. Indeed, let A D 0, then eA .t; s/ D I. Consequently, keA .t; s/k D 1 at all t  s, which contradicts the estimate (2.2.13). Now let t0 2 T be a right-scattered arbitrary point on T; i.e., .t0 / > 0. Then at the point t0 , the system (2.2.12) can be represented as x.t0 C .t0 //  x.t0 / D Ax.t0 /: .t0 /

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Method of Dynamic Integral Inequalities

Hence it follows that eA .t0 C .t0 /; t0 / D I C .t0 /A, and according to (2.2.13) we get kI C .t0 /Ak  K exp.˛.t0 //; i.e., 1 C .t0 /kAk  K. Hence it follows that .t0 / 

KC1 kAk

for any right-scattered point t0 2 T, i.e., the scale T has a bounded graininess. Assume that the graininess of the scale T is bounded from above, i.e., .t/  h, 1 h > 0, at all t 2 T. Let A D 2h I. Obviously, the matrix I C .t/A is invertible at all t 2 T and consequently A is regressive on T. We shall now show that the zero solution of the system x D Ax

(2.2.14)

is uniformly exponentially stable. Note that the matrix A is diagonally regressive, and therefore the following relation holds: Zt keA .t; s/k D exp

ˇ log ˇ1  lim u u&. /

ˇ

u ˇ 2h



s

Zt  exp s

1   D exp 2h



 1  .t  / : 2h

This estimate completes the proof of Theorem 2.2.7. Theorem 2.2.8 Let the time scale T be unbounded from above, with bounded graininess. If the system x D Ax;

A 2 Rnn ;

(2.2.15)

is uniformly exponentially stable, then there exists " > 0 such that the system x D Bx;

B 2 Rnn ;

(2.2.16)

is uniformly exponentially stable as well, as soon as kA  Bk  ". Proof Let for the system (2.2.15) constants K > 0 and ˛ > 0 be found, such that keA .t; s/k  K exp.˛.t  s//

at all t  s:

(2.2.17)

Method of Dynamic Integral Inequalities

47

Rewrite the system(2.2.16) as x D Ax C .B  A/x:

(2.2.18)

Applying the formula of variation of constants for the transfer matrix of the system (2.2.16), we arrive at the expression Zt eB .t; s/ D eA .t; s/ C

eA .t; u C .u//.B  A/ s

 eB .u; s/ u;

(2.2.19)

at all t  s:

Introduce the notation f .t/ D exp.˛.ts//keB .t; s/k for the fixed value s 2 T. Then, taking into account the estimate (2.2.17) and the relation (2.2.19), we obtain Zt exp.˛.u//f .u/ u

f .t/  K C KkA  Bk

(2.2.20)

s

at all t  s. Since the graininess of the scale T is bounded, there exists H > 0 such that .t/  H at all t 2 T. From the estimate (2.2.20), it follows that Zt f .t/  K C KkA  Bk exp.˛H/

f .u/ u s

at all t  s. Applying the Gronwall inequality to this estimate and assuming f .s/ D 1, we find that f .t/  KeM .t; s/

at all t  s;

where M D KkA  Bk exp.˛H/. Then, taking into account that e˛ .t; s/  exp.˛.t  s// at all t  s, for any ˛ > 0, we get keB .t; s/k  K exp..˛ C M/.t  s//

at all t  s:

Choose " > 0 so that K" exp.˛H/ < ˛. For any matrix B 2 Rnn such that kA  Bk  ", we have the estimate keB .t; s/k  K exp..˛ C K" exp.˛H//.t  s// at all t  s. Since ˛ C K" exp.˛H/ < 0, the state x D 0 of the system (2.2.16) is uniformly exponentially stable.

48

Method of Dynamic Integral Inequalities

2.2.3 Elements of Floquet theory Consider the linear homogeneous system x .t/ D A.t/x.t/

x.t0 / D x0 ;

(2.2.21)

where A.t/ is a p-periodic matrix at all t 2 T; and the time scale T is p-periodic as well. Recall that the time scale T is p-periodic if for any p 2 Œ0; 1/, the following conditions are satisfied: (a) if t 2 T, then t C p 2 T; (b) .t/ D .t C p/ at any t 2 T. The matrix A.t/W T ! Rnn is p-periodic if A.t/ D A.t C p/ at all t 2 T and p 2 Œ0; 1/. The solution x.t/ of the system (2.2.21) can be represented in the form x.t/ D ˆA .t; t0 /x0 , where Zt ˆA .t; t0 / D I C

Zt A. 1 /  1 C

t0

A. 1 / t0

Zt t0

A. 2 /  2  1 C : : : t0

Z 1 A. 1 /

C

Z 1

Z i1 A. 2 / : : :

t0

A. i /  i : : :  1 C : : : t0

This expression is a generalization of the classical result of the theory of linear differential equations for the case of the system (2.2.21) specified on a time scale. If the matrix A.t/ of the system (2.2.21) commutes with its integral, i.e., Zt

Zt A. /  D

A.t/ s

A. /  A.t/ s

at all s; t 2 Œt0 ; 1/ \ T, then the solution of the system (2.2.21) can be represented in the form x.t/ D eA .t; t0 /x0 . In this case the matrix A.t/ and the matrix exponent eA commute, i.e., A.t/eA .t; t0 / D eA .t; t0 /A.t/, while A.t/ˆA .t; t0 / ¤ ˆA .t; t0 /A.t/: The proof of the above statements can be found in DaCunha [1], with further references. Now let us discuss some auxiliary results for the generalized Floquet theory. Lemma 2.2.2 Let a nonsingular constant n  n-matrix M and a constant p > 0 be specified. The solution RW T ! Rnn of the matrix exponential equation eR .t0 C

Method of Dynamic Integral Inequalities

49

p; t0 / D M is determined by the formula 1 s=p .M  I/: s&.t/ s

R.t/ D lim

(2.2.22)

If the time scale T has constant graininess over the interval Œt0 ; t0 C p \ T, then the matrix R.t/ is constant. Rt Rt Proof Note that the relation R.t/ R. / D R. / R.t/ holds at all t; t0 2 T. t0

t0

Represent the fundamental matrix ˆR .t; t0 / of the equation z .t/ D R.t/z.t/;

z.t0 / D z0

(2.2.23)

in the form ˆR .t; t0 / D eR .t; t0 /. This relation can be easily verified by the direct computation e R .t; t0 / D R.t/eR .t; t0 / D eR .t; t0 /R.t/ at all t 2 T. Using the expression for R.t/, we obtain eR .t; t0 / D M

tt0 p

:

To prove the above, note that eR .t0 ; t0 / D M 0 D I. Then   tCst0 tt0 1 M p M p s&.t/ s   

tt0 s 1 M p  I M p D R.t/eR .t; t0 /: D lim s&.t/ s

e R .t; t0 / D lim

Hence it follows that eR .t0 C p; t0 / D M

t0 Cpt0 p

D M:

Lemma 2.2.2 is proved. We now go over to the main result of this section: the generalized Floquet theorem for p-periodic system on a time scale. We shall now prove the following result. Theorem 2.2.9 Let system (2.2.21) with an n  n-matrix A.t/ p-periodic on a time scale T be specified. If there exist p-periodic and invertible at all t 2 T matrix RW T ! Rnn and transformation L.t/ 2 C1rd .T; Rnn /, then the transfer matrix ˆA .t; / of the system (2.2.21) is representable in the form of the expansion ˆA .t; / D L.t/ eR .t; /L1 . / at all t; 2 T.

(2.2.24)

50

Method of Dynamic Integral Inequalities

Proof Assume M D ˆA .t0 C p; t0 / and calculate the matrix R according to the formula (2.2.22). We get eR .t0 C p; t0 / D ˆA .t0 C p; t0 /: Determine the matrix L.t/ by the formula L.t/ D ˆA .t; t0 / e1 R .t; t0 /. It is easy to see that the matrix L.t/ 2 Crd 1 .T; Rnn / and is invertible at any t 2 T. Therefore 1 1 ˆA .t; t0 / D L.t/eR .t; t0 / and ˆA .t0 ; t/ D e1 R .t; t0 /L .t/ D eR .t0 ; t/L .t/. Hence, taking into account the expression for the matrix L.t/, we obtain the expansion of the fundamental matrix (2.2.24). Now show that the matrix L.t/ is p-periodic. According to the expression for L.t/, we have L.t C p/ D ˆA .t C p; t0 /e1 R .t C p; t0 / D ˆA .t C p; t0 C p/ˆA .t0 C p; t0 /eR .t0 ; t0 C p/eR .t0 C p; t C p/ D ˆA .t; t0 /ˆA .t0 C p; t0 /e1 R .t0 C p; t0 /eR .t0 ; t/ D ˆA .t; t0 /e1 R .t; t0 / D L.t/: Theorem 2.2.9 is proved. Theorem 2.2.10 Let for the system (2.2.21) the fundamental matrix ˆA .t; t0 / admit the expansion (2.2.24). Then the solution x.t/ D ˆA .t; t0 /x0 of the system (2.2.21) is p-periodic if and only if the function z.t/ D L1 .t/x.t/ is a solution of the system z .t/ D R.t/z.t/;

z.t0 / D x0 :

(2.2.25)

Proof Let x.t/ D ˆA .t; t0 /x0 be a solution of the system (2.2.21), then x.t/ D L.t/eR .t; t0 /x0 . Let z.t/ D L1 .t/x.t/ D L1 .t/L.t/eR .t; t0 /x0 D eR .t; t0 ; x0 /. Now suppose that the vector function z.t/ D L1 .t/x.t/ is a solution of the system (2.2.25). Then z.t/ D eR .t; t0 /x0 . Assuming x.t/ D L.t/z.t/, we obtain x.t/ D L.t/eR .t; t0 /x0 D ˆA .t; t0 /x0 , i.e., x.t/ is a solution of the system (2.2.21). Theorem 2.2.11 The equilibrium state x D 0 of the system (2.2.21) is uniformly stable (exponentially stable, asymptotically stable respectively) if and only if the state z D 0 of the system (2.2.25) possesses a corresponding type of stability. Proof The conclusion of Theorem 2.2.11 follows from the fact that the systems (2.2.21) and (2.2.25) are connected by the Lyapunov transformation, and both systems are determined in a finite-dimensional space, where the norms of vectors are equivalent. Now consider the heterogeneous linear system x .t/ D A.t/x.t/ C f .t/;

x.t0 / D x0 ;

(2.2.26)

Method of Dynamic Integral Inequalities

51

where A.t/ is an n  n–p-periodic matrix, A.t/ 2 R.T; Rnn /, f .t/ 2 Crd .T; Rn1 / \ R.T; Rn1 / is a p-periodic function on T. Note that the solution x.t/ of the system (2.2.26) is p-periodic if and only if x.t0 C p/ D x.t0 / at any t0 2 T. Theorem 2.2.12 For the system (2.2.26), let the abovementioned conditions be satisfied for the matrix A.t/ and the vector function f .t/. Then at any t0 2 T and f .t/ 2 Crd .T; Rn1 / \ R.T; Rn1 /, there exists an initial value x.t0 / D x0 of the solution x.t/ such that x.t/ will be p-periodic if and only if a nonzero vector z.t0 / D z0 does not exist for any t0 2 T; such that the initial problem z .t/ D A.t/z.t/;

z.t0 / D z0 ;

(2.2.27)

has a p-periodic solution. Proof For any t0 2 T and any p-periodic function f .t/, the solution of the system (2.2.26) satisfies the relation Zt x.t/ D ˆA .t; t0 /x0 C

ˆA .t; . //f . /  :

(2.2.28)

t0

Since the solution x.t/ is p-periodic when x.t0 / D x.t0 C p/ for any t0 2 T, we have tZ 0 Cp

ŒI  ˆA .t0 C p; t0 /x0 D

ˆA .t0 C p; . //f . /  :

(2.2.29)

t0

Next we shall need the following result. Lemma 2.2.3 For the system (2.2.27) for any t0 2 T, let an initial value z.t0 / D z0 ¤ 0 be chosen so that the solution z.t/ is p-periodic. Then, and only then, the equation eR .t0 C p; t0 / D ˆA .t0 C p; t0 / will have at least one eigenvalue equal to 1. In view of the above statement, it is necessary to show that the equation (2.2.29) has a solution for any initial value x.t0 / D x0 and any p-periodic function f .t/ if and only if eR .t0 C p; t0 / does not have an eigenvalue equal to 1. Let for some t1 2 T the equation eR .t1 C p; t1 / D ˆA .t1 C p; t1 / have no eigenvalue equal to 1. This is equivalent to the condition detŒI  ˆA .t1 C p; t1 / ¤ 0:

52

Method of Dynamic Integral Inequalities

Since ˆA is p-periodic and invertible, and ˆA .t0 C p; t1 C p/ D ˆA .t0 ; t1 /, we find for t0 2 T and the p-periodic function f .t/ that 1

tZ 0 Cp

x0 D ŒI  ˆA .t0 C p; t0 /

ˆA .t0 C p; . //f . /  : t0

Now assume that the equation (2.2.29) has a solution at any t0 and any p-periodic function f .t/. Specify an arbitrary value t0 2 T and denote f0 D f .t0 /. Determine a p-periodic vector f 2 Crd .T; Rn1 / by the formula f .t/ D ˆA ..t/; t0 C p/f0 ;

t 2 Œt0 ; t0 C p/ \ T:

(2.2.30)

Taking into account the p-periodicity and the formula (2.2.30), we get tZ 0 Cp

tZ 0 Cp

ˆA .t0 C p; . //f . /  D t0

f0  D pf0 : t0

Hence it follows that (2.2.29) is equivalent to the relation ŒI  ˆA .t0 C p; t0 /x0 D pf0 :

(2.2.31)

By construction of f .t/ for the corresponding value f0 , the equation (2.2.31) has a solution for the specified x0 . Consequently, detŒI  ˆA .t0 C p; t0 / ¤ 0, and the equation eR .t0 C p; t0 / D ˆA .t0 C p; t0 / does not have an eigenvalue equal to 1. Therefore, according to Lemma 2.2.3, the system (2.2.27) does not have a p-periodic solution.

2.3 Stability of Nonlinear Dynamic Equations In this section the application of dynamic integral inequalities for the stability analysis of solutions to nonlinear dynamic equations is considered.

2.3.1 Estimations of solutions We shall now consider the estimates of solutions to the systems of integral and differential dynamic equations to be applied in subsequent discussion. Consider an

Method of Dynamic Integral Inequalities

53

integral dynamic equation of the form Zt x.t/ D g.t/ C B.t/

U.s; x.s//s;

t 2 Œa; C1/T ;

(2.3.1)

a

where x; gW I ! Rn , BW I ! R, UW I  Rn ! Rn are rd-continuous functions, I D Œa; C1/ \ T, a 2 T. Let there exist a solution of equation (2.3.1) on I. Lemma 2.3.1 Assume that there exist rd-continuous functions L; MW IRC ! RC such that: (1) kU.t; x/k  L.t; kxk/ at all t 2 I, x 2 Rn ; (2) 0  L.t; u/  L.t; v/  M.t; v/.u  v/ at all t 2 I, u  v  0. Then for any solution x.t/ of the integral equation (2.3.1) at all t 2 Œa; C1/T , the following estimate holds Zt kx.t/  g.t/k  jB.t/j

L.s; kg.s/k/ep .t; .s//s; a

where p.t/ D jB.t/jM.t; kg.t/k/. Proof Let the function x.t/ be a solution of the equation (2.3.1). Denote Zt y.t/ D

U.s; x.s//s

(2.3.2)

x.t/ D g.t/ C B.t/y.t/:

(2.3.3)

a

and from (2.3.1) obtain

Differentiating (2.3.2) and taking into account (2.3.3), we obtain y .t/ D U.t; g.t/ C B.t/y.t//;

y.a/ D 0:

Then ky .t/k D kU.t; g.t/ C B.t/y.t//k  L.t; kg.t/ C B.t/y.t/k/  L.t; kg.t/k C jB.t/jky.t/k/ D L.t; kg.t/k C jB.t/jky.t/k/  L.t; kg.t/k/ C L.t; kg.t/k/  L.t; kg.t/k/ C M.t; kg.t/k/jB.t/j ky.t/k

(2.3.4)

54

Method of Dynamic Integral Inequalities

 Rt  Rt   and, since ky.t/k D  y .s/s  ky .s/ks, we find that (2.3.4) implies the a

a

estimate Zt ky.t/k 

Zt



ky .s/ks  a

L.s; kg.s/k/s a

(2.3.5)

Zt C

jB.s/jM.s; kg.s/k/ky.s/ks: a

Denoting f .t/ D

Rt

L.s; kg.s/k/s, p.t/ D jB.t/jM.t; kg.t/k/, we rewrite the

a

inequality (2.3.5) as Zt ky.t/k  f .t/ C

p.s/ky.s/ks; a

here f  .t/ D L.t; kg.t/k/  0, f .a/ D 0. Taking into account Lemma 2.1.2, at all t 2 Œa; C1/T , we obtain the inequality Zt ky.t/k  f .a/ep .t; a/ C

L.s; kg.s/k/ep .t; .s//s; a

hence Zt L.s; kg.s/k/ep .t; .s//s:

kx.t/  g.t/k  jB.t/j ky.t/k  jB.t/j a

Lemma 2.3.1 is proved. Lemma 2.3.2 Assume that there exists an rd-continuous function SW Œa; C1/T  RC ! RC such that at all t 2 Œa; C1/T and x; y 2 Rn , the following inequality holds: kU.t; x C y/  U.t; x/k  S.t; kxk/kyk:

Method of Dynamic Integral Inequalities

55

Then for any solution x.t/ of the integral equation (2.3.1) at all t 2 Œa; C1/T , the following estimate holds Zt kx.t/  g.t/k  jB.t/j

kU.s; g.s//kep .t; .s//s;

(2.3.6)

a

where p.t/ D jB.t/jS.t; kg.t/k/. Proof In the same manner as in the proof of Lemma 2.3.1, for Zt y.t/ D

U.s; x.s//s a

we obtain the estimate ky .t/k D kU.t; g.t/ C B.t/y.t//k  kU.t; g.t//k C S.t; kg.t/k/jB.t/j ky.t/k; which implies that Zt ky.t/k 

Zt



ky .s/ks  a

kU.s; g.s//ks a

Zt C

(2.3.7)

jB.s/jS.s; kg.s/k/ky.s/ks: a

If we assume f .t/ D

Rt

kU.s; g.s//ks, p.t/ D jB.t/jS.t; kg.t/k/, then the

a

inequality (2.3.7) can be written as Zt ky.t/k  f .t/ C

p.s/ky.s/ks: a

Then, taking into account Lemma 2.1.2, we get Zt kU.s; g.s//kep .t; .s//s;

ky.t/k  a

56

Method of Dynamic Integral Inequalities

hence we have Zt kU.s; g.s//kep .t; .s//s:

kx.t/  g.t/k  jB.t/j ky.t/k  jB.t/j a

Lemma 2.3.2 is proved. Corollary 2.3.1 Assume that there exist rd-continuous functions L; MW I  RC ! RC such that: (1) k f .t; x/k  L.t; kxk/ at all t 2 I, x 2 Rn ; (2) 0  L.t; u/  L.t; v/  M.t; v/.u  v/ at all t 2 I, u  v  0. Then for the solution x.t/ of the initial problem for the system (2.2.1) at all t 2 Œt0 ; C1/T , the following estimate holds: Zt kx.t/  x0 k 

L.s; kx0 k/ep .t; .s//s;

(2.3.8)

t0

where p.t/ D M.t; kx0 k/. Proof Rewrite the equations (2.2.1) as Zt x.t/ D x0 C

f .s; x.s//s: t0

Using Lemma 2.3.1 at g.t/  x0 , B.t/  1, a D t0 , we arrive at the estimate (2.3.8). Corollary 2.3.2 Assume that there exists an rd-continuous function SW Œa; C1/T  RC ! RC such that at all t 2 Œa; C1/T and x; y 2 Rn , the following inequality holds: kU.t; x C y/  U.t; x/k  S.t; kxk/kyk: Then for the solution x.t/ of the initial problem for the system (2.2.1) at all t 2 Œt0 ; C1/T , the following estimate holds: Zt kx.t/  x0 k 

k f .s; x0 /kep .t; .s//s;

(2.3.9)

t0

where p.t/ D S.t; kx0 k/. Proof Proceeding as in the proof of Corollary 2.3.1, we apply Lemma 2.3.2 at g.t/  x0 , B.t/  1, a D t0 and obtain the estimate (2.3.9).

Method of Dynamic Integral Inequalities

57

2.3.2 Theorems on stability For the system of dynamic equations (2.2.1), we present the following general results. Theorem 2.3.1 Assume that for the equations (2.2.1), there exist rd-continuous functions L; MW I  RC ! RC such that L.t; 0/  0 and the following inequalities hold: (1) k f .t; x/k  L.t; kxk/ at all t 2 I, x 2 Rn ; (2) 0  L.t; u/  L.t; v/  M.t; v/.u  v/ at all t 2 I, u  v  0. Then, if there exist constants M > 0, ı0 > 0 such that C1 Z M.s; ı/s  M ˛

for any 0  ı  ı0 , then the equilibrium state x D 0 of the system (2.2.1) is uniformly stable. Proof Using Corollary 2.3.1 and Lemma 2.3.1, for the solution x.t/ of the system (2.2.1), we obtain the estimate kx.t/k  kx0 k C kx.t/  x0 k Zt  kx0 k C

L.s; kx0 k/ep .t; .s//s t0

Zt  kx0 k C

Zt L.s; kx0 k/ exp

(2.3.10)

p. / s  .s/

t0

Zt D kx0 k C

!

!

Zt L.s; kx0 k/ exp

M. ; kx0 k/ s:  .s/

t0

Choose an arbitrary " > 0 and assume  " " ; ; ı0 : ı D ı."/ D min 2 2MeM 

58

Method of Dynamic Integral Inequalities

In view of the condition (2) of Theorem 2.3.1, we obtain that L.t; u/  M.t; 0/u at all t 2 Œt0 ; C1/T and u  0. Hence follows the inequality Zt

Zt L.s; u/s  u

t0

C1 Z M.s; 0/s  u M.s; 0/s  Mu: ˛

t0

Extending the estimate (2.3.10), at all kx0 k  ı and t 2 Œt0 ; C1/T , we get " kx.t/k  C 2 "  C 2

Zt L.s; kx0 k/exp

M. ; kx0 k/ s

t0

 .s/

Zt

Z L.s; kx0 k/exp

t0

"  C eM 2 

!

Zt

C1 ˛

! M. ; kx0 k/ s

Zt L.s; kx0 k/exps t0

" C eM Mkx0 k  "; 2

Hence follows the uniform stability of the zero equilibrium state of the system (2.2.1). Theorem 2.3.1 is proved. Theorem 2.3.2 Assume that for the system (2.2.1), there exists an rd-continuous function SW Œ˛; C1/T  RC ! RC such that at all t 2 Œ˛; C1/T and x; y 2 Rn , the following inequality holds: k f .t; x C y/  f .t; x/k  S.t; kxk/kyk: Then, if there exist constants M > 0 and ı0 > 0 such that C1 Z S.u; ı/u  M ˛

for any 0  ı  ı0 , then the equilibrium state x D 0 of the system (2.2.1) is uniformly stable. Proof Assuming y D x, at all t 2 I and x 2 Rn , we obtain the inequality k f .t; x/k  S.t; kxk/kxkI

Method of Dynamic Integral Inequalities

59

hence, according to Lemma 2.3.2, we have kx.t/k  kx0 k C kx.t/  x0 k Zt  kx0 k C

k f .s; x0 /kep .t; .s//s t0

Zt  kx0 k C kx0 k

S.s; kx0 k/ exp

(2.3.11)

p. / s  .s/

t0

Zt D kx0 k C kx0 k

!

Zt

!

Zt S.s; kx0 k/ exp

S. ; kx0 k/ s;  .s/

t0

where p.t/ D S.t; kx0 k/, t 2 Œt0 ; C1/T . Choose an arbitrary " > 0 and assume ı D ı."/ D minf"=2; "=.2MeM /; ı0 g. Extending the estimate (2.3.11) at all kx0 k  ı and t 2 Œt0 ; C1/T , we arrive at " kx.t/k  C kx0 k 2

C1 C1 Z Z S.s; kx0 k/exp S. ; kx0 k/ s ˛

"  C kx0 keM 2

˛ C1 Z S.s; kx0 k/ s ˛

"  C eM Mkx0 k  "; 2 hence follows the uniform stability of the zero equilibrium state of the system (2.2.1). Theorem 2.3.2 is proved. Remark 2.3.1 In the case when T D R, the integral and the -derivatives on T coincide with the Riemann integral and the Euler derivative. Therefore Theorems 3.5.1 and 3.5.7 from the monograph by Dragomir [1] are obtained automatically as corollaries of Theorems 2.3.1 and 2.3.2, respectively. Now let T D Z. In this case the initial problem for the system (2.2.1) will take the form x. / D f . ; x. //; x. 0 I 0 ; x0 / D x0 ;

0 2 I;

2 I;

(2.3.12)

x0 2 R ; n

(2.3.13)

where x 2 Rn , x. / D x. C 1/  x. /, I D f˛; ˛ C 1; ˛ C 2; : : : g, ˛ 2 Z, f W I  Rn ! Rn , f . ; 0/  0. In addition, for the problem (2.3.12)–(2.3.13), let all

60

Method of Dynamic Integral Inequalities

the solution existence and uniqueness conditions be satisfied over an infinite interval at any initial data . 0 ; x0 / 2 I  Rn . Corollary 2.3.3 Assume that for the system (2.3.12), there exist functions L; MW I  RC ! RC such that L. ; 0/  0 and the following inequalities hold true: (1) k f . ; x/k  L. ; kxk/ at all 2 I, x 2 Rn , (2) 0  L. ; u/  L. ; v/  M. ; v/.u  v/ at all 2 I, u  v  0. If there exist constants M > 0 and ı0 > 0 such that C1 X

M. ; ı/  M

D˛

for any 0  ı  ı0 , then the equilibrium state x D 0 of the system (2.3.12) is uniformly stable. Corollary 2.3.4 Assume that for the system (2.3.12), there exists a function SW I  RC ! RC such that at all t 2 I and x; y 2 Rn , the following inequality holds: k f .t; x C y/  f .t; x/k  S.t; kxk/kyk: Then, if there exist constants M > 0 and ı0 > 0 such that C1 X

S. ; ı/  M

D˛

for any 0  ı  ı0 , then the equilibrium state x D 0 of the system (2.3.12) is uniformly stable. Example 2.3.1 Consider a system of dynamic equations on a time scale T of the form x .t/ D A.t/x.t/;

(2.3.14)

 0 a.t/ where x D .x1 ; x2 /T , x1 ; x2 W T ! R, A.t/ D a.t/ , I D Œ˛; C1/ \ T, ˛ 2 T. 0 Assume that the function aW T ! R is continuous, regressive, a.t/ ¤ 0 at all t 2 I and C1 Z ja.s/js  M: ˛

It is clear that the function f .t; x/ D A.t/x is rd-continuous; let us show that it is regressive. To this end it is sufficient to show that at any fixed t 2 I, the operator

Method of Dynamic Integral Inequalities

61

F D In C .t/A.t/W R2 ! R2 , acting by the formula F.x/ D x C .t/A.t/x, is invertible. For any z 2 R2 , the equation F. / D z has a unique solution D .z2 =.1 C .t/a.t//; z1 =.1 C .t/a.t///T ; which means the invertibility of the operator F and, consequently, the regressivity of the function f .t; x/. Since k f .t; x/k D ja.t/jkxk, all the conditions of Theorem 8.24 from the monograph by Bohner and Peterson [1] are satisfied, i.e., there exists a unique solution of the system (2.3.14) over an infinite open interval at any initial data. We shall verify that the conditions of Theorem 2.3.1 are satisfied. It is easy to see that L.t; u/ D ja.t/ju  0 at u  0 and, in addition, 0  L.t; u/  L.t; v/ D ja.t/j.u  v/ at u  v  0 and M.t; v/ D ja.t/j. The functions L.t; u/, M.t; v/ are rd-continuous; therefore, all the conditions of Theorem 2.3.1 are satisfied, and the equilibrium state x D 0 of the system (2.3.14) is uniformly stable. In particular, for continuous .t/, the function a.t/ D j  je .t; ˛/ D

e .t; ˛/ 1 C .t/

is continuous and regressive at > 0. Since C1 Z

C1 Z  e .s; ˛/s D e .s; ˛/jC1 ˛

˛

˛

e .s; ˛/s D  1 C .s/

D  lim e .s; ˛/  e .˛; ˛/ D 1; s!C1

the equilibrium state x D 0 of the system (2.3.14) is uniformly stable.

2.3.3 Stability of quasilinear equations Further, nonlinear dynamic inequalities are applied for the equations of the form x D A.t/x C f .t; x/;

f .t; 0/ D 0;

(2.3.15)

where A 2 R.T; Rnn / with the value n 2 N, f W T  Rn ! Rn , F.t/ D f .t; x.t// satisfies the condition F 2 Crd .T/ as soon as x is a differentiable function. These assumptions ensure the existence of a unique solution x D x.I t0 ; x0 / of the system (2.3.15) under the initial conditions x.t0 / D x0 , where t0 2 T and x0 2 Rn .

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Method of Dynamic Integral Inequalities

According to the formula (1.6.9), this solution can be represented as Zt x.t/ D x.tI t0 ; x0 / D eA .t; t0 /x0 C

eA .t; . //f . ; x. // :

(2.3.16)

t0

Case m 2 N n f1g. Assume that m 2 N n f1g and make two assumptions about the system (2.3.15): kf .t; x/k  a.t/ kxkm at t  t0 ; x 2 Rn ; where a 2 Crd .T; RC /;

(2.3.17)

and keA .t; s/k  '.t/ .s/ at t  s  t0 ; where ';

2 Crd .T; RC /:

(2.3.18)

Under these assumptions we will obtain the conditions for the stability, uniform stability, and asymptotic stability of unperturbed motion of the system (2.3.15). Then the cases m D 1 and m > 1 are considered. If m D 1, then the known Gronwall inequality on a time scale is applied. If m > 1, then the dynamic version of Stakhurskaya’s inequality is applied (see Lemma 2.1.8). Lemma 2.3.3 Assume that the condition (2.3.17) is satisfied at m D 1 and the inequality (2.3.18) holds. Then any solution of the system (2.3.15) satisfies the estimate kx.tI t0 ; x0 /k  '.t/ .t0 /e'

a

.t; t0 / kx0 k at all t  t0 :

(2.3.19)

Proof First of all, note that under the conditions of Lemma 2.3.3, all the conditions of Corollary 2.1.2 are satisfied. Let x.t/ be a solution of the system (2.3.15), and consequently, at all t  t0 from the relation (2.3.16) and the conditions (2.3.17), (2.3.18), we obtain the estimate Zt kx.tI t0 ; x0 /k  '.t/ .t0 / kx0 k C

'.t/ .. //a. / kx. I t0 ; x0 /k t: t0

By the substitution y D kx.I t0 ; x0 /k =', transform this inequality to the form Zt y.t/ 

.t0 / kx0 k C

'. / .. //a. /y. / at all t  t0 : t0

Applying Corollary 2.1.2 to the above inequality, we get the estimate y.t/ 

.t0 / kx0 k e'

a

.t; t0 /

at all t  t0 :

Method of Dynamic Integral Inequalities

63

Taking into account the above substitution of the function y, we arrive at the inequality (2.3.19). Lemma 2.3.3 is proved. Theorem 2.3.3 Assume that for the system (2.3.15), the condition (2.3.17) is satisfied at m D 1 and the inequality (2.3.18) holds: (1) If at all s  t0 , there exists K.s/ > 0 such that '.t/e'

a

.t; s/  K.s/

at all t  s  t0 ;

then the unperturbed motion of the system (2.3.15) is stable; (2) if there exists K > 0 such that '.t/ .s/e'

a

.t; s/  K

at all t  s  t0 ;

then the ˚unperturbed motion of the system (2.3.15) is uniformly stable;

D 0; then the unperturbed motion of the (3) if lim '.t/e'  a .t; s/ t!1 system (2.3.15) is asymptotically stable. Proof We shall prove the assertion (1). Let " > 0 and t0 2 T. Determine ı."; t0 / D "K 1 .t0 /

1

.t0 /

and assume that kx0 k < ı. Then according to Lemma 2.3.3, we obtain kx.tI t0 ; x0 /k < '.t/ .t0 /e'

a

.t; t0 /ı 

.t0 /K.t0 /ı D ":

We shall next prove the assertion (2). Let " > 0. Determine ı."/ D "K 1 and assume that kx0 k < ı. Proceeding as above, we apply Lemma 2.3.3 and find that kx.tI t0 ; x0 /k < '.t/ .t0 /e'

a

.t; t0 /ı  Kı D ":

We shall further prove the assertion (3). Since 'e'  a .; s/ vanishes, this expression is bounded. In view of the assertion (1) of the theorem, the unperturbed motion is stable. Let ı0 D 1 and assume that kx0 k < ı0 . Then, according to Lemma 2.3.3, we obtain kx.tI t0 ; x0 /k < '.t/ .t0 /e'

a

.t; t0 / ! 0

as soon as t ! 1. Lemma 2.3.4 Assume that for the system (2.3.15), the condition (2.3.17) at m > 1 and the condition (2.3.18) are satisfied. Then the solution of the system (2.3.15)

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Method of Dynamic Integral Inequalities

satisfies the estimate kx.tI t0 ; x0 /k  n

'.t/ .t0 / kx0 k 1  .m  1/ kx0 km1

m1 .t /D.t; t / 0 0

o1=.m1/

(2.3.20)

at all t  t0 , for which .m  1/ kx0 km1

m1

.t0 /D.t; t0 / < 1;

where Zt D.t; t0 / D

' m . / .. //a. / : t0

Proof We first note that in this assertion, all the conditions of Lemma 2.1.6 are satisfied. Let x be a solution of the system (2.3.15). Then, due to the relation (2.3.16) at all t  t0 , the following estimate holds: kx.tI t0 ; x0 /k  '.t/ .t0 / kx0 k Zt C

'.t/ .. //a. / kx. I t0 ; x0 /km  : t0

Hence the function y D kx.I t0 ; x0 /k =' satisfies the inequality Zt y.t/ 

.t0 / kx0 k C

' m . / .. //a. /ym . / at all t  t0 : t0

According to Lemma 2.1.6, we have the inequality y.t/  

.t0 / kx0 k Rt

1 C .m  1/ . m

a

m1 .t / kx km1 /. /

0 0

 1=.m1/ ;

t0

which is satisfied as long as the denominator remains positive. Since g D 

g  g 1 C g

at all g  0;

turning back to the definition of y, we find that the inequality (2.3.20) is proved.

Method of Dynamic Integral Inequalities

65

The estimate (2.3.20) allows new stability conditions to be established for the unperturbed motion of the quasilinear system (2.3.15) on a time scale in the same way as for the system of ordinary differential equations (cf. Martynyuk et al. [1], p. 181–189). Theorem 2.3.4 Assume that for the system (2.3.15), the condition (2.3.17) at m > 1 and the condition (2.3.18) are satisfied: (1) If at all s  t0 , there exists a K.s/ > 0 such that '.t/  K.s/ at all t  s  t0 and D.t0 / D lim D.t; t0 / < 1; t!1

(2.3.21)

then the unperturbed motion of the system (2.3.15) is stable; (2) if there exist K1 ; K2 o> 0 such that '.t/ .s/  K1 at all t  s  t0 and n m1 .s/ lim D.t; s/  K2 at all s  t0 , then the unperturbed motion of the t!1 system (2.3.15) is uniformly stable; (3) if the condition (2.3.21) is satisfied and lim '.t/ D 0; then the unperturbed t!1 motion of the system (2.3.15) is asymptotically stable. Proof We shall first prove the assertion (1) of Theorem 2.3.4. Let " > 0 and t0 2 T. Determine n ı."; t0 / D min Œ2.m  1/ m1 .t0 /D.t0 /1=.m1/ ; o " 1 .t0 /K 1 .t0 /21=.m1/ and assume that kx0 k < ı. Then, according to Lemma 2.3.4, we obtain '.t/ .t0 /ı

kx.tI t0 ; x0 /k
0 be such that the denominator in the inequality (2.3.20) is positive, and assume that kx0 k < ı0 . Then, according to Lemma 2.3.4, '.t/ .t0 /ı0 kx.tI t0 ; x0 /k < ˚

1=.m1/ ! 0 m1 m1 1  .m  1/ı0 .t0 /D.t; t0 / as soon as t ! 1. Case m 2 R n f1g. Consider the quasilinear dynamic equation (2.3.15) where x 2 Rn , t 2 T, and the matrix-valued function AW T ! Rnn and the vector function f W T  Rn ! Rn satisfy the following hypotheses: (H1 ) (H2 )

functions A.t/ and f .t; x/ are rd-continuous and A 2 R.T; Rnn /; function f .t; x/ satisfies Lipschitz condition with respect to spatial variable in Rn , i.e., there exists L > 0 such that k f .t; x1 /  f .t; x2 /k  Lkx1  x2 k

(H3 )

for all .t; x1 /; .t; x2 / 2 T  Rn ; there exist functions ˛.t/; '.t/; .t/ 2 Crd .T; RC / and a constant m > 1 such that: (a) k f .t; x/k  ˛.t/kxkm ; (b) keA .t; t0 /k  '.t/ .t0 /, for all t  t0 , belonging to T, and x 2 Rn , where eA .t; t0 / denotes the matrix exponential function of the linear dynamic equation x D A.t/x.

It should be noted that the conditions of the hypotheses (H1 ) and (H2 ) ensure existence and uniqueness of solution for the dynamic equation with given initial conditions. Further, under the hypotheses (H1 ) – (H3 ), we investigate the problem on stability, uniform stability, and asymptotic stability of zero solution for the dynamic equation (2.3.15).

Method of Dynamic Integral Inequalities

Designate h.t/ D

67

..t//' m .t/˛.t/, Zt

D.t; a; / D a

1 1 lim

&.s/

.1 C h.s/

!

1 m1 .a/ m1 /m1

s:

The following lemma provides estimate of solution to the equation (2.3.15) by means of the inequality (2.1.21). Lemma 2.3.5 For the equation (2.3.15), let the hypotheses (H1 )–(H3 ) be satisfied. Then for arbitrary t0 2 T and x0 2 Rn , the following estimate of solution x.tI t0 ; x0 / to the equation (2.3.15) holds h i1=1m kx.tI t0 ; x0 /k  '.t/ .t0 /kx0 k 1  D.t; t0 ; kx0 k/

(2.3.22)

for all t from Œt0 ; C1/ \ T, for which D.t; t0 ; kx0 k/ < 1. Proof As noted, the hypotheses (H1 )–(H2 ) ensure the existence and uniqueness of solution x.tI t0 ; x0 / to the equation (2.3.15) found by the Cauchy formula: Zt x.tI t0 ; x0 / D eA .t; t0 /x0 C

eA .t; .s//f .s; x.sI t0 ; x0 //s;

(2.3.23)

t0

where the integration is made on the scale T within the limits from t0 to t. From (2.3.23) and the hypothesis (H3 ), we have the estimate of the norm x.tI t0 ; x0 / (further denoted as x.t/) Zt kx.t/k  '.t/ .t0 /kx0 k C

'.t/ ..s//˛.s/kx.s/km s: t0

Having designated u.t/ D

kx.t/k , '.t/

a.t/ D

.t0 /kx0 k, we get the inequality

Zt u.t/  a.t/ C

h.s/um .s/s;

for all

t  t0 :

t0

Since the functions in this inequality satisfy all conditions of Lemma 2.1.9, we get the estimate  1=1m u.t/  a.t/ 1  D.t; t0 ; kx0 k/ ; which is valid for all t, such that D.t; t0 ; kx0 k/ < 1. Lemma 2.3.5 is proved.

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Method of Dynamic Integral Inequalities

Further sufficient conditions of stability, uniform stability, and asymptotic stability of zero solution to dynamic equations of (2.3.15) type are established in terms of generalized nonlinear dynamic inequality. Theorem 2.3.5 If for the equation (2.3.15) for all s  t0 , there exists K.s/ such that '.t/  K.s/ for all t  s  t0 and e D.t0 ; / D lim D.t; t0 ; / < 1;

(2.3.24)

t!1

for all t0 2 T and  > 0, the solution x D 0 of the equation (2.3.15) is stable. Proof We study properties of the function D.t; a; /, defined above. Direct computation shows that the function D.t; a; / increases on the set RC , uniformly in t and a. Consequently, the function e D.a; / from (2.3.24) does not decrease on RC , uniformly in a, and, moreover, e D.a; 0/ D 0. Consider the set ƒ of from Œ0; 12  such that the equation e D.a; / D possesses the largest solution  D  .a/ for all a 2 T. Let D sup ƒ, then for all a 2 T and  <  .a/, we have 1 e D.a; /  e D.a;  .a// D  : 2

(2.3.25)

Now let us choose some " > 0 and t0 2 T. Set n "21=1m o ı D min  .t0 /; .t0 /K.t0 / and show that if kx0 k < ı, then kx.tI t0 ; x0 /k < "

(2.3.26)

for all t  t0 . Since the function e D.a; / does not decrease on RC , by the estimates (2.3.25) for all t  t0 from the scale, we have D.t; t0 ; kx0 k/  lim D.t; t0 ; kx0 k/ D e D.t0 ; kx0 k/  e D.t0 ; ı/  t!1

1 : 2

(2.3.27)

From (2.3.27) we conclude that by Lemma 2.3.5 for all t  t0 from the scale, the estimate (2.3.22) is valid. Using (2.3.22) and the method of choosing of ı, we arrive

Method of Dynamic Integral Inequalities

69

at the estimates h i1=1m kx.tI t0 ; x0 /k  '.t/ .t0 /kx0 k 1  D.t; t0 ; kx0 k/ h i1=1m  K.t0 / .t0 /kx0 k 1  D.t; t0 ; kx0 k/ 

1 1=1m  K.t0 / .t0 /kx0 k 1  2 < K.t0 / .t0 /ı  K.t0 / .t0 / 

(2.3.28)

1 21=1m 1 "21=1m  1=1m D "; .t0 /K.t0 / 2

which are valid for all t  t0 from the scale. This completes the proof. Theorem 2.3.6 If for the equation (2.3.15), there exist a positive constant K1 and a continuous nondecreasing function K2 ./ such that '.t/ .s/  K1 for all t  s  t0 and e D.s; / D lim D.t; s; /  K2 ./ t!1

for all s  t0 and  > 0, then the solution x D 0 of the equation (2.3.15) is uniformly stable. Proof Let " > 0, t0 2 T. Due to the properties of function K2 ./, there exists a value of the parameter  from the interval .0; 12  such that the equation K2 ./ D  possesses the largest solution ./. Designate by 1 the largest of the mentioned 1 values of parameter . We set ı D minf.1 /; ".2 m1 k1 /1 g and show that if kx0 k < ı, then kx.tI t0 ; x0 /k < ", for all t  t0 . By the condition of the theorem, for all t  t0 from the time scale, we have D.t; t0 ; kx0 k/  lim D.t; t0 ; kx0 k/  K2 .kx0 k/ < K2 .ı/ t!1

< K2 ..1 // D 1 

1 < 1: 2

(2.3.29)

From (2.3.29) we conclude that the estimate (2.3.22) is fulfilled for all t  t0 from the time scale. Therefore, h i1=1m kx.tI t0 ; x0 /k  '.t/ .t0 /kx0 k 1  D.t; t0 ; kx0 k/ h i1=1m  K1 kx0 k 1  D.t; t0 ; kx0 k/

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Method of Dynamic Integral Inequalities

h i1=1m  K1 kx0 k 1  e D.t0 ; kx0 k/  1=1m 1  K1 kx0 k 1  K2 .kx0 k/ < K1 ı2 m1  " for all t  t0 from the scale. The theorem is proved. Theorem 2.3.7 If for the equation (2.3.15), the conditions e D.s; / D lim D.t; s; / < 1 t!1

are satisfied for all s  t0 and  > 0, and lim '.t/ D 0, then the solution x D 0 of t!1 the equation (2.3.15) is asymptotically stable. Besides, the domain of attraction of solution x D 0 contains a sphere B.0;  .t0 //, where  .t0 / is the largest solution of the equation e D.t0 ; / D , 2 .0; 1/. Proof Let " > 0, t0 2 T. Since the value of function '.t/ vanishes for t ! 1, the function is bounded. Then, by Theorem 2.3.5, the solution x D 0 of the equation (2.3.15) is stable. Let us show that there exists a ı0 > 0 such that if kx0 k < ı0 , then the limit equality lim kx.tI t0 ; x0 /k D 0 holds true. It can be easily t!C1

verified that the function D.t; s; / increases in the last variable on RC . Therefore, the function e D.s; / does not decrease in  on RC . Then, there exists a 2 .0; 1/, for which the equation e D.a; / D possesses the largest solution designated by  .a/. We set ı0 D  .t0 /, then for all t  t0 and kx0 k < ı0 , the following inequalities hold true: D.t; t0 ; kx0 k/  e D.t0 ; kx0 k/  e D.t0 ; ı0 / D < 1: By Lemma 2.3.5 for solution x.tI t0 ; x0 / of the equation (2.3.15), the estimate (2.3.22) is valid. Using the above inequality and inequality (2.3.22) we get h i1=1m kx.tI t0 ; x0 /k  '.t/ .t0 /kx0 k 1  D.t; t0 ; kx0 k/ h i1=1m  '.t/ .t0 /kx0 k 1  e D.t0 ; kx0 k/ h i1=1m D.t0 ; ı0 / < '.t/ .t0 /ı0 1  e ! 0; whenever t ! 1. Thus, the neighborhood of the point x D 0 with the radius  .t0 / is contained in the domain of attraction of the solution x D 0 of the equation (2.3.15). Example 2.3.1 Consider the system of dynamic equations (2.3.15) on time scale, satisfying the hypotheses (H1 ) – (H3 ) for any real m > 1, for the following values of

Method of Dynamic Integral Inequalities

functions '.t/,

71

.t/, ˛.t/ :

'.t/ D Me .t; 0/;

.t/ D e .0; t/;

˛.t/ D Ae .t; 0/:

(2.3.30)

Here A and M are positive constants, and the real numbers and satisfy positive regressivity conditions 1 C .t/ > 0;

1 C .t/ > 0;

for all t 2 T:

(2.3.31)

Assume that the scale T has a bounded graininess function .t/, i.e., there exists   0, such that .t/   for all t 2 T. Applying Theorem 2.3.7 one can easily establish additional conditions, which the constants and must satisfy to, so that the solution x D 0 of equation (2.3.15) be asymptotically stable under assumptions (2.3.30) and (2.3.31). Such result is contained in Corollary 2.3.5. Corollary 2.3.5 Let the equation (2.3.15) satisfy the assumptions (H1 ) – (H3 ), and the functions '.t/, .t/, and ˛.t/ from these assumptions, in their turn, satisfy the assumptions (2.3.30) and (2.3.31). Then, if there exist positive constants ı1 ; ı2 ; ı3 such that for all t 2 T, the following conditions are fulfilled: (1) 1 C .t/  ı1 ; (2) ep .t; 0/  eı2 .t; 0/; (3) ı2 2 RC , where p.t/ D lim

&.t/

.1C /m1 .1C /1 ,

then the solution x D 0 of equation (2.3.15)

is asymptotically stable. Proof Let us show that under conditions of Corollary 2.3.5, all hypotheses of Theorem 2.3.7 are satisfied. Compute h.t/. Using the properties of exponential function, we find h.t/ D

..t//' m .t/˛.t/

m D M m Ce .0; .t//em

.t; 0/e .t; 0/ D M C

em

.t; 0/e .t; 0/ D e ..t/; 0/

D Mm C

em em1 .t; 0/e .t; 0/

.t; 0/e .t; 0/ D Mm C e ..t/; t/e .t; 0/ 1 C .t/

D Mm C

ep .t; 0/ : 1 C .t/

Consider further the function  8  1 0; therefore dˆ d!  0. Consequently, the function ˆ.!/ does not increase on the set RC . Hence ˆ.!/  ˆ.0/ D m  1 for all !  0. Now we estimate the function D.t; a; / Zt D.t; a; / D a

 1 1 lim

&.s/

.1 C h.s/

1 m1 .a/ m1 /

 s m1

Zt D

h.s/

m1

h.s/

m1

.a/m1 lim ˆ. h.s/

&.s/

m1

.a/m1 /s

a

Zt 

.a/m1 .m  1/s

a

Zt D .m  1/

m1

.a/

m1

h.s/s a

Zt D .m 

1/e m1 .0; a/m1

Mm C a

ep .s; 0/ s: 1 C .s/

In view of conditions (1) and (2) of the corollary, we can continue the obtained estimate as follows: Zt D.t; a; /  .m  1/M

m

Ce m1 .0; a/m1 a

D

1  m m m1 M Ce .0; a/m1 ı1 ı2

1 eı .s; 0/s ı1 2

Zt ı2 eı2 .s; 0/s a

Zt   1  m m m1 m1 D eı2 .s; 0/ s M Ce .0; a/ s ı1 ı2 a

 1  m m m1 D M Ce .0; a/m1 eı2 .t; 0/  eı2 .a; 0/ : ı1 ı2

Method of Dynamic Integral Inequalities

73

By the definition of exponential function and due to condition (3) of the corollary, we get the estimate ( Zt jeı2 .t; 0/j D exp 0

) ln.1  ı2 / s  e0 D 1; lim

&.s/

i. e., the function eı2 .t; 0/ is bounded on the set Œ0; C1/ T. Thus, lim D.t; s; / < C1 for all s  t0 and  > 0. Therefore, by Theorem 2.3.7 t!1 the solution x D 0 of the equation (2.3.15) is asymptotically stable. This completes the proof. Note that the conditions of asymptotic stability obtained in Corollary 2.3.5 for zero solution of dynamic equation of certain type cover some known results for T D R.

2.3.4 Exponential stability Now consider the system of equations (cf. Martynyuk-Chernienko [1]) x .t/ D .t/g.t; x.t// C .1  .t//f .t; x.t//;

(2.3.32)

where x 2 Rn , gW T  Rn ! Rn ; f W T  Rn ! Rn . If T D R, then for any t 2 R, .t/ D inf.t; 1/ D t and .t/  0 at all t 2 T. In this case, from (2.3.32) we obtain the nonlinear system of ordinary differential equations xP .t/ D f .t; x.t//;

t 2 R:

(2.3.33)

If T D Z, then for any t 2 Z, .t/ D inf.s 2 ZW s > t/ D inf.tC1,tC2; : : : / D tC1 and .t/  1 at all t 2 T. In this case from the dynamic equation (2.3.32), we obtain the system of difference equations x.t/ D g.t; x.t//;

t 2 Z:

Thus, the equation (2.3.32) is a general form of nonlinear systems describing time-continuous-discrete processes. Rewrite the system of equations (2.3.32) as x .t/ D f .t; x.t// C .t/.g.t; x.t//  f .t; x.t///

(2.3.34)

and assume that the zero solution of the system (2.3.33) is exponentially stable.

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Method of Dynamic Integral Inequalities

Assume that in the system (2.3.34), the vector function f .t; x/ D A.t/x at all t 2 T and x 2 S./ D fx 2 Rn W kxk < g, and for the system x .t/ D A.t/x;

t2T

(2.3.35)

the fundamental matrix ˆA .t; / D ˆA .t; s/ˆA .s; / is known at all  s  t,

; s; t 2 T. Denote the vector function Q.t; x/ D g.t; x/  A.t/x and assume that Q.t; x/ D 0 if and only if x D 0 at all t 2 T. Theorem 2.3.8 Assume that in the system (2.3.34), the vector function f .t; x/ D A.t/x and the following conditions are satisfied: (1) the zero solution of the system (2.3.35) is exponentially stable with the constants q and K; (2) there exists a constant L > 0 such that the vector function Q.t; x/ satisfies the estimate kQ.t; x/k  Lkxk at all .t; x/ 2 T  S./; (3) for the given graininess .t/ > 0 of the time scale T, the inequality q  .t/KL > 0 is satisfied at all t 2 T. Then the state x D 0 of the system (2.3.34) is exponentially stable. Proof Under the conditions (1) and (2) of Theorem 2.3.5 for any solution x.t/ of the system (2.3.34), the representation Zt x.t/ D ˆA .t; /x. / C

ˆA .t; .s//.s/Q.s; x.s//s

(2.3.36)



holds at all t  . From the relation (2.3.36), we obtain the estimate Zt kx.t/k  kˆA .t; /x. /k C

kˆA .t; .s//.s/Q.s; x.s//ks

Zt  Kkx. /keq .t; / C

KLeq .t; .s//.s/kx.s/ks

Zt  Kkx. /keq .t; / C

.s/KL eq .t; s/kx.s/ks: 1  q.s/

(2.3.37)

Method of Dynamic Integral Inequalities

75

As eq1.t; / > 0 at all t  , then, in view of the fact that q 2 RC , we extend the estimate (2.3.37) as follows: kx.t/k  Kkx. /k C eq .t; /

Zt

.s/KL kx.s/k s: 1  q.s/ eq .s; /

(2.3.38)

Applying the Gronwall inequality on a time scale to the estimate (2.3.14), we get kx.t/k  Kkx. /ke .t/KL .t; /; 1q.t/ eq .t; / or kx.t/k  Kkx. /keq .t; /e D Kkx. /keq˚

.t/KL 1q.t/

.t/KL 1q.t/

.t; /

.t; /

(2.3.39)

D Kkx. /keqC.t/KL .t; / at all t  . According to the condition (3) of Theorem 2.3.8, we obtain .q.t/KL/ 2 RC , and therefore from the estimate kx.t/k  Kkx. /ke.q.t/KL/ .t; / it follows that the state x D 0 of the system (2.3.34) is exponentially stable on T. Remark 2.3.2 If the condition (3) of Theorem 2.3.5 is satisfied at 0 < .t/ <  , where  D const < C1, then  is a limiting value of the graininess of the time scale, at which the property of the exponential stability is retained in the system (2.3.34), if it existed in the linear approximation of the system (2.3.32).

2.3.5 Scalar quasilinear equation Now consider the scalar quasilinear equation on a time scale T x .t/ D p.t/x.t/ C f .t; x/;

x.t0 / D x0 ;

(2.3.40)

where t0 2 T and x.t/ D x.t; t0 ; x0 / is a solution of the initial problem (2.3.40). Assume that p.t/ 2 R and f .t; x/W T  R ! R, and uniformly with respect to

76

Method of Dynamic Integral Inequalities

t 2 Œt0 ; 1/ \ T, the following condition is satisfied: f .t; x/ D 0: x!0 x

(2.3.41)

lim

Introduce the notation 8 < log j1 C .t/p.t/j .t/ ˇp .t/ D : p.t/

at .t/ > 0; at .t/ D 0

at all t 2 Œt0 ; 1/ \ T, and write the estimate for the function ep .t; t0 / in the form er.tt0 /  jep .t; t0 /j  eq.tt0 / ;

(2.3.42)

which holds at all t 2 Œt0 ; 1/ \ T under the conditions that p.t/ 2 R, t0 2 T, and r  ˇp .t/  q at all t 2 Œt0 ; 1/ \ T, where r, q are some positive constants. Theorem 2.3.9 Assume that the equation (2.3.40) is defined on a scale T unbounded from above, with the graininess .t/. Let the function p 2 R and let the function f .t; x/ be continuous at all .t; x/ 2 Œt0 ; 1/ \ T  S and satisfy the condition (2.3.41). In addition, there exists a constant M > 0 such that 1 M j1 C .t/p.t/j

at all t 2 Œt0 ; 1/ \ T:

Then: (1) if q D lim sup ˇp .t/ < 0, then the solution x D 0 of the equation (2.3.40) is t!1

exponentially stable; (2) if q D supfˇp .t/W t 2 Œt0 ; 1/ \ Tg < 0, then the solution x D 0 of the equation (2.3.40) is uniformly exponentially stable; (3) if lim inf p.t/ > 0, then the solution x D 0 of the equation (2.3.40) is unstable t!1

on Œt0 ; 1/ \ T. Proof Applying the formula of variation of parameters (1.6.9) to the equation (2.3.40), we obtain Zt x.t/ D ep .t; t0 /x0 C

ep .t; .s//f .s; x.s// s:

(2.3.43)

t0

Under the condition (2.3.41) for the given " > 0, there exists ı > 0 such that jf .t; x/j  "jxj at all t 2 Œt0 ; 1/ \ T and jxj < ı. Let jx0 j < ı, then from the

Method of Dynamic Integral Inequalities

77

relation (2.3.43) for those values t 2 Œt0 ; 1/ \ T, for which jx.t/j < ı, we have Zt jep .t; t0 /x.t/j  jx0 j C

jep ..s/; t0 /j jf .s; x.s//j s t0

Zt  jx0 j C " t0

(2.3.44) 1 jep .s; t0 /j jx.s/j s: j1 C .s/p.s/j

Applying the Gronwall inequality to this estimate, we get jx.t/j  jx0 jep .t; t0 /eg .t; t0 /

(2.3.45)

" . In view of the j1 C .t/p.t/j condition of Theorem 2.3.9, for the function eg .t; t0 /, it is easy to obtain the estimate

at all t 2 Œt0 ; 1/ \ T and at jx.t/j < ı. Here g D

Rt

eg .t; t0 /  e

t0

M" s

D eM".tt0 / :

(2.3.46)

Taking into account the inequality (2.3.46), from (2.3.45), we get the estimate jx.t/j  ep .t; t0 /eM".tt0 / jx0 j:

(2.3.47)

Let q D lim sup ˇp .t/ < q1 < 0, then there exists 2 Œt0 ; 1/ \ T such that t!1

ˇp .t/  q1 < 0 at all t 2 Œ ; 1/ \ T. Define k D estimate (2.3.47), obtain the following inequality jx.t/j  ke.q1 CM"/.tt0 / jx0 j;

jep . ; t0 /j eq1 . t0 /

and from the

(2.3.48)

which is satisfied at all t 2 Œ ; 1/ \ T and at jx.t/j < ı. Since q1 < 0, the value "1 > 0 can be chosen so that q1 C "M < 0 at all 0 < " < "1 . If the initial value jx0 j is chosen smaller than ı, and at all t 2 Œ ; 1/ \ T, the estimate jx.t/j < ı holds, then all the abovementioned estimates are correct, and the inequality (2.3.48) implies the first assertion of Theorem 2.3.9. The second assertion of Theorem 2.3.9 is proved in a similar way. Next we shall prove the instability. Under the conditions (2.3.41) and the condition of the theorem lim inf p.t/ > 0, there exist some values p > 0 and t!1

t1 2 Œt0 ; 1/ \ T such that p.t/  p at t  t1 , and, if 0 < " < p is given, then one can find ı1 > 0 such that jf .t; x/j < "jxj at t 2 Œt0 ; 1/ \ T and jxj < ı1 . Now assume that the solution x D 0 of the equation (2.3.40) is stable on Œt0 ; 1/\ T. In this case there exists ı2 D ı2 .ı1 / > 0 such that 0 < ı2 < ı1 and jx.t/j < ı1 , as soon as jx1 j < ı2 , where x1 D x.t1 ; t0 ; x0 /. Let 0 < x1 < ı2 , then x.t1 / D x1 > 0.

78

Method of Dynamic Integral Inequalities

Assume that x.t/ > 0 at all t 2 Œt1 ; 1/ \ T. If this is not so, then there exists a value t2 2 Œt1 ; 1/ \ T such that x.t2 /  0 and x.t/ > 0

at Œt1 ; t2 / \ T:

(2.3.49)

From the equation (2.3.40), we have x .t/  p.t/x.t/  "jx.t/j  . p.t/  "/x.t/  .p  "/x.t/ at all t 2 Œt1 ; t2 / \ T:

(2.3.50)

Hence follows x.t2 / > 0, which contradicts the inequalities (2.3.49). Consequently, x.t/ > 0 on Œt1 ; 1/\T; and the estimate (2.3.50) holds on Œt1 ; 1/\T. From (2.3.50) it follows that x.t/ D x.t; t1 ; x1 /  ep" .t; t1 /x1 at t 2 Œt1 ; 1/\T, but this contradicts the assumption that jx.t/j  ı1 at all t 2 Œt1 ; 1/ \ T. Thus the instability of the solution x D 0 of the equation (2.3.40) is proved. Corollary 2.3.6 In the equation (2.3.40), let p.t/  p D const, and p 2 R. If the condition (2.3.41) is satisfied and there exists a constant M  > 0 for which 1  M j1 C .t/pj

at all t 2 Œt0 ; 1/ \ T;

then: (1) if lim sup .t/ <  2p , then the solution x D 0 of the equation (2.3.40) is t!1

exponentially stable; (2) if supf.t/W t 2 Œt0 ; 1/ \ Tg <  2p , then the solution x D 0 of the equation (2.3.40) is uniformly exponentially stable.

2.4 Preservation of Stability Under Perturbations 2.4.1 Linear systems under parametric perturbations We shall first consider the system (2.2.2) under parametric perturbations x .t/ D ŒA.t/ C P.t/x.t/;

x.t0 / D x0 ;

(2.4.1)

where A 2 Crd .T; Rnn / and A.t/ is regressive on T. For P.t/  0 and at all t 2 T, let the state x D 0 of the system (2.4.1) be uniformly stable (uniformly exponentially stable). It is of interest to specify the restrictions on the matrix P.t/, under which the state x D 0 of the system (2.4.1) possesses the same type of stability.

Method of Dynamic Integral Inequalities

79

Theorem 2.4.1 Assume that the state x D 0 of the system (2.2.2) is uniformly stable. Then there exists a constant ˇ > 0 such that if Z1 kP.s/k s  ˇ

at all 2 T, then the state x D 0 of the system (2.4.1) is uniformly stable under parametric perturbations. Proof For any t0 2 T and x0 2 Rn , the solution of the system (2.4.1) can be represented in the form Zt x.t/ D ˆA .t; t0 /x0 C

ˆA .t; .s//P.s/x.s/ s;

(2.4.2)

t0

where ˆA .t; t0 / is a fundamental matrix of the system (2.2.2). Since the state x D 0 of the system (2.2.2) is uniformly stable, there exists a constant > 0 such that kˆA .t; t0 /k  at all t; t0 2 T at t  t0 . Taking this into account, from the relation (2.4.2), we obtain the estimate Zt kx.t/k  kx0 k C

kP.s/kkx.s/k s

(2.4.3)

t0

at all t  t0 , t0 2 T. Applying the Gronwall inequality to the estimate (2.4.3), we get the following estimate kx.t/k  kx0 ke kPk .t; t0 / ! Log.1 C .s/ kP.s/k/ s .s/

Zt D kx0 k exp t0

Z1 D kx0 k exp t0

Z1  kx0 k exp

! Log.1 C .s/ kP.s/k/ s .s/ !

kP.s/ks  kx0 k e ˇ

t0

at all t  t0 . The value > 0 can be chosen for any t0 2 T and x0 2 Rn . Consequently, for any " > 0, one can choose ı."/ > 0 so that kx.t/k < " at all

80

Method of Dynamic Integral Inequalities

t  t0 , as soonas kx 0 k < ı."/. In particular, if " > ı."/, then ˇ > 0 can be taken " 1 as ˇ D ln ı."/ . Theorem 2.4.2 Assume that the state x D 0 of the system (2.2.2) is uniformly exponentially stable, i.e., kˆA .t; t0 /k  e .t; t0 / for some constants ; > 0,  2 RC , and  is uniformly regressive on T. Then there exists a constant ˇ  > 0 such that if kP.t/k  ˇ  at all t  t0 , .t; t0 / 2 T, then the state x D 0 of the system (2.4.1) is uniformly exponentially stable. Proof From the formula (2.4.2) at kˆA .t; t0 /k  e .t; t0 /, t; t0 2 T and  2 RC , we obtain the estimate Zt kx.t/k  e .t; t0 /kx0 k D

e .t; .s//kP.s/kkx.s/k s;

t  t0 :

(2.4.4)

t0

In view of the uniform regressivity condition for  , i.e., 0 < ı 1 < 1  .t/ , for some ı > 0 at all t 2 T, the estimate (2.4.4) yields kx.t/k  kx0 ke ˇ ı .t; t0 /e .t; t0 / D e ˚ ˇ ı .t; t0 /;

t  t0 :

(2.4.5)

It is obvious that  ˚ ˇ  ı 2 RC and must be negative at all t 2 T. Note that ˇ  ı > 0 by construction, and this expression is positively regressive. Since RC is a subgroup of the set R, we have  ˚ ˇ  ı 2 RC . The value ˇ  can be calculated from the condition  <  ˚ ˇ  ı < 0 by the formula 0 < ˇ <

ı.1  .t/ /

at all t 2 T. Thus, since was chosen irrespective of t0 2 T and x0 2 Rn , the conclusion of Theorem 2.4.2 follows immediately. Note that the system (2.4.1) is linear, and therefore the property of uniform exponential stability of the state x D 0 of the system (2.4.1) holds in the whole.

2.4.2 Quasilinear dynamic equations Now consider the dynamic equations x .t/ D A.t/x.t/ C f .t; x/;

(2.4.6)

Method of Dynamic Integral Inequalities

81

where A.t/ 2 Crd .T; Rnn /, the vector function f .t; x/W T  Rn ! Rn and is rdcontinuous on T, f .t; x/ D 0 if and only if x D 0 at all t 2 T. For the solution x.t/ of the system (2.4.6), passing through the point .t0 ; x0 / 2 T  S, S D fx 2 Rn W kxk < H, H D const > 0g, the relation Zt x.t/ D ˆA .t; t0 /x0 C

ˆA .t; .s//f .s; x.s// s

(2.4.7)

t0

holds at all t  t0 . For f .t; x/  0 in the system (2.4.6), let the state x D 0 of the system x .t/ D A.t/x.t/;

x.t0 / D x0 ;

(2.4.8)

be exponentially stable, i.e., for the fundamental matrix ˆA .t; t0 / of the system (2.4.8), the following estimate holds: kˆA .t; t0 /k  e .t; t0 /;

t  t0 :

(2.4.9)

We shall establish the conditions under which this type of stability is preserved in the system (2.4.6). Theorem 2.4.3 Assume that for the system (2.4.6), the following conditions are satisfied: (1) the state x D 0 of the system (2.4.8) is exponentially stable; (2) there exists a constant L > 0 such that k f .t; x/k  Lkxk

at all .t; x/ 2 T  SI

(3)  L > 0. Then the state x D 0 of the system (2.4.6) is exponentially stable on T. Proof From (2.4.7) we have Zt kx.t/k  e .t; t0 /kx0 k C t0

L e .t; s/kx.s/k s; 1  .s/

t  t0 :

(2.4.10)

Performing simple transformations of the estimate (2.4.10) and applying the Gronwall inequality, we obtain kx.t/k  kx0 ke .t; t0 /e

L 1 .t/

.t; t0 / D kx0 ke ˚

D kx0 ke C L .t; t0 / D kx0 ke.  L/ .t; t0 /

L 1 .t/

.t; t0 /

82

Method of Dynamic Integral Inequalities

at all t  t0 . Since .  L/ 2 RC , the state x D 0 of the system (2.4.6) is exponentially stable on T. Corollary 2.4.1 For the system (2.4.1), let the following conditions be satisfied: (1) the state x D 0 of the system (2.4.8) is exponentially stable; (2) sup kP.t/k D L < C1; t2T

(3)  L > 0. Then the state x D 0 of the system (2.4.1) is exponentially stable in the whole on T. Now consider the system (2.4.6) under parametric perturbations x .t/ D ŒA.t/ C P.t/x.t/ C f .t; x/;

x.t0 / D x0 ;

(2.4.11)

and establish conditions for preserving the exponential stability of the state x D 0 of this system. Theorem 2.4.4 Assume that for the system (2.4.11), the following conditions are satisfied: (1) the state x D 0 of the system (2.4.8) is uniformly exponentially stable on T; (2) P.t/ 2 Crd .T; Rnn / and there exists a constant ˇ > 0 such that kP.t/k  ˇ at all t 2 T; (3) f .t; x/ 2 Crd .T; Rn / and there exists a constant L > 0 such that k f .t; x/k  Lkxk at all .t; x/ 2 T  S. Then, if the constants ˇ and L are sufficiently small, then there exist and  > 0 . 2 RC / such that for the solution x.t/ of the system (2.4.11), the estimate kx.t/k  kx0 ke  .t; t0 /

(2.4.12)

holds at all t  t0 . Proof For the solution x.t/ of the system (2.4.11), the following relation holds Zt x.t/ D ˆA .t; t0 /x0 C

ˆA .t; .s//ŒP.s/x.s/ C f .s; x.s// s

(2.4.13)

t0

for any t  t0 and x0 2 R. In view of the conditions (1)–(3) of Theorem 2.4.4 and the fact that  is uniformly regressive .0 < ı 1 < 1  .t/ / at all t 2 T, the relation (2.4.13) yields "

Zt

kx.t/k  e .t; t0 / kx0 k C

# ı.ˇ C L/e .t0 ; s/kx.s/k s

t0

(2.4.14)

Method of Dynamic Integral Inequalities

at all t  t0 . For the functions becomes

83

.t/ D e .t0 ; t/kx.t/k, the inequality (2.4.14) Zt

.t/  kx0 k C

ı.ˇ C L/ .s/ s;

t  t0 :

t0

Applying the Gronwall inequality to this estimate, we obtain .t/  kx0 ke ı.ˇCL/ .t; t0 /; hence kxk  kx0 ke ı.ˇCL/ .t; t0 / e .t; t0 / D kx0 ke ˚ ı.ˇCL/ .t; t0 /: To complete the proof, it is necessary to show that  ˚ ı.ˇ C L/ 2 RC

and

 ˚ ı.ˇ C L/ < 0:

Since ı.ˇ C L/ > 0, we have ı.ˇ C L/ 2 RC and since RC is a subgroup of R, we have  ˚ ı.ˇ C L/ 2 RC . Then the condition <  ˚ ı.ˇ C L/ < 0

implies 0 < .1  .t/ / ı.ˇ C L/ < and 0 < ˇ < .1.t//  L. Hence it follows ı that

L >0 .1  .t/ / ı at all t 2 T, i.e., L<

.1  .t/ / ı

at all t 2 T. Thus, the following values of ˇ, L, and  satisfy the conditions of Theorem 2.4.4: 0 0. Theorem 3.1.1 Let the following conditions be satisfied: (1) for the given values t0 2 T, x0 2 Rn and a > 0, Ia D .t0  a; t0 C a/ and S.b/ D fx 2 Rn W kx  x0 k < bg, here inf T  t0  a, sup T  t0 C aI (2) the vector function f W Ia  S.b/ ! Rn is rd-continuous and bounded (the conditions H1 and H4 ); (3) there exists a constant L > 0 such that k f .t; x1 /  f .t; x2 /k  Lkx1  x2 k at all .t; x1 /; .t; x2 / 2 T  S.b/. Then the initial problem (3.1.1) has exactly one solution on Œt0  ˛; t0 C ˛, where ˛ D minfa; Mb ; 1" L g, " > 0. If t0 is right-scattered and ˛ < .t0 /, then the unique solution x.t/ exists over the interval Œt0  ˛; .t0 /. The proof of this proposition is based on the application of the Banach fixed point theorem for the operator Zt F. /.t/ D x0 C

f .s; .s// s;

2 C;

t0

where C is a space of all continuous functions 2 C and g.t/ D f .t; .t// is an rd-continuous function on T. We shall state a result on the global existence of solution to the system (3.1.1) as follows. Theorem 3.1.2 Let the following conditions be satisfied: (1) the vector function f W T  Rn ! Rn satisfies the conditions H1  H2 at all t 2 TI (2) for any pair .t; x/ 2 T  Rn , there exists a neighborhood Ia  S.b/ such that f .t; x/ is bounded by Ia  S.b/ and satisfies the condition k f .t; x1 /  f .t; x2 /k  L.t; x/kx1  x2 k at all .t; x1 /; .t; x2 / 2 Ia  S.b/, L.t; x/ > 0. Then the initial problem (3.1.1) has exactly one solution over a maximal open interval on T.

88

Lyapunov Theory for Dynamic Equations

Corollary 3.1.1 If the conditions (1)–(2) of Theorem 3.1.2 are satisfied and there exist functions p.t/ > 0 and q.t/ > 0 at all t 2 T, such that k f .t; x/k  p.t/kxk C q.t/ at all .t; x/ 2 T  Rn , then any solution of the problem (3.1.1) exists at all t 2 T.

3.2 Auxiliary Functions for Dynamic Equations Consider the system of dynamic equations x D f .t; x.t//;

x.t0 / D x0 ;

(3.2.1)

where x 2 Rn , f 2 Crd .T  Rn ; Rn /, t0 2 T and x .t/ is a -derivative of the state vector x.t/ of the system (3.2.1) on a time scale T. It is assumed that for the system (3.2.1), the conditions are satisfied under which its solution x.t/ D x.t; t0 ; x0 / exists and is unique at all t  t0 , t0 2 T. Auxiliary functions of Lyapunov type play a key role in the study of the behavior of solutions to systems of the form (3.2.1), which cannot be integrated. Consider three classes of auxiliary functions applied for the dynamic equations (3.2.1).

3.2.1 Scalar functions Assume that for the system of dynamic equations (3.2.1), the auxiliary function v.t; x/ 2 Crd .T  Rn ; RC / is constructed by some method. Let v.t; x/ be -differentiable with respect to t 2 T and continuously differentiable with respect to x 2 D  Rn , where D is some open domain in Rn . Applying the formula of -differentiation of a complex function (see Keller [1]), for the total -derivative of the function v.t; x/ along the solutions of the system (3.2.1), we obtain (see also Al-Lian [1]) ˇ ˇ v  .t; x.t//ˇ (3.2.1)

D

vt .t; x..t///

Z1 C

vx0 .t; x.t/ C .t/x .t//dx .t/

0

D

vt .t; x..t///

Z1 C 0

vx0 .t; x.t/ C .t/f .t; x.t/// d  f .t; x.t//:

(3.2.2)

Lyapunov Theory for Dynamic Equations

89

Here vt is calculated as a -derivative of the function v.t/ D v.t; x.t// on a time scale T, and vx0 is an ordinary partial derivative of the function v.t; x/ with respect to x. The scalar function v.t; x/ solving along with the -derivative (3.2.2) the stability problem for the state x D 0 of the system (3.2.1) is called a Lyapunov function for the system of differential equations (3.2.1). Example 3.2.1 (1) For the function v.t; x/ D v.x/ D x2 , x 2 R, and equation x D f .t; x/, f W T  R ! R, the total -derivative is of the form v  .x.t// D 2x.t/f .t; x.t// C .t/f 2 .t; x.t//: (2) For the function v.t; x/ D v.x/ D kxk2 the total -derivative in view of system (3.2.1) is of the form v  .x.t// D 2xT.t/f .t; x.t// C .t/k f .t; x.t//k2 :

3.2.2 Vector functions In some cases, for the system of dynamic equations (3.2.1), it is possible to construct auxiliary functions vj .t; x/, j D 1; 2; : : : ; m, whose totality makes the vector auxiliary function V.t; x/ D .v1 .t; x/; : : : ; vm .t; x//T:

(3.2.3)

The total -derivative of the function (3.2.3) by virtue of the system (3.2.1) is calculated by the formula V  .t; x.t// D .v1 .t; x.t//; : : : ; vm .t; x.t///T;

(3.2.4)

where vj .t; x.t// at each j 2 Œ1; m is calculated by the formula (3.2.2). It is clear that by means of the vector ˛ 2 Rm , the vector functions (3.2.3) and (3.2.4) can be transformed to the scalar functions v.t; x; ˛/ D ˛ TV.t; x/

(3.2.5)

v  .t; x.t/; ˛/ D ˛ TV  .t; x.t//

(3.2.6)

and

respectively. Example 3.2.2 For the function (3.2.3) with vi .t; x/ D vi .xi / D kxi k2 , i D 1; 2; : : : ; m, the total -derivative by virtue of system (3.2.1) is of the form (3.2.4)

90

Lyapunov Theory for Dynamic Equations

with vi .xi .t// D 2xTi .t/fi .t; x/ C .t/k fi .t; x/k2 ; where xi is a subvector of x 2 Rn , i. e., xi 2 Rni and

m P

ni D n, and fi .t; x/W

iD1

T  Rn ! Rni . The vector function V.t; x/ 2 Crd .T  Rn ; Rm / along with the -derivative (3.2.4), solving the stability problem for the state x D 0 of the system (3.2.1), is called the vector Lyapunov function for the system of dynamic equations (3.2.1). The sign definiteness of the vector function (3.2.3) and its -derivative are determined based on the analysis of the functions (3.2.5) and (3.2.6).

3.2.3 Matrix-valued functions A new class of auxiliary functions for the system of dynamic equations (3.2.1) are functions of the form U.t; x/ D Œvij .t; x/;

i; j D 1; 2; : : : ; m;

(3.2.7)

where vii .t; x/ 2 Crd .T  Rn ; RC / at all i D 1; 2; : : : ; m and vij .t; x/ 2 Crd .T  Rn ; R/ at all i ¤ j, i; j 2 Œ1; m. The total -derivative of the function (3.2.7) by virtue of the system (3.2.1) is calculated by the formula U  .t; x.t// D Œvij .t; x.t//;

i; j D 1; 2; : : : ; m;

(3.2.8)

where vij .t; x.t// at all i; j 2 Œ1; m is calculated by the formula (3.2.2). On the basis of the function (3.2.7), the following functions are constructed: v.t; x; / D  TU.t; x/;

 2 Rm C

(3.2.9)

and v  .t; x.t/; / D  TU  .t; x.t//:

(3.2.10)

Let the elements vij .t; x/ of the function (3.2.7) satisfy the conditions: (a) vij .t; x/ are locally Lipschitz with respect to x at all t 2 T; (b) vij .t; x/ D 0 at all t 2 T, if only x D 0; (c) vij .t; x/ D vji .t; x/ at all t 2 T and i; j D 1; : : : ; m. Along with the function (3.2.7), we shall use the scalar function (3.2.9) and the comparison function of K-class. Recall that the real function a.r/ belongs to

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91

K-class (KR-class), if it is defined, continuous, and strictly increasing at 0  r  r1 .0  r < C1/ and a.0/ D 0. Definition 3.2.1 The matrix-valued function (3.2.7) is said to be: (a) positive (negative) semidefinite on T  N, N Rn , if v.t; x; /  0 (v.t; x; /  0) at all .t; x; / 2 T  N  Rm C , respectively; (b) positive definite on T  N; N Rn ; if there exists a function a 2 K-class such that v.t; x; /  a.kxk/ at all .t; x; / 2 T  N  Rm CI (c) decreasing on TN; if there exists a function b 2 K-class, such that v.t; x; /  b.kxk/ at all .t; x; / 2 T  N  Rm CI (d) radially unbounded on T  N; if v.t; x; / ! C1 at kxk ! C1; at .t; x; / 2 T  N  Rm C. Lemma 3.2.1 The matrix-valued function UW T  Rn ! Rmm is positive definite on T if and only if the function (3.2.7) is representable in the form  TU.t; x/ D  TUC .t; x/ C a.kxk/;

t 2 T;

where UC .t; x/ is a positive semidefinite matrix-valued function and a 2 K-class. Lemma 3.2.2 The matrix-valued function UW T  Rn ! Rmm is decreasing on T if and only if the function (3.2.7) is representable in the form  TU.t; x/ D  TU .t; x/ C b.kxk/;

t 2 T;

where U .t; x/ is a negative semidefinite matrix-valued function and b 2 K-class. Example 3.2.3 Let U.t; x/ D U.x/ D xxT. The total -derivative of the function is U  .x.t// D f .t; x/xT.t/ C x.t/f T.t; x/ C .t/f .t; x/f T.t; x/: The matrix-valued function U.t; x/ is a Lyapunov function, if, along with the total -derivative (3.2.8), it solves the stability problem for the state x D 0 of the system (3.2.1) on a time scale. The function (3.2.8) allows construction of the vector Lyapunov function L.t; x; / D AU.t; x/; where A is an m  n–constant matrix and  2 Rm C is a vector with positive components. In this case the total -derivative of the function L.t; x; / by virtue of the system (3.2.1) is calculated by the formula L .t; x.t/; / D AU  .t; x.t//; where U  .t; x/ is calculated componentwise, according to the formula (3.2.2).

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Lyapunov Theory for Dynamic Equations

Remark 3.2.1 If in the matrix-valued function (3.2.7) the elements vij .t; x/  0 at all i ¤ j D 1; 2; : : : ; m, then U.t; x/ is a matrix-valued function on the main diagonal of which the components of a vector Lyapunov function are located.

3.3 Theorems of Stability and Instability 3.3.1 General systems of dynamic equations The actual application of the matrix-valued function (3.2.1) for the stability analysis of the state x D 0 of the system (3.1.1) is associated with the construction of its elements vii .t; x/ 2 Crd .T  Rn ; RC / vij .t; x/ 2 Crd .T  Rn ; R/

for all i D 1; 2; : : : ; mI for all .i ¤ j/ 2 Œ1; m;

which satisfy special estimates. Lemma 3.3.1 If the elements vij .t; /, i; j D 1; 2; : : : ; m, of the matrix-valued function U.t; x/ satisfy the estimates ij

ij .kxi k/ ji .kxj k/

 vij .t; x/  ij

ij .kxi k/ ji .kxj k/;

where ii ; ii > 0 and ij ; ij at .i ¤ j/ 2 Œ1; m are arbitrary constants ij ; ji 2 K.KR/–class, then for the function (3.2.9) the following two-sided estimate holds: T T 1 .kxk/Y GY

1 .kxk/

 v.t; x; / 

T T 2 .kxk/Y GY

2 .kxk/;

1 .kxk/

D.

11 .kx1 k/; : : : ;

1m .kxm k//

T

2 .kxk/

D.

12 .kx1 k/; : : : ;

m2 .kxm k//

T

(3.3.1)

where ; ;

Y D diagŒ1 ; : : : ; m ; G D Œ ij ;

G D Πij ;

i; j D 1; 2; : : : ; m:

The proof of this proposition is carried out via the forward substitution by estimates of the elements vij .t; / into the expression (3.2.2) with further transformations. Estimates of the form (3.3.1) are applied later, when establishing stability conditions for the state of the system (3.4.1).

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93

Theorem 3.3.1 Assume that for the system (3.1.1) there exist: (1) a matrix-valued function UW T  N ! Rmm , N Rn , and a vector  2 Rm C such that the function v.t; x; / D  TU.t; x/ is locally Lipschitz with respect to x at all t 2 T; (2) comparison functions i1 ; i2 ; i3 2 K-class and m  m-matrices Aj , j D 1; 2; such that at all .t; x/ 2 T  N (a) 1T.kxk/A1 1 .kxk/  v.t; x; /I (b) v.t; x; /  2T.kxk/A2 2 .kxk/I (c) there exists an m  m matrix A3 D A3 ..t// such that v  .t; x; / 

T 3 .kxk/A3

3 .kxk/

at all .t; x/ 2 T  NI (d) there exists 0 <  2 M such that 12 ŒAT3 ..t// C A3 ..t//  A3 . / at all 0 < .t/ <  . If the matrices A1 and A2 are positive definite and the matrix A3 D A3 . / is negative semidifinite, then the state x D 0 of the system (3.3.1) is stable under the conditions 2(a) – 2(c) and uniformly stable under the conditions 2(a) – 2(d). Proof From the fact that A1 and A2 are positive-definite matrices, it follows that

m .A1 / > 0 and M .A2 / > 0; where m .A1 / and M .A2 / are the minimal and the maximal eigenvalues of the matrices A1 and A2 , respectively. In view of this fact, the estimates (a), (b) from the condition 2 of Theorem 3.3.1 can be represented in the form

m .A1 / N 1 .kxk/  v.t; x; /  M .A2 / N 2 .kxk/ at all .t; x/ 2 T  N; where N 1 ; N 2 2 K-class such that N 1 .kxk/ 

T 1 .kxk/ 1 .kxk/;

N 2 .kxk/ 

T 2 .kxk/ 2 .kxk/

at all x 2 N. Let " > 0 and supposition S.t/ be as follows: There exists ı D ı."/ > 0 such that the condition kx0 k < ı implies kx.tI t0 ; x0 /k < ": Let S D ft 2 Œt0 ; 1/W S.t/ not trueg: Show that under conditions of Theorem 3.3.1, the set S D ¿. Assume the converse S ¤ ¿. From the fact that S is closed and nonempty, it follows that

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Lyapunov Theory for Dynamic Equations

inf S D t 2 S . Show that the supposition S.t / holds. Let t D t0 ; then S.t0 / holds, since kx.t0 I t0 ; x0 /k < " at kx0 k < "; because x.t0 I t0 ; x0 / D x0 . Now let t ¤ t0 . Show that in this case S.t / holds as well. Indeed, choose ı1 D ı1 ."/ so that M .A2 / N 2 .ı1 / < m .A1 / N 1 ."/: Now let ı D min."; ı1 / be such that kx.t I t0 ; x0 /k  " and kx.tI t0 ; x0 /k < " at t 2 Œt0 ; t / and kx0 k < ı. From the conditions 2(c) and 2(d) of Theorem 3.3.1, we obtain  vC .t; x; /  M .A3 / N 3 .kxk/  0  at all .t; x; / 2 T  N  Rm C . Hence at t D t , we get

m .A1 / N 1 ."/  m .A1 / N 1 .kx.t I t0 ; x0 /k/  v.t ; x.t /; /   v.t0 ; x0 ; / < M .A2 / N 2 .ı/

(3.3.2)

at kx0 k < ı. From the contradiction (3.3.2), it follows that S.t / holds, therefore t 62 S . Consequently, S D ¿. Theorem 3.3.1 is proved. Corollary 3.3.1 (cf. Hoffacker and Tisdell [1]) Let the vector function f in the system (3.2.1) satisfy the assumptions H1 –H4 on T  N, N Rn , and let there exist at least one pair of indices .p; q/ 2 Œ1; m; for which .vpq .t; x/ ¤ 0/ 2 U.t; x/ and the function v.t; x; / D eTU.t; x/e D v.t; x/ at all .t; x/ 2 T  N satisfies the conditions: (a) 1 .kxk/  v.t; x/I (b) v.t; x/  2 .kxk/I (c) v  .t; x/j(3.2.1)  0 at all 0 < .t/ <  2 M; where 1 ; 2 are some functions of K-class. Then the state x D 0 of the system (3.2.1) is stable under the conditions (a) and (c) and uniformly stable under the conditions (a)–(c). Example 3.3.1 Consider the system of dynamic equations on a time scale T with the graininess function .t/:   x D f .x; y/; (3.3.3) y D g.x; y/: where f .0; 0/ D 0; g.0; 0/ D 0 and xf .x; y/yg.x; y/  ˛ 2 < 0 for all .x; y/ 2 R2 . For the function v.x; y/ D x2 C y2 on the time scale T, the total -derivative is of the form ˇ ˇ D 2.x f .x; y/  yg.x; y// C .t/f f 2 .x; y/ C g2 .x; y/g: (3.3.4) v  .x; y/ˇ (3.3.3)

Lyapunov Theory for Dynamic Equations

95

Hence there exists a neighborhood ˇ of x D y D 0 in which the solution .x.t/; y.t// of system 3.3.3 is stable if v  .x; y/ˇ.3:3:3/  0. Theorem 3.3.2 Assume that for the system (3.2.1) there exist: (1) a matrix-valued function UW T  Rn ! Rmm and a vector  2 Rm C such that the function v.t; x; / D  TU.t; x/ is locally Lipschitz with respect to x at all t 2 T; (2) comparison functions i1 ; i2 ; i3 2 K–class and mm matrices Bj , j D 1; 2; 3; such that: (a) 1T.kxk/B1 1 .kxk/  v.t; x; /I (b) v.t; x; /  2T.kxk/B2 2 .kxk/ at all .t; x; / 2 T N  Rm CI (c) an m  m matrix B3 D B3 ..t// such that there exists v  .t; x; / 

T 3 .kxk/B3

3 .kxk/

C w.t;

3 .kxk//

at all .t; x; / 2 TN Rm C , where the function w.t; / satisfies the condition lim

jw.t; 3 .kxk//j D 0 as k k 3 k2

3k

!0

uniformly with respect to t 2 TI (d) 0 <  2 M such that 1 T ŒB ..t// C B3 ..t//  B3 . / 2 3 at any 0 < .t/ <  , then, if the matrices B1 and B2 are positive definite and the matrix B3 D B3 . / is negative definite, then: (a) under the conditions 2(a), 2(c) the state x D 0 of the system (3.2.1) is asymptotically stable on TI (b) under the conditions 2(a)–2(c) the state x D 0 of the system (3.2.1) is uniformly asymptotically stable on T. Proof Consider the supposition fS1 .t/W S.t/ at t 2 Œt0 ; 1/ and lim kx.tI t0 ; x0 /k D 0; t!1

if kx0 k < ı.t0 /g: By the same arguments as in the proof of Theorem 3.3.1, it is easy to prove the assumptions of Theorem 3.3.2. Corollary 3.3.2 (cf. Hoffacker and Tisdell [1]) Let the vector function f in the system (3.2.1) satisfy the assumptions H1 –H4 on T  N, N Rn , and there exists

96

Lyapunov Theory for Dynamic Equations

at least one pair of indices .p; q/ 2 Œ1; m; for which .vpq .t; x/ ¤ 0/ 2 U.t; x/ and the function v.t; x; / D eTU.t; x/e D v.t; x/ at all .t; x/ 2 T  N satisfies the conditions: (a) 1 .kxk/  v.t; x/I (b) v.t; x/  2 .kxk/I (c) at all 0 < .t/ <  2 M the following inequality holds v  .t; x/j(3.2.1)  

3 .kxk/

C m.t;

3 .kxk//

and lim

jm.t;

3 .kxk//j 3 .kxk/

D 0 as

uniformly with respect to t 2 T; where class K.

1;

2;

3

3

!0

are comparison functions of

Then under the conditions (a), (c) the state x D 0 of the system (3.2.1) is asymptotically stable, and under the conditions (a)–(c), the state x D 0 of the system (3.2.1) is uniformly asymptotically stable. Example 3.3.2 Consider the system of dynamic equations (3.3.3). From the expression of total -derivative (3.3.4) of the function v.x; y/ D x2 C y2 , by virtue of system (3.3.3), it follows that outside some neighborhood Q of the state x D y D 0, there exists a value of the graininess  of the time scale T; such that 2˛ 2 C .t/f f 2 .x; y/ C g2 .x; y/g < 0 for all 0 < .t/ <  and all .x; y/ 2 R2 n Q. In this case solutions x.t/, y.t/ of the system (3.3.3) will be attracted to the boundary of domain Q. For a varying graininess of the scale T, the property of solution attraction to the boundary of domain Q is new and does not occur in system (3.3.3) on R. Recall that the two comparison functions 1 ; 2 2 K have the same order of growth if there exist constants ˛i , ˇi , i D 1; 2, such that ˛i

i .r/



j .r/

 ˇi i .r/;

i ¤ j;

i; j D 1; 2:

Definition 3.3.1 The state x D 0 of the dynamic system (3.2.1) is exponentially stable on Œt0 ; 1/, if for the given t1 2 Œt0 ; 1/ there exist constants ı1 > 0, > 0 and K D K.t1 / > 0 such that the condition kx.t1 /k < ı1 implies the estimate kx.tI t1 ; x1 /k  Ke .tt0 / kx.t1 /k at all t 2 Œt1 ; 1/. If K does not depend on t1 , then the state x D 0 of the system (3.2.1) is uniformly exponentially stable.

Lyapunov Theory for Dynamic Equations

97

Before formulating the sufficient conditions for the exponential stability of the state x D 0 of the system (3.2.1), we shall introduce the notation and give the estimate of an exponential function, similar to that found in the paper by Rashford, Siloti, and Wrolstad [1]. Let ˛ > 0, k 2 R and let at all t 2 Œt0 ; 1/ the following function be determined: ( ˇk .t/ D

1 .t/ log j1 C .t/˛k.t/j;

if

.t/ > 0;

˛k.t/;

if

.t/ D 0:

If k 2 R, t0 2 T and there exist constants c1 ,c2 such that c1 < ˇk .t/ < c2 at all t 2 Œt0 ; 1/ \ T, then ec1 .tt0 /  jek .t; t0 /j  ec2 .tt0 /

(3.3.5)

at all t 2 Œt0 ; 1/ \ T. We shall now give a result on the exponential stability of the equilibrium state x D 0 of the system (3.2.1). Theorem 3.3.3 Assume that the vector function f .t; x/ in the system (3.2.1) satisfies the assumptions H1 –H4 on T  N, N Rn . Let there exist (1) a matrix-valued function UW T  N ! Rmm and a vector  2 Rm C such that the function v.t; x; / D  TU.t; x/ is locally Lipschitzian with respect to x at all t 2 TI (2) comparison functions i1 ; i2 ; i3 2 K-class, with i1 and i2 having the same order of growth, m  m-matrices Ai , i D 1; 2; and a constant r > 1 such that at all .t; x/ 2 T  N (a) uTA1 u  v.t; x; / at all .t; x/ 2 T  N, where u D .kxkr=2 ; : : : ; kxkr=2 /T 2 Rm C; (b) v.t; x; /  1T.kxk/A2 1 .kxk/; (3) an m  m matrix A3 D A3 .t/ such that v  .t; x; / 

T 2 .kxk/A3

2 .kxk/

C ˆ.t; v.t; x; //

at all .t; x/ 2 T  NI (4) for the given graininess .t/ of the time scale T, there exist constants ˛ > 0 and M > 0 such that 1 M j1 C .t/˛ M .t/j

at all t 2 Œt0 ; 1/;

where M .t/ D M .A3 .t// is the maximal eigenvalue of the matrix A3 .t/; ˆ.t; v/ D 0 uniformly with respect to t 2 T, in the range of values .t; x/ 2 (5) lim v!0 v T  N.

98

Lyapunov Theory for Dynamic Equations

Then, if the matrices A1 and A2 are positive definite, and if ˛ M .t/ 2 R and lim sup ˇ˛ M .t/ D q < 0, then the equilibrium state x D 0 is exponentially stable, t!1

and if supfˇ˛ M .t/W t 2 Œt0 ; 1/g D q < 0, then it is uniformly exponentially stable on Œt0 ; 1/. Proof We shall first transform the conditions (2) of Theorem 3.3.3 to the form convenient for further computations. From the fact that A1 and A2 are positivedefinite matrices, it follows that m .A1 / > 0 and M .A2 / > 0; where m .A1 / and

M .A2 / are the minimal and the maximal eigenvalues of the matrices A1 and A2 , respectively. Taking this into account, we present the estimates (a), (b) from the condition (2) of Theorem 3.3.3 in the form m m .A1 /kxkr  v.t; x; /  M .A2 / N 1 .kxk/ at all .t; x/ 2 T  N, where N 1 2 K-class, such that N 2 .kxk/  2T.kxk/ all x 2 N. We next present the condition (3) of Theorem 3.3.3 in the form ˇ v  .t; x; /ˇ(3.2.1)  M .t/

2 .kxk/

C ˆ.t; v.t; x; //;

(3.3.6) 2 .kxk/

at

(3.3.7)

1 T ŒA .t/ C A3 .t/ 2 3 T and 2 2 K such that 2 .kxk/  2 .kxk/ 2 .kxk/ at all x 2 N. Since the functions  1 and 2 have the same order of growth, there exist constants ˛ > 0 and ˛ > 0 such that where M .t/ D M .A3 .t// is the maximal eigenvalue of the matrix

˛

1 .kxk/



2 .kxk/

 ˛

1 .kxk/

(3.3.8)

at all x 2 N. Taking into account the estimates (3.3.6) and (3.3.8), we reduce the inequality (3.3.7) to the form v  .t; x; /  ˛ M .t/v.t; x; / C ˆ.t; v.t; x; // at all .t; x/ 2 T  N. Denote k.t/ D ˛ M .t/, v.t/ D v.t; x.t/; /, where x.t/ D x.t; t1 ; x1 / is a solution of the system 2.2.1 at .t1 ; x1 / 2 int .T  N/ and consider the comparison equation v  .t/ D k.t/v.t/ C ˆ.t; v.t//;

(3.3.9)

whose solution v.t/ D v.t; t1 ; v1 / is assumed to be unique for the initial problem (3.3.9), (3.3.10) v.t1 ; x.t1 /; / D v1 :

(3.3.10)

Lyapunov Theory for Dynamic Equations

99

Note that the existence conditions for the unique maximal solution of the problem (3.3.9), (3.3.10) are similar to the assumptions H1 –H3 . Applying the formula of variation of constants, we find the solution of the equation (3.3.9) in the form Zt v.t/ D ek .t; t1 /v1 C

ek .t; . //ˆ. ; v. //  ;

(3.3.11)

t1

where t1 2 Œt0 ; 1/. According to the property of the exponential function, the relation (3.3.11) can easily be transformed as follows: Zt ek .t; t1 /v.t/ D v1 C

ek .. /; t1 /ˆ. ; v. // 

t1

by multiplying the left-hand and the right-hand parts of this relation by the expression ek .t; t1 /. From the condition (3) of Theorem 3.3.3, it follows that for any "1 , there exists  > 0 such that jˆ.t; v/j  "1 jvj at all t 2 Œt0 ; 1/, if jvj < . It is easy to show 1 that for the values of x for which kxk  1 M .A2 / 1 ./, the inequality jv.t/j <  holds at all t 2 Œt1 ; 1/, and then the following estimate is correct: Zt jek .t; t1 /v.t/j  jv1 j C

eg . ; . //jv1 jg. / 

t1

(3.3.12)

Zt D jv1 j C jv1 j

eg .. /; /g. /  ; t1

where g.t/ D

"1 . The inequality (3.3.12) implies j1 C .t/k.t/j

jek .t; t1 /v.t/j  jv1 j  jv1 jŒeg .t; t/  eg .t1 ; t/ D jv1 jeg .t; t1 /: Hence, taking into account (3.3.12), we obtain jv.t/j  jv1 jjek .t; t1 /jeg .t; t1 /;

(3.3.13)

as soon as jv.t/j < . Under the conditions (3.3.5) for the function eg .t; t1 /, we get the estimate Rt

eg .t; t1 / D e

"1 j1C.t/k.t/j

.t; t1 /  e

t1

M"1 

D eM"1 .tt1 / :

(3.3.14)

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Lyapunov Theory for Dynamic Equations

In view of (3.3.13), the inequality (3.3.14) becomes jv.t/j  jek .t; t1 /jeM"1 .tt1 / jv1 j:

(3.3.15)

Let q D lim sup ˇk .t/ < q1 < 0. Then one can find t 2 Œt0 ; 1/ such that t!1

ˇk .t/  q1 < 0 at all t 2 Œt ; 1/. Taking into account the estimate (3.3.14), we transform the inequality (3.3.15) to the form jv.t/j  jek .t; t /jjek .t ; t1 /jeM"1 .tt1 / jv1 j 

 eq1 .tt / jek .t ; t1 /jeM"1 .tt1 / jv1 j: Denote ~.t1 / D

(3.3.16)

jek .t ; t1 /j and rewrite (3.3.16) as follows: eq1 .t t1 /

jv.t/j  ~.t1 /eq1 .tt1 / eM"1 .tt1 / jv1 j  ~.t1 /e.q1 C"1 M/.tt1 / jv1 j;

t 2 Œt1 ; 1/:

(3.3.17)

Taking into account the lower estimate for the function v.t; x; / in the condition (2)(a) of Theorem 3.3.3, we obtain from the inequality (3.3.17) that kx.t/k  Ke

q1 C"1 M .tt1 / r

kx.t1 /k;

t 2 Œt1 ; 1/;

1=r

where K D m1=r m .A1 /~ 1=r .t1 /. Since q1 < 0, we can choose "1 so that D 1 .q1 C"M/r1 < 0 at all 0 < "  "1 . Now if we assume that kx1 k < 1 M .A2 / 1 ./, then kx.t/k  Ke .tt1 / kx1 k;

t 2 Œt1 ; 1/;

(3.3.18)

at all t 2 Œt1 ; 1/. According to Definition 3.3.1, the estimate (3.3.18) for any solution of the system 2.2.1 means the exponential stability of the state x D 0. We now assume that q D supfˇk .t/W t 2 Œt0 ; 1/g < 0. Hence it follows that ˇk .t/  q at all t 2 Œt0 ; 1/ and, taking into account the inequalities (3.3.5), for any t1 2 Œt0 ; 1/, we obtain jek .t; t1 /j  eq.tt1 / at all t 2 Œt1 ; 1/. This estimate, together with the estimate (3.3.14), yields the inequality jv.t/j  ~.t1 /eq.tt1 / eM"1 .tt1 / jv1 j  ~.t1 /e.qC"1 M/.tt1 / jv1 j jek .t ; t1 /j where ~.t1 / D q.t t / uniformly with respect to t1 2 Œt0 ; 1/. 1 e Like in the previous case, we obtain from the estimate (3.3.19) that kx.t/k  Ke .tt1 / kx1 k;

t 2 Œt1 ; 1/;

(3.3.19)

Lyapunov Theory for Dynamic Equations

101

1=r

where K D m1=r M .A1 /~ 1=r .t1 /. Since q < 0, then the value " can be chosen so that D .q1 C "M/r1 < 0 at all 0 < "  "1 . It is obvious that at kx1 k < ı1 we have kx.t/k  Ke .tt1 / kx1 k;

t 2 Œt1 ; 1/;

at all t 2 Œt1 ; 1/ uniformly with respect to t1 . Hence it follows that the state x D 0 of the system 2.2.1 is uniformly exponentially stable on T. Example 3.3.3 Let a system of dynamic equations be given: x 1 D ˛.t/x1 C ˛.t/x2 C ˆ1 .t; x1 ; x2 /; x 2 D ˛.t/x1  ˛.t/x2 C ˆ2 .t; x1 ; x2 /:

(3.3.20)

p   1 1 C 1  2.t/k .t/ , where .t/ is the graininess of the time Here ˛.t/ D 2.t/ scale T, k .t/ 2 R is an rd-continuous function on T. Nonlinear functions are of the form ˆ1 .t; x1 ; x2 / D m.t/x1 .x21 C x22 /, ˆ2 .t; x1 ; x2 / D m.t/x2 .x21 C x22 /, where m.t/ 2 Crd .T/ is a bounded function on T. For the function v.x1 ; x2 / D x21 C x22 with the -derivative v  .t/ D  v .x1 .t/; x2 .t// in the form  2   2 v  .t/ D 2x1 x 1 C .t/.x1 / C 2x2 x2 C .t/.x2 /

it is easy to find that ˇ v  .t/ˇ(3.3.20) D k .t/v.t/ C .2m.t/  2˛.t/.t/m.t//v 2 .t/ C m2 .t/v 3 .t/:

(3.3.21)

Therefore, the equation (3.3.21) becomes v  .t/ D k .t/v.t/ C ˆ .t; .t/; v.t//; where ˆ D .2m.t/  2˛.t/.t/m.t//v 2 .t/ C .t/m2 .t/v 3 .t/. It is easy to see that the condition (5) of Theorem 3.3.3 for the function ˆ .t; v.t// is satisfied. If for the function ˇk .t/ the following conditions are satisfied: q D lim sup ˇk .t/ < 0 .q D supfˇk .t/W t 2 Œt0 ; 1/g < 0/, then the equilibrium state x1 D x2 D 0 of the system (3.3.20) is exponentially stable (uniformly exponentially stable), respectively. Corollary 3.3.3 Let the vector function f .t; x/ in the system (3.2.1) satisfy the assumptions H1 –H4 . Assume that there exist a function vW T  Rn ! RC and comparison functions 1 ; 2 2 K-class, where 1 and 2 has the same order of growth, constants a > 0, r > 1 and a function k.t/ 2 R such that

102

Lyapunov Theory for Dynamic Equations

(1) akxkr ˇ v.t; x/  1 .kxk/ at all .t; x/ 2 T  B, where 0 2 B Rn ; (2) v  .t; x/ˇ(3.2.1)  k.t/ 2 .kxk/ C ˆ.t; .t/; 2 /, where ˆW T  RC  RC ! R, and lim

ˆ.t; .t/;

2 !0

2

2/

D 0 uniformly with respect to t;

(3) there exist constants ˛ > 0, M 2 RC such that at all t 2 Œt0 ; 1/ 1  M: j1 C .t/˛k.t/j Then (a) if ˛k.t/ 2 R and lim sup ˇk .t/ D ˇ < 0, then the state x D 0 of the system of t!1

dynamic equations (3.2.1) is exponentially stable; (b) if ˛k.t/ 2 R and supfˇk .t/W t 2 Œt0 ; 1/g D ˇN < 0, then the state x D 0 of the system (3.2.1) is uniformly exponentially stable. Example 3.3.4 Consider the scalar dynamic equation x .t/ D ˛.t/x.t/ C m.t/x3 ;

(3.3.22)

p  1  1  1  .t/k.t/ , m.t/ 2 Crd .T/. .t/ Along with the equation (3.3.22), consider the function v.x/ D x2 with the -derivative

where ˛.t/ D 

v  .x.t// D 2x.t/x .t/ C .t/.x .t//2 : It is easy to find that ˇ v  .x.t//ˇ(3.3.22) D k.t/v.x.t// C ˆ.t; .t/; v.t//;

(3.3.23)

where ˆ.t; .t/; v.t// D m.t/v 2 .x.t// C .t/m2 .t/v 3 .x.t//. It is obvious that lim

v!0

ˆ.t; .t/; v.t// D0 v.t/

with respect to t 2 T.

In terms of restrictions on the function ( 1 .t/ log j1 C .t/˛.t/j; ˇk .t/ D ˛.t/;

if .t/ > 0; if .t/ D 0;

Lyapunov Theory for Dynamic Equations

103

under the condition 1 M j1 C .t/˛.t/j

at all t 2 Œt0 ; 1/;

where M D const > 0, it is easy to determine the conditions for the exponential stability on T of the state x D 0 of the dynamic equation (3.3.22) on the basis of Corollary 3.3.3. Let D.0; "/ be a sphere of radius " in Rn with center at the point x D 0. Theorem 3.3.4 Assume that for the system (3.2.1): (1) there exist a matrix-valued function UW T  Rn ! Rmm and a vector  2 Rm C such that the function v.t; x; / D  TU.t; x/ is locally Lipschitz with respect to x at all t 2 T; (2) there exist some k < 1 and a set ‚ 2 D.0; "/ such that at all .t; x/ 2 T  ‚: (a) 0 < v.t; x; /  k < 1; (b) there exist 3 2 K and an m  m matrix C3 D C3 ..t// such that v  .t; x; /  3T.kxk/C3 3 .kxk/ at .t; x/ 2 T  ‚; (c) there exists an m  m matrix C3 . /  12 ŒC3T..t// C C3 ..t//,  2 M, Rt t 2 T and M .C .s//s ! C1 as t ! C1; t0

(3) the point x D 0 belongs to the boundary ‚; (4) v.t; x; / D 0 on T  .@‚ \ D.0; "//. Then the state x D 0 of the system (3.2.1) is unstable. Proof The condition (3) implies that for any ı > 0 an x0 2 ‚ \ D.0; "/ is found such that v.t0 ; x0 ; / > 0. We shall estimate the variation of function v.t; x.t/; / along the solution x.t/ D x.t; t0 ; x0 / 2 ‚: Zt k  v.t; x.t/; /  v.t0 ; x0 ; / 

v  .s; x.s/; / s

t0

Zt 

3 .kx0 k/

m .s/s

t0

for all t 2 T. Here 3 2 K1 is such that 3 .kxk/  3T.kxk/ 3 .kxk/ and m .t/ is the minimal eigenvalue of the matrix C3 . /. This inequality contradicts the condition (2)(a) of Theorem 3.3.4, and the solution x.t/ must leave the set ‚ after some instant t 2 T. According to the condition (4) of Theorem 3.3.4, the solution x.t/ cannot leave ‚ through @‚ 2 D.0; "/ and therefore x.t/ leaves the sphere D.0; "/. This proves the instability of the solution x D 0 to the system (3.2.1).

104

Lyapunov Theory for Dynamic Equations

Corollary 3.3.4 (cf. Hoffacker and Tisdell [1]) Let the vector function f in the system (3.2.1) satisfy the assumptions H1 –H4 on T  D.0; "/ and let there exists at least one pair .p; q/ 2 Œ1; m, for which .vpq .t; x/ ¤ 0/ 2 U.t; x/; and the function v.t; x; e/ D eTU.t; x/e D v.t; x/ at all .t; x/ 2 T  N satisfies the conditions: (a) 1 .kxk/  v.t; x/, 1 2 K;  (b) at all 0 < .t/ <  < M the following inequality holds: vC .t; x; /j(3.2.1)  .kxk/, 2 K; 3 3 (c) the point .x D 0/ 2 @‚; (d) v.t; x/ D 0 on T  .@‚ \ D.0; "//. Then the equilibrium state x D 0 of the system (3.2.1) is unstable. Example 3.3.5 Consider the equations of perturbed motion on T with the graininess function 0 < .t/ < C1 x D y.x C y/;

x.t0 / D x0 ;

y D x.x C y/; y.t0 / D y0 :

(3.3.24)

For the function v.x; y/ D x2 C y2 at T D R, we obtain v.x.t/; P y.t// D 0

at all t 2 R

(3.3.25)

and v  .x.t/; y.t//j(3.3.24) D .t/.x C y/2 .x2 C y2 /: From (3.3.25) it follows that x D y D 0 of the system corresponding to (3.3.24) at T D R is stable, while x D y D 0 of the system (3.3.24) is unstable at any graininess function 0 < .t/ < C1. Example 3.3.6 Let a system of dynamic equation be given: x D x  y.x2 C y2 /; y D y C x.x2 C y2 /;

x.t0 / D x0 ; y.t0 / D y0 :

(3.3.26)

For the function v.x; y/ D x2 C y2 at T D R, we have v.x.t/; P y.t// D 2.x2 C y2 /

at all t 2 R

(3.3.27)

and on T v  .x.t/; y.t//j(3.3.26) D 2.x2 C y2 / C .t/Œx2 C y2 C .x2 C y2 /3 :

(3.3.28)

Lyapunov Theory for Dynamic Equations

105

From the analysis of (3.3.27) and (3.3.28), it follows that x D y D 0 of the system corresponding to (3.3.26) at T D R is asymptotically stable. If the scale T has the graininess .t/  1, i.e., T D Z, then at the initial values .x0 ; y0 / from the domain x20 C y20 < 1, the zero solution of the system (3.3.26) will be asymptotically stable on Z. If .t/  2, which corresponds to the time scale T D 2N0 D fk0 ; k0 C 2; k0 C 4; : : : g, then v  .x.t/; y.t//j(3.3.26) D 2.x2 C y2 /3 and the equilibrium state x D y D 0 of the system (3.3.26) is unstable.

3.3.2 Stability of linear systems Linear systems of dynamic equations in the form x .t/ D A.t/x.t/;

(3.3.29)

x.t0 / D x0 ;

(3.3.30)

t0 2 T;

where A 2 R, i.e., the n  n-matrix A is rd-continuous on T and regressive, are very important in the study of dynamic equations of the general form (3.2.1). At the same time note that for the system of dynamic equations (3.2.1), the known Lyapunov theorem of stability by first approximation does not hold. Recall that the matrix A is regressive if the matrix I C .t/A.t/ is invertible at all t 2 T. The class of all such regressive and rd-continuous matrix functions is denoted by R D R.T/ D R.T; Rnn /: The matrix A 2 Crd .T; Rnn / is regressive if and only if its eigenvalues i .t/, 1  i  n, are regressive. Let the matrices A.t/ and AT.t/ be regressive on T. Further the following notation is used: .A ˚ AT/.t/ D A.t/ C AT.t/ C .t/A.t/AT.t/; .A/.t/ D ŒI C .t/A.t/1 A.t/ D A.t/ŒI C .t/A.t/1 ; .A  AT/.t/ D .A ˚ .AT//.t/ at all t 2 T~ . It is easy to show that the sequence .R.T; Rnn /; ˚/ is an Abelian group. Hence it follows that if A; AT 2 R.T; Rnn /, then A ˚ AT 2 R.T; Rnn /.

106

Lyapunov Theory for Dynamic Equations

In order to apply Theorems 3.3.1–3.3.4 for the analysis of stability of the system (3.3.29), we will consider some specific auxiliary functions v.t; x; /. For the total -derivative of the function v.t; x; / D v.t; x/ D xTx by virtue of the system (3.3.29), we obtain the expression ˇ v  .t; x/ˇ(3.3.29) D .xT.t// C xT..t//x .t/ D xT.t/AT.t/x.t/ C xT.t/ŒI C .t/AT.t/A.t/x.t/ D xT.t/ŒAT.t/ C A.t/ C .t/AT.t/A.t/x.t/

(3.3.31)

D xT.t/.A ˚ AT/.t/x.t/: Let B 2 Crd .T; Rnn / be an n  n matrix of the quadratic form xTB.t/x and let B .t/ D B.t/ at t 2 T. For the total -derivative of the auxiliary function v.t; x/ D xTB.t/x by virtue of the system (3.3.29), we get the expression T

ˇ v  .t; x/ˇ(3.3.29) D xT.t/ŒAT.t/B.t/ C .I C .t/AT.t//.B .t/ C B.t/A.t/ C .t/B .t/A.t//x.t/ D xT.t/ŒAT.t/B.t/ C B.t/A.t/ C .t/AT.t/B.t/A.t/

(3.3.32)

C .I C .t/AT.t/B .t/.I C .t/A.t///x.t/: In what follows the expressions (3.3.31) and (3.3.32) are applied to establish stability conditions for the state x D 0 of the system (3.3.29). To analyze the solution x D 0 of system (3.3.29), we shall first apply the function v.t; x/ D xT.t/x.t/. Theorem 3.3.5 Assume that the system (3.3.29) is defined on the time scale T whose graininess .t/ is bounded from above, i.e., .t/ 2 M, where M is a compact set. If the matrix D0 .t; .t// in the expression ˇ v  .t; x.t//ˇ(3.3.29) D xT.t/D0 .t; .t//x.t/; where D0 .t; .t// D .A ˚ AT/.t/, is negative semidefinite (negative definite) at all 0 < .t/   2 M, then the state x D 0 of the system (3.3.29) is stable (asymptotically stable) in the whole on T. Proof For any t0 2 T and x0 2 D.0; "/, we consider the solution x.t/ D x.t; t0 ; x0 / of the system (3.3.29) and the behavior of the function v.t; x/ D xT.t/x.t/ along this solution ˇ v  .t; x/ˇ(3.3.29) D xT.t/.A ˚ AT/.t/x.t/:

Lyapunov Theory for Dynamic Equations

107

Under the conditions of Theorem 3.3.5, we have Zt x .t/x.t/  x .t0 /x.0 / D T

xT.s/.A ˚ AT/.s/x.s/ s  0 for t  t0 :

T

t0

Hence it follows that xT.t/x.t/  xT.t0 /x.t0 / < " for all t  t0 . This proves the stability of the state x D 0. If the matrix D0 .t; .t// is negative definite, then a ˇ > 0 is found such that .A ˚ AT/.t/  ˇI;

where

 ˇ 2 RC ;

and, consequently, xT.t/x.t/  xT.t0 /x.t0 /  ˇxT.t/x.t/ for all t 2 T and t  t0 . This implies that kx.t/k2  kx.t0 /k2 eˇ .t; t0 /;

t  t0 ;

and therefore the state x D 0 is asymptotically stable. Theorem 3.3.6 Assume that for the system (3.3.29) the following conditions are satisfied: (1) there exists an auxiliary function v.t; x/ D xTB.t/x, where B 2 Crd 1 .T; Rnn / such that ˛kx.t/k2  xT.t/B.t/x.t/  ˇkx.t/k2 at all t 2 T, ˛; ˇ > 0 are some constants; (2) there exists  2 M such that the matrix D1 .t; .t// in the expression ˇ v  .t; x/ˇ(3.3.29) D xT.t/D1 .t; .t//x.t/; where D1 .t; .t// D .I C .t/AT.t//B .t/.I C .t/A.t// C C AT.t/B.t/ C B.t/A.t/ C .t/AT.t/B.t/A.t/; is semidefinite negative (definite negative) on T at all 0 < .t/   . Then the state x D 0 of the system (3.3.29) is uniformly stable in the whole (uniformly asymptotically stable in the whole) on T.

108

Lyapunov Theory for Dynamic Equations

Proof Under the conditions (1), (2) of Theorem 3.3.6, for the function v.t; x/ D xTB.t/x.t/, t 2 T, all the conditions of Theorems 3.3.1 and 3.3.2 are satisfied. Hence follows the conclusion of Theorem 3.3.6. Theorem 3.3.7 Assume that for the system (3.3.29) the condition (1) of Theorem 3.3.6 is satisfied and there exists  2 M such that for the function v.t; x/ D xT.t/B.t/x.t/ the inequality ˇ v  .t; x/ˇ(3.3.29)   kx.t/k2 ;

t 2 T;

(3.3.33)

2 RC , then the state ˇ x D 0 of the system (3.3.29) is uniformly asymptotically stable on T. is satisfied at all 0 < .t/   , ˛; ˇ; 2 RC and 

Proof From the estimate (3.3.33) and the condition (1) of Theorem 3.3.6, it follows that ˇ v  .t; x/ˇ(3.3.29) D ŒxT.t/B.t/x.t/   xT.t/B.t/x.t/ ˇ at all t  t0 . Since 

(3.3.34)

2 RC , we find from the estimate (3.3.34) that ˇ v.t; x.t//  v.t0 ; x0 /e ˇ .t; t0 /;

t  t0 :

Hence follows the estimate  1=2 ˇ e .t; t0 / ; kx.t/k  kx.t0 /k ˛ ˇ

t  t0 ;

which holds at any t0 2 T and x.t0 / 2 Rn . Theorem 3.3.7 is proved. Theorem 3.3.8 Assume that for the system (3.3.29) the following conditions are satisfied: (1) there exists an auxiliary function v.t; x/ D xT.t/B.t/x.t/, B.t/ 2 Crd 1 .T; Rnn / such that (a) 0 < v.t; x/  k < 1 for all .t; x/ 2 T  ‚, (b) v.t; x/ D 0 on T  .@‚ \ D.0; "//; (2) there exists  2 M such that ˇ v  .t; x/ˇ(3.3.29) D xT.t/D1 .t; .t//x.t/ for all .t; x/ 2 T  D.0; "/ and 0 < .t/   I

Lyapunov Theory for Dynamic Equations

109

(3) an n  n-matrix D1 .t; .t// is such that D1 . .t// D

1 T .D .t; .t// C D1 .t; .t/// 2 1

has minimal eigenvalue m .D1 . .t/// for which Zt

m .D1 . .s///s ! 1

as t ! 1;

t 2 T:

t0

Then the state x D 0 of the system (3.3.29) is unstable. Proof Condition (1)(b) implies that for any ı > 0, an x0 2 ‚ \ D.0; "/ is found such that v.t0 ; x0 / > 0. We shall consider the behavior of the function v.t; x/ D xT.t/B.t/x.t/ along the solution x.t/ D x.t; t0 ; x0 / of the system (3.3.29). With this in mind, the condition (2) of Theorem 3.3.8 is rewritten as ˇ v  .t; x/ˇ(3.3.29) D xT.t/D1 .t; .t//x.t/ D

 1 T D .t; .t// C D1 .t; .t// 2 1

 m .D1 . .t///xT.t/x.t/: Hence it follows that as long as the solution x.t/ 2 ‚, the following estimate holds: Zt k  v.t; x.t//  v.t0 ; x0 / 

m .D1 . .s///xT.s/x.s/ s

t0

Zt  xT0 x0

m .D1 . .s/// s:

t0

According to the condition (3), the function v.t; x.t// is unboundedly increasing, while the condition (1)(a) assumes its boundedness. Therefore, the solution x.t/ cannot leave the set ‚ through the boundary @‚ 2 D.0; "/ and will leave the sphere D.0; "/. Hence, the solution x D 0 of the system (3.3.29) is unstable. Then for solutions of the initial problem (3.3.29)–(3.3.30), we shall determine the Wazhewsky inequality known in the theory of nonautonomous linear systems of differential equations.

110

Lyapunov Theory for Dynamic Equations

Lemma 3.3.2 (The Wazhewsky inequality) For any solution x.t; t0 ; x0 / of the initial problem (3.3.29)–(3.3.30) at all t 2 Œt0 ; 1/ \ T, the following inequality holds: Rt

kx.t0 /ke

t0

m .s/ s

Rt

 kx.t/k  kx.t0 /ke

t0

M .s/ s

;

(3.3.35)

where k  k is the Euclidean norm of the vector x.t/, m .s/, m .s/ are the smallest and the largest eigenvalues of the Hermitian-symmetrized matrix .AT ˚ A/H D

1 T Œ.A ˚ A/ C .AT ˚ A/T; 2

where AT ˚ A D AT C A C .t/ATA. Proof Consider the function v.x/ D xTx, ( xTx D kxk2 , where k  k is the Euclidean norm) and calculate its -derivative ˇ ˇ v  .x.t//ˇ D .x /Tx..t// C xTx D xT.AT .t/ ˚ A.t//x (3.3.29) (3.3.36) D 2xT.AT ˚ A/H x D 2xT.AT ˚ A/H x: For the symmetrized Hermitian matrix .AT ˚ A/H at any t 2 Œt0 ; 1/ \ T, the following estimates hold:

m .t/xTx  xT.AT ˚ A/x  M .t/xTx:

(3.3.37)

Taking into account (3.3.36), we find from the estimate (3.3.37) that 2 m .t/kxk2  .kxk2 /  2 M .t/kxk2 :

(3.3.38)

Integrating this inequality between the limits from t0 to t, we obtain the two-sided estimate (3.3.35) which holds for the trivial solution x D 0 of the system (3.3.29) as well. From the equality (3.3.36) and the upper bound (3.3.38), it follows that v  .x.t//  2 M .t/v.x.t//;

t 2 Œt0 ; 1/ \ T:

Along with the inequality (3.3.39) consider the scalar comparison equation y .t/ D 2 M .t/y.t/;

y.t1 / D y1 ;

where t1 2 Œt0 ; 1/, and y1 2 RC , where y1 D v.x.t1 //.

(3.3.39)

Lyapunov Theory for Dynamic Equations

111

Let M .t/ 2 R and ( ˇ .t/ D

1 .t/ log j1 C 2.t/ M .t/j; if .t/ > 0; 2 M .t/;

if .t/ D 0;

at all t 2 Œt0 ; 1/. Theorem 3.3.9 Let the time scale T and the system of dynamic equations (3.3.29) be such that: (1) the graininess .t/ of the time scale T is bounded, i.e., 0 < .t/ <  at all t 2 Œt0 ; 1/; 1 (2) there exists a constant M > 0 such that j1C2.t/  M at all t 2 Œt0 ; 1/; M .t/j (3) lim sup ˇ .t/ < 0. ( ˇ .t/  ˇ < 0 at all t 2 Œt0 ; 1/. t!1

Then the zero solution of the system (3.3.29) is exponentially stable (uniformly exponentially stable) on Œt0 ; 1/, respectively. Proof Let v.t; x.t// D xTx, ˆ.t; ; v/ D 0 at all t 2 Œt0 ; 1/. Now consider the inequality (3.3.39) under the condition that M D sup M .t/. t2T

In this case the comparison equation becomes u .t/ D 2 M u.t/;

u.t1 / D u1 :

(3.3.40)

The exponential stability conditions for the comparison equation (3.3.40) are known as follows. Corollary 3.3.5 (cf. Gard and Hoffacker [1]) Let ( ˇ M .t/ D

1 .t/ log j1 C 2.t/ M j; .t/ > 0; 2 M .t/;

.t/ D 0;

where ˇ M .0/ D M and q D lim sup ˇ M .t/ < 0:

(3.3.41)

t!1

Then the zero solution of the equation (3.3.40) is exponentially stable. Corollary 3.3.6 The zero solution of the equation (3.3.29) is exponentially stable if for an arbitrary t0 2 T the following conditions are satisfied: 1 (a) lim sup

!1  t0

Zt

log j1 C 2s M j t < 0I s!.t/ s lim

t0

(b) at any 2 T there exists t 2 T, such that 1 C 2.t/ M D 0.

112

Lyapunov Theory for Dynamic Equations

The inequality v.x.t//  u.tI t1 ; u1 / holds at all t 2 Œt0 ; 1/. Under the conditions (3.3.41) or the conditions (a), (b) of Corollary 3.3.6, owing to the comparison principle, we conclude on the exponential stability of the state x D 0 of the system (3.3.29). We shall now apply a Lyapunov function of the form v.t; x/ D xTB.t/x, where B 2 Crd .T; Rnn /, and use Theorem 3.3.3. Theorem 3.3.10 Let the time scale T and the system of dynamic equations (3.3.29) be such that the following conditions are satisfied: (1) 0 < .t/ <  at all t 2 Œt0 ; 1/; (2) there exists a matrix B 2 Crd 1 .T; Rnn / solving the generalized Lyapunov equation AT.t/B.t/ C .I C .t/AT.t//  .BT.t/ C B.t/A.t/ C .t/B .t/A.t// D C; where C is a given constant symmetric and positive-definite matrix; (3) the matrix B.t/ is symmetric, and the quadratic form xTB.t/x satisfies the estimates

M .t/xTx  xTB.t/x  M .t/xTx at all t 2 Œt0 ; 1/, where m .t/ > 0, M .t/ > 0 are the maximal and the minimal eigenvalues of the matrix B.t/; (4) there exists a constant M > 0 such that 1 M j1 C .t/ .t/j

at all t 2 Œt0 ; 1/;

where .t/ D M .C/ 1 M .B.t//;  (5) lim sup ˇ .t/ D ˇ < 0 ( ˇ .t/  ˇ < 0 at all t 2 Œt0 ; 1/. t!1

Then the zero solution of the system (3.3.29) is exponentially stable (uniformly exponentially stable on Œt0 ; 1//, respectively. Proof Under the conditions (2), (3) of Theorem 3.3.10, it is easy to obtain the estimate v  .t; x.t//   m .C/ 1 M .B.t//v.t; x/ at all .t; x/ 2 RC  Rn and the comparison equation y .t/ D  y.t/;

y.t0 / D y0 :

(3.3.42)

Lyapunov Theory for Dynamic Equations

113

Let ( ˇ .t/ D

1 .t/ log j1 C .t/ .t/j; at .t/ > 0; .t/;

at .t/ D 0:

Applying the arguments from the proof of Theorem 3.3.9 to the comparison equation (3.3.42), we arrive at the conclusion that the state of the system (3.3.29) is exponentially stable. Example 3.3.8 Consider the second-order pseudolinear system of dynamic equations x 1 D a11 .t; x/x1 C a12 .t; x/x2 ; x 2 D a21 .t; x/x1 C a22 .t; x/x2 ;

(3.3.43)

where aij .t; x/ 2 Crd .T; R/, i; j D 1; 2. Assume that there exists an rd-continuous function a.t/, a.t/ > 0 at all t 2 T, such that a11 .t; x/  a.t/; a12 .t; x/  a.t/; a21 .t; x/  a.t/;

(3.3.44)

a22 .t; x/  a.t/ at all .t; x/ 2 T  N. For the function v.x/ D xTx, x 2 R2 ; calculate the -derivative  2   2 v  .x/ D 2x1 x 1 C .t/.x1 / C 2x2 x2 C .t/.x2 / :

Taking into account the condition (3.3.44), for the function v  .x/, we obtain the estimate ˇ v  .x/ˇ(3.3.43)  k.t/v.x/;

(3.3.45)

where k.t/ D 2a.t/.1  .t/a.t//. The comparison equation for the inequality (3.3.45) is y .t/ D k.t/y.t/;

y.t1 / D y1 ;

and the conditions for the exponential stability of the solution y D 0 follow from the conditions of Theorem 3.3.9.

114

Lyapunov Theory for Dynamic Equations

Namely, let there exist M > 0 such that 1 M j1 C .t/k.t/j at all t 2 Œt0 ; 1/ and 8 < 1 Œlog j1 C .t/k.t/j; ˇk .t/ D .t/ : k.t/;

.t/ > 0; .t/ D 0:

If lim sup ˇk .t/ D ˇ < 0, then the state x D 0 of the system (3.3.43) is t!1

exponentially stable, and if supfˇk .t/W t 2 Œt C 0; 1/g D ˇN < 0, then the state x D 0 of the system (3.3.43) is uniformly exponentially stable.

3.4 Existence and Construction of Lyapunov Functions 3.4.1 Converse theorem For the linear nonautonomous system (3.3.29) with exponentially stable solutions, the following result holds. Theorem 3.4.1 Assume that the system (3.3.29) is defined on an unbounded time scale T with the bounded graininess .t/. Let for each pair of values .t0 ; x0 / 2 T  Rn at all t  t0 ; t0 2 T there exists a solution x.t; t0 ; x0 / of the problem (3.3.29)–(3.3.30), for which the following estimate holds: kx.t; t0 ; x0 /k  kec.tt0 / kx0 k;

(3.4.1)

where k > 0 and c is some constant. Then for the system (3.3.29), there exists a Lyapunov function v.t; x/W TRn ! R such that (a) kxk  v.t; x/  kkxk at all .t; x/ 2 T  Rn I (b) jv.t; x1 /  v.t; x2 /j  kkx1  x2 k at any t 2 T and for any pair .x1 ; x2 / 2 Rn I (c) for any fixed .t0 ; x0 / 2 TRn the function v .t; x/ is defined along the solutions x.t/ D x.t; t0 ; x0 / and v .t; x/  c .tv /.t; x/ < 0 at all t  t0 , where the function c .t/ is determined by the formula 8 < exp..t/c/  1 ; .t/ c D : c

at t < .t/I at t D .t/I

Lyapunov Theory for Dynamic Equations

115

(d) the function v.t; x/ is right-continuous with respect to .t; x/ 2 T  Rn , i.e., lim

.Qt;Qx/!.t;x/

jv.Qt; xQ /  v.t; x/j D 0:

Proof For a fixed t 2 T, the following set is introduced: At D f 2 Œ0; 1/W t C 2 Tg:

(3.4.2)

Since 0 2 At , the set At is not empty. Let x.t/W T ! Rn be a solution of the initial problem (3.3.29) with the initial condition x.t/ D y. We introduce the function v.t; x/ D sup kx.t C ; t; y/kec ;

(3.4.3)

2At

which is defined at any .t; x/ 2 T  Rn . By direct calculation it is easy to show that the function (3.4.3) satisfies the conditions (a), (b) of Theorem 3.4.1. In the proof of the property (c) of the function (3.4.3), the two cases are considered: Case A.

Let .t/ D t and h 2 At . Then v .t; x/ D

lim

h!0; h2At

v .t C h/  v .t/ : h

In this case the graininess .t/ D 0; therefore, we have v .t; x/  lim v .t/ h!0

Case B.

ech  1 D v .t/ c .t/ < 0: h

Let .t/ > t, then .t/ ¤ 0 and v .t/ D

follows v .t/  v .t/

(3.4.4)

v ..t//  v .t/ . Hence .t/

ec.t/  1 D v .t/ c .t/ < 0: .t/

(3.4.5)

Collecting the estimates (3.4.4) and (3.4.5), we obtain the property (c) of the function v.t; x/. The property (d) is proved by consideration of two cases: t 2 T is right-scattered and t 2 T is right-dense.

116

Lyapunov Theory for Dynamic Equations

3.4.2 Solution of dynamic Lyapunov equation The relation (3.3.32) shows that the following equation is an analog of the matrix differential Lyapunov equation for the nonautonomous linear system (3.3.29): AT.t/B.t/ C B.t/A.t/ C .t/AT.t/B.t/A.t/ C .I C .t/AT.t//B .t/.I C .t/A.t// D C.t/;

(3.4.6)

where C.t/ is an n  n-symmetric positive-definite matrix. The equation (3.4.6), in which .I C .t/AT.t//B .t/.I C .t/A.t// D 0;

(3.4.7)

will be called a quasistationary Lyapunov equation for the dynamic system (3.3.29). This equation is written as AT.t/B.t/ C B.t/A.t/ C .t/AT.t/B.t/A.t/ D M.t/;

(3.4.8)

where M.t/ is an n  n-symmetric positive-definite matrix. Note that the matrix equation (3.4.8) for T D R becomes AT.t/B.t/ C B.t/A.t/C D M.t/;

M D M T;

(3.4.9)

and for T D Z, it becomes AT.t/B.t/ C B.t/A.t/ C AT.t/B.t/A.t/ D M.t/;

M D M T:

(3.4.10)

The equation (3.4.9) is a particular case of the matrix differential Lyapunov equation (see Zubov [1]) dB C AT.t/B.t/ C B.t/A.t/ D M.t/; dt

M D M T;

for the system (3.3.29) on R and the Lyapunov function v.t; x/ of the form xTB.t/x, x 2 Rn . Theorem 3.4.2 In order that any solution of the system of dynamic equations (3.3.29) be uniformly exponentially stable, it is necessary and sufficient that there exist two quadratic forms v.t; x/ D xT.t/B.t/x.t/; W.x/ D xTMx;

B.t/ 2 Crd .T; Rnn / M D M T;

Lyapunov Theory for Dynamic Equations

117

satisfying the conditions: (1) the function v.t; x/ satisfies the inequalities ˛kx.t/k2  v.t; x/  ˇkxk2 ;

˛; ˇ > 0I

(2) the function W.x/ is a total -derivative of the function v.t; x/ and satisfies the inequalities a1 kxk2  W.x/  a2 kxk2 ;

ai > 0I

(3) the function B.t/ of the form xTB.t/x is a solution of the equation (3.4.8); (4) the -derivative of the n  n-matrix B.t/ satisfies the equation (3.4.7). Proof Necessity. Let the state x D 0 of the system (3.3.29) be uniformly exponentially stable. Then for its solutions, the estimate (3.4.1) holds and, according to Theorem 3.4.1, there exists a function v.t; x/ satisfying the conditions (a)–(d) of this theorem. These conditions imply the conditions (1)–(2) of Theorem 3.4.2. Sufficiency. Under the conditions (1)–(3) of Theorem 3.4.2 for the total derivative of the function v.t; x/ D xT.t/B.t/x.t/, by virtue of the system (3.3.29), we obtain ˇ v  .t; x.t//ˇ(3.3.29) D xTMx;

x 2 Rn ;

and, according to Theorem 3.3.7, the state x D 0 of the system (3.3.29) is uniformly exponentially stable. Theorem 3.4.1 is proved. We now turn to the system (3.3.29) and the auxiliary function v.t; x/ D xTB.t/x, x 2 Rn . From the expression (3.3.32) of the total -derivative ˇ of the function v.t; x/ by virtue of the system (3.3.29), it follows that v  .t; x.t//ˇ(3.3.29) < 0, if AT.t/B.t/ C B.t/A.t/ C .t/AT.t/B.t/A.t/ C .I C .t/AT.t//B .t/.I C .t/A.t// D C.t/;

(3.4.11)

where C.t/ is an n  n-symmetric positive-definite matrix at all t 2 T. The dynamic Lyapunov equation (3.4.11) includes the continuous and the discrete forms of Lyapunov equations for linear systems in the case T D R and T D Z. It is clear that a direct solution of this equation is hard to obtain. At the same time, under some auxiliary assumptions about the system (3.3.29), this equation can be solved.

118

Lyapunov Theory for Dynamic Equations

3.4.3 Lyapunov function for linear periodic system Consider the following linear equations of perturbed motion: x 1 .t/ D A11 .t/x1 .t/ C A12 .t/x2 .t/;

x1 .t0 / D x10 2 Rn1 ;

x 2 .t/ D A21 .t/x1 .t/ C A22 .t/x2 .t/;

x2 .t0 / D x20 2 Rn2 ;

(3.4.12)

where xi .t/ 2 Rni , i D 1; 2, t 2 T, and Aij .t/ are linear matrices of corresponding dimensions. Let the system of perturbed motion equations (3.4.12) be p-periodic i. e., the following conditions are satisfied: (1) for any t 2 T, t C p 2 T; (2) .t/ is p-periodic, i. e., .t C p/ D .t/;

for all t 2 TI

(3) the matrix-valued functions Aij .t/ are p-periodic, i. e., Aij .t C p/ D Aij .t/;

for all t 2 T:

Consider a two-index system of functions (see (3.2.7)) 

 v11 .t; x1 / v12 .t; x1 ; x2 / ; U.t; x1 ; x2 / D v21 .t; x1 ; x2 / v22 .t; x2 / where vij .t; xi / D xTi Pij .t/xj , i; j D 1; 2, Pij .t/ 2 Crd .T; Rni  Rnj / are matrix-valued p-periodic functions with values in Rni  Rnj , Pij D Pji and Pii .t/ are symmetric positive-definite matrices for all t 2 T. Assume that the elements of the matrix-valued function U.t; x1 ; x2 / satisfy the two-sided estimates ci .t/kxi k2  vii .t; xi /  ci .t/kxi k2 ;

i D 1; 2

kPij .t/kkxi kkxj k  vij .t; xi ; xj /

(3.4.13)

 kPij .t/kkxi kkxj k;

i ¤ j;

for all .t; xi ; xj / 2 T  Rni  Rnj . Here, ci .t/ and ci .t/ are positive functions defined for all t 2 T. Along with U.t; x1 ; x2 /, we use the scalar function v.t; x1 ; x2 / D eTU.t; x1 ; x2 /e;

e D .1; 1/T:

(3.4.14)

Lyapunov Theory for Dynamic Equations

119

Estimates (3.4.13) for the elements of U.t; x1 ; x2 / are used to derive a two-sided estimate for v.t; x1 ; x2 /:

m .Q1 .t//uTu  v.t; x1 ; x2 /  M .Q2 .t//uTu; where u D .kx1 k; kx2 k/T,  c1 .t/ kP12 .t/k ; kP12 .t/k c2 .t/   c1 .t/ kP12 .t/k Q2 .t/ D ; kP12 .t/k c2 .t/ 

Q1 .t/ D

and m .Q1 .t// and M .Q2 .t// are the minimum and maximum eigenvalues of Q1 .t/ and Q2 .t/, respectively. Let Si 2 Rni  Rnj be linear symmetric positive-definite matrices. Assume that the functions Pij .t/, i; j D 1; 2 satisfy the system of equations  .I C .t/A11 .t// P 11 .t/.I C .t/A11 .t// C A11 .t/P11 .t/ C P11 .t/A11

C .t/A11 .t/P11 .t/A11 .t/ C .I C .t/A11 .t// P12 .t/A21 .t/ C .I C

A21 .t/P12 .t/.I

C .t/A11 .t// D S1 ;

.t/A22 .t// P 22 .t/.I

C .t/A22 .t// C A22 .t/P22 .t/ C P22 .t/A22

C .t/A22 .t/P22 .t/A22 .t/ C .I C .t/A22 .t// P12 .t/A12 .t/ C

A12 .t/P12 .t/.I

(3.4.15)

(3.4.16)



C .t/A22 .t// D S2

 .I C .t/A11 .t// P 12 .t/.I C .t/A22 .t// C A11 .t/P12 C P12 .t/A22 .t/

C .t/A11 .t/P12 .t/A22 .t/ C A21 .t/P22 .t/ C P11 .t/A12 .t/ C

.t/.A21 .t/P22 .t/A22 .t/

C

A11 .t/P11 .t/A12 .t//

(3.4.17)

D 0;

where S1 and S2 are symmetric positive-definite matrices. Then the generalized total derivative of the scalar function v.t; x1 ; x2 / is ˇ v  .t; x1 ; x2 /ˇ(3.4.12) D .x1 ; .S1  .t/A21 .P22 C P 22 /A21     .t/.I C .t/A11 /P 12 A21  .t/A21 P12 .I C .t/A11 / /x1 /H1    .x2 ; .S2  .t/A12 .P11 C .t/P 11 /A12  .t/A12 P12 .I C .t/A22 /

 .t/.I C .t/A22 / P 12 A12 /x2 /H2   C 2.t/.x1 ; .A21 .P12 C .t/P 12 /A12 C .I C .t/A11 P11 A12

C A21 P 22 .I C .t/A22 //x2 /H1 :

(3.4.18)

120

Lyapunov Theory for Dynamic Equations

Consider the homogeneous equations  .I C .t/A11 .t// P 11 .t/.I C .t/A11 .t// C A11 .t/P11 .t/

C P11 .t/A11 C .t/A11 .t/P11 .t/A11 .t/ D 0;  .I C .t/A22 .t// P 22 .t/.I C .t/A22 .t// C A22 .t/P22 .t/

C P22 .t/A22 C .t/A22 .t/P22 .t/A22 .t/ D 0;  .I C .t/A11 .t// P 12 .t/.I C .t/A22 .t// C A11 .t/P12

C P12 .t/A22 .t/ C .t/A11 .t/P12 .t/A22 .t/ D 0:

(3.4.19)

(3.4.20)

(3.4.21)

Assume that A11 .t/ and A22 .t/ are regressive matrices for any t 2 T. Let t0 2 T. Denote by ˆ11 .t/, ˆ22 .t/, and ˆ12 .t/ the Cauchy matrices of equations (3.4.18)–(3.4.20), respectively. They satisfy ˆ11 .t0 / D I, ˆ22 .t0 / D I, and ˆ12 .t0 / D I, and the matrices .I  ˆij .t0 C p// are invertible. The corresponding Green functions are defined as 8 ˆ ˆij .t/.I  ˆij .t0 C p//1 ˆ1 ˆ ij .. //; ˆ ˆ < t0  < t  t0 C p; ij .t; / D 1 1 ˆ ˆ ˆˆij .t C p/.I  ˆij .t0 C p// ˆij .. //; ˆ : t0  t   t0 C p: Under these assumptions, the p-periodic solutions to equations (3.4.19) and (3.4.20) can be represented as tZ 0 Cp

11 .t; /.P12 . /A21 . /.I C . /A11 . //1

P11 .t/ D  t0

(3.4.22)

C .I C . /A11 . //1 A21 . /P12 . / C .I C . /A11 . //1 S1 .I C . /A11 . //1 / : tZ 0 Cp

22 .t; /.P12 . /A12 . /.I C . /A22 . //1

P22 .t/ D  t0

C .I C . /A22 . //1 A12 . /P12 . / C .I C . /A22 . //1 S2 .I C . /A22 . //1 / :

(3.4.23)

Lyapunov Theory for Dynamic Equations

121

Substituting these expressions into (3.4.17) gives an integrodifferential equation for the off-diagonal element of the matrix-valued Lyapunov function  .I C .t/A11 .t// P 12 .t/.I C .t/A22 .t// C A11 .t/P12

C P12 .t/A22 .t/ C .t/A11 .t/P12 .t/A22 .t/ tZ 0 Cp

 .I C .t/A11 .t//1 A21 .t/22 .t; /

 t0

 .P12 . /A12 . /.I C . /A22 . //1 C .I C . /A22 . //1 A12 . /P12 . /

(3.4.24)

C .I C . /A22 . //1 S2 .I C . /A22 . //1 / C 11 .t; /.P12 . /A21 . / C .I C . /A11 . //1 A21 . /P12 . // C .I C . /A11 . //1 S1 .I C . /A11 . //1   A21 .t/.I C .t/A22 .t//1  D 0: Since we are interested in a p-periodic solution to equation (3.4.24), we use the corresponding Green’s function to find tZ 0 Cp

P12 .t/ D

tZ 0 Cp

 1 I C .s/A11 .s/ A21 .t/22 .s; /

12 .t; s/ t0

t0

  P12 . /A12 . /.I C . /A22 . //1 C .I C . /A22 . //1 A12 . /P12 . / C .I C . /A22 . //1    S2 .I C . /A22 . //1 C 11 .s; / P12 . /A21 . /   1 C .I C . /A11 . //1 A21 . /P12 . / C I C . /A11 . /  1  S1 I C . /A11 . / A21 .s/.I C .s/A22 .s//1  s: Changing the order of integration yields a generalized integral equation for the element of the matrix-valued Lyapunov function tZ 0 Cp

P12 .t/ D f .t/ C

G.t; /P12 . /  ; t0

(3.4.25)

122

Lyapunov Theory for Dynamic Equations

where tZ 0 Cp tZ 0 Cp

Œ12 .t; s/.I C .s/A11 .s//1 A21 .s/22 .s; /

f .t/ D t0

t0

 .I C . /A22 . //1 S2 .I C . /A22 . //1 C 12 .t; s/11 .s; /  .I C . /A11 . //1 S1 .I C . /A11 . //1  A21 .s/.I C .s/A22 .s//1 / s and G.t; /W H12 ! H12 is a linear operator defined by the formula tZ 0 Cp

12 .t; s/Œ.I C .s/A11 .s//1 A21 .s/22 .s; /

G.t; /X D t0

 .X  A12 . /.I C . /A22 . //1 C .I C . /A22 . //1 /  A12 . /X/ C 11 .s; /.XA21 . /.I C . /A11 . //1 C .I C . /A11 . //1 /A21 . /X  /A21 .s/.I C .s/A22 .s//1 s: Thus, the construction of the elements of the matrix-valued Lyapunov function is reduced to solving the (generalized) Fredholm integral equation (3.4.24): if the solution to this equation is known, the diagonal elements of the matrix-valued Lyapunov function are determined by formulas (3.4.22) and (3.4.23). Stability conditions for the trivial solution to system (3.4.12) can be obtained from the general theorems for the generalized Lyapunov direct method. Using estimates for scalar function v.t; x1 ; x2 / and its total derivative along solutions to system (3.4.12), we formulate the following result. Theorem 3.4.3 Assume that the system (3.4.12) is such that the following conditions are satisfied: (1) the functions P11 .t/, P22 .t/, and P12 .t/ defined by (3.4.22), (3.4.23), and (3.4.25) are such that there are constants m > 0 and M > 0 satisfying c11 .t/ > m;

c11 .t/c22 .t/  kP12 .t/k2 > m;

c11 .t/ C c22 .t/ < M; for all t 2 T;

sup kP12 .t/k < M t2T

Lyapunov Theory for Dynamic Equations

123

(2) there are positive constants ˇ1 , ˇ2 , and " such that S1  .t/A21 .t/.P22 .t/ C .t/P 22 .t//A21 .t/  .t/.I C .t/A11 .t//P 12 .t/A21 .t/   .t/A21 .t/P 12 .t/.I C .t/A11 .t//  ˇ1 I;

S2  .t/A12 .t/.P11 .t/ C .t/P 11 .t//A12 .t/  .t/A12 .t/P 12 .t/.I C .t/A22 .t//  .t/.I C .t/A22 .t// P 12 .t/A12 .t/  ˇ2 I; .t/kA21 .t/.P12 .t/ C .t/P 12 .t//A12 .t/ C .I C .t/A11 .t//P 11 .t/A12 .t/ C A21 .t/P 22 .t/.I C .t/A22 .t//k 

p ˇ1 ˇ2  "

for all t 2 T. Then the equilibrium state x1 D 0, x2 D 0 of the system (3.4.12) is asymptotically stable. Proof Under conditions (1) and (2), all the assumptions of Theorem 3.3.2 are satisfied if the constructed function U.t; x1 ; x2 / is used as a matrix-valued function. The negative definiteness of the total -derivative of v.t; x1 ; x2 / along solutions to the system (3.4.12) follows from the Cauchy-Schwarz inequality and assumption (2) of Theorem 3.4.3. Let us present corollaries of Theorem 3.4.3 when T D R or T D Z. Corollary 3.4.1 Let the system (3.4.12) be given on the time scale T D R. If there are positive-definite matrices S1 and S2 such that

m .P11 .t// m .P22 .t//  kP12 .t/k2 > 0; where the matrix P12 .t/ satisfies the integral equation Zp P12 .t/ D f .t/ C

G.t; /P12 . / d ; 0

124

Lyapunov Theory for Dynamic Equations

and the matrices P11 .t/ and P22 .t/ are defined by the formulas Zp P11 .t/ D 

11 .t; /.P12 . /A21 . / C A21 . /P12 . / C S1 /d ;

0

Zp P22 .t/ D 

22 .t; /.P12 . /A12 . / C A12 . /P12 . / C S2 /d ;

0

then the trivial solution of the system (3.4.12) is asymptotically stable. Corollary 3.4.2 Let the system (3.4.12) be given on the time scale T D hZ, h > 0. If there are positive-definite matrices S1 and S2 such that

m .P11 .t// m .P22 .t//  kP12 .t/k2 > 0; D1 D S1  hAT21 .t/.P22 .t/ C hP22 .t//A21 .t/  h.I C hA11 .t//  P12 .t/A21 .t/  hAT21 .t/PT12 .t/.I C hA11 .t//T > 0; D2 D S2  hAT12 .t/.P11 .t/ C hP11 .t//A12 .t/  hAT12 .t/P12 .t/  .I C hA22 .t//  h.I C hA22 .t//T PT12 .t/A12 .t/ > 0; hkA21 .t/.P12 .t/ C hP12 .t//A12 .t/ C .I C hA11 .t//P11 .t/A12 .t/ p C AT21 .t/P22 .t/.I C hA22 .t//k < m .D1 / m .D2 /; where the matrix P12 .t/ satisfies the equation P12 .t/ D f .t/ C

p1 X

G.t; /P12 . /;

D0

and the matrices P11 .t/ and P22 .t/ are defined by the formulas P11 .t/ D 

p1 X

11 .t; /.P12 . /A21 . /.I C hA11 . //1

D0

C .I C hA11 . //1 A21 . /P12 . / C .I C hA11 . //1 S1 .I C hA11 . //1 /;

Lyapunov Theory for Dynamic Equations

P22 .t/ D 

p1 X

125

22 .t; /.P12 . /A12 . /.I C hA22 . //1

D0

C .I C hA22 . //1 A12 . /P12 . / C .I C hA22 . //1 S2 .I C hA22 . //1 /; then the trivial solution of the system (3.4.12) is asymptotically stable.

3.5 Stability Under Structural Perturbations 3.5.1 Description of structural perturbations for dynamic equations Assume that the time scale T is unbounded from above and possesses bounded granularity .t/; :: 0 < .t/ < M .M D const < 1/ for all t 2 T. The object of our investigation is a system of dynamic equations on time scale under structural and parametric perturbations. We designate the class of systems under investigation by D.T/; Di .T/ denotes the i-th subsystem whose totality composes system D.T/. We assume on system D.T/ (on subsystems Di .T/, respectively) as follows: H1 . Subsystems Di .T/ with the state vector xi .t/ 2 Rni ,

m P

ni D n, have a unique

iD1

equilibrium state xi .t/ D 0 for all t 2 T and all i D 1; 2; : : : ; m. H2 . Parametric and/or external perturbations in system D.T/ are characterized by the matrix P D .pT1 ; pT2 ; : : : ; pTm /T 2 Rmq . The set of all admissible matrices P is determined by P D fPW P1  P. /  P2 ; 2 Tg;

(3.5.1)

where P1 and P2 are preassigned constant matrices. The set P can be zero, i. e., P D f0g. In this case parametric and/or external perturbations in system D.T/ are absent. H3 . The family of vector mappings F D f f 1 ; f 2 ; : : : ; f N g has the vector functions f k W T  Rn  Rsq ! Rn as its elements, where k D 1; 2; : : : ; N, so that fi 2 Fi , where Fi D f fi1 ; fi2 ; : : : ; fiN g and fik 2 Crd .T  R  R1q ; Rni / for all k 2 f1; Ng and n D n1 C n2 C    C nm , i D 1; 2; : : : ; m. H4 . Dynamics of the i-th interacting subsystem Di .T/ in system D.T/ is described by the finite dimensional system of dynamic equations x i .t/ D fi .t; x.t/; pi /;

i D 1; 2; : : : ; m;

(3.5.2)

126

Lyapunov Theory for Dynamic Equations

where xi 2 Rni and the symbol  denotes the -derivative of the state vector xi .t/ of subsystem Di .T/, fi .t; 0; 0/ D 0 for all t 2 T. The number N in the definition of the families F and Fi , i D 1; 2; : : : ; m, and the parameter k D k.t/, varying on the set N D f1; 2; : : : ; Ng for all t 2 T, describe structural changes of system D.T/. The number N is the number of all possible structures of system D.T/. H5 . Dynamics of the i-th isolated subsystem in system D.T/ is described by the dynamic equations x i .t/ D gi .t; xi .t//;

i D 1; 2; : : : ; m;

(3.5.3)

where xi 2 Rni , the vector function gi W T  Rni ! Rni and is determined by the correlation gi .t; xi .t// D fi .t; xi ; 0/;

i D 1; 2; : : : ; m;

where xi D .0T; : : : ; 0T; xTi ; 0T; : : : ; 0T/T. It is clear that the functions 'i , specified by the expression 'i .t; x; pi / D fi .t; x; pi /  gi .t; xi /;

t 2 T;

for all i D 1; 2; : : : ; m, describe the action of the whole system D.T/ on the subsystem Di .T/. This can be written by the equations x i .t/ D gi .t; x.t// C 'i .t; x.t/; pi /;

i D 1; 2; : : : ; m:

(3.5.4)

Designate by ˆi a set of all possible 'i , ˆi D f'i1 ; 'i2 ; : : : ; 'iN g, where j D fi .t; x; pi /  gi .t; xi /, j D 1; 2; : : : ; N; i D 1; 2; : : : ; m. For the description of structural changes in system D.T/, we introduce the structural parameter eij W T ! Œ0; 1, which is the .i; j/-element of the structural matrix Ei W T ! Rni Nni corresponding to the i-th interacting subsystem (3.5.2). In general case the matrices Ei are of the form j 'i .t; x; pi /

Ei D Œei1 Ii ; ei2 Ii ; : : : ; eiN Ii ;

Ii D diag .1; 1; : : : ; 1/ 2 Rni ni :

Note that, possibly but not necessarily, if eij .t/ D 1 for all t 2 T, then eik .t/ D 0 for all t 2 T and k ¤ j. Functions ˆi W T  Rn  R1q ! RNni , i D 1; 2; : : : ; m, describe all possible interactions of the subsystem (3.5.4) in system D.T/. This is shown by the system of dynamic equations x i .t/ D gi .t; xi .t// C Ei .t/'i .t; x.t/; pi /;

i D 1; 2; : : : ; m:

(3.5.5)

Lyapunov Theory for Dynamic Equations

127

Let Ei W T ! RnNn be specified by the formula 0 1 E1 .t/ 012 013 : : : 01m B 021 E2 .t/ 023 : : : 02m C C E.t/ D B @ A ::::::::::: 0m1 0m2 0m3 : : : Em .t/ i D 1; 2; : : : ; m;

0ij 2 Rni nj ;

j D 1; 2; : : : ; N:

The matrix E.t/ describes possible structural changes in system D.T/ and is called the structural matrix of the system D.T/ on time scale. The set of all possible matrices E.t/ is designated by Es .t/ and is referred to as the structure of the system D.T/: 8
0) time scale T (its graininess function .t/ D .t/t is such that .tC!/ D .t/ for all t 2 T) and A.t/ D I D fA1 .t/; A2 .t/; : : : ; AN .t/g, where the functions Ak 2 Crd R.T; Rnn / (which belong to the set of rd-continuous functions regressive on T (see Hilger [1])) (k D 1; 2; : : : ; N) are assumed to be periodic with period D p!=q, where p and q are some natural numbers. Let the system (3.5.9) be decomposable into two subsystems of order n1 and n2 (n1 C n2 D n): x 1 D A11 x1 C A12 .t/x2 ; x 2 D A21 .t/x1 C A22 x2 ; where xi 2 Rni (x D .xT1 ; xT2 /T), Aii are constant ni  ni -matrices, and A12 .t/ 2 J1 D fA112 .t/; A212 .t/; : : : ; AN12 .t/g; A21 .t/ 2 J2 D fA121 .t/; A221 .t/; : : : ; AN21 .t/g: The functions h1 and h2 defined by h1 .t/ D ..A112 /T.t/; .A212 /T.t/; : : : ; .AN12 /T.t//T; h2 .t/ D ..A121 /T.t/; .A221 /T.t/; : : : ; .AN21 /T.t//T

(3.5.10)

Lyapunov Theory for Dynamic Equations

129

appear on the right-hand side of the system of dynamic equations x 1 D A11 x1 C S1 .t/h1 .t/x2 ; x 2 D S2 .t/h2 .t/x1 C A22 x2 :

(3.5.11)

The matrix S.t/ expressed as  S1 O12 ; O21 S2

 SD

Oij 2 Rni nj ;

is called the structural matrix of the system (3.5.9), and the set S of all possible matrices S.t/ is called the structural set of the system (3.5.9). If  A11 O12 ; O21 A22

 A0 D

Oij 2 Rni nj ;

h D .hT1 ; hT2 /T;

then the original system can be described as x D A0 x C S.t/h.t/x;

S.t/ 2 S for all t 2 T:

(3.5.12)

We will establish stability conditions for the solution x D 0 of the system (3.5.12) for each S 2 S. To solve the problem posed, we will use Lyapunov’s direct method for dynamic equations to set up a matrix-valued Lyapunov function U.t; x/ D Œvij .t; x/2i;jD1 with the following components for the system (3.5.12) vij .t; x/ D xTi Pij xj ; where Pij are constant square matrices of order nj , and P12 is an n1  n2 -matrix, which is a -differentiable function of t, and P21 D PT12 . The matrix-valued function U.t; x/ can be used to set up a scalar Lyapunov function v.t; x; / D  TU.t; x/;

 2 R2C :

(3.5.13)

The function (3.5.13) makes it possible to establish sufficient stability conditions for the solution x D 0 of the system (3.5.12) (see Theorem 3.5.1 below). To obtain these conditions, we will formulate auxiliary lemmas. Consider a linear homogeneous matrix-valued dynamic equation .In1 C .t/AT1 /X  .t/.In2 C .t/AT2 / C AT1 X.t/ C X.t/A2 C .t/AT1 x.t/A2 D 0;

(3.5.14)

130

Lyapunov Theory for Dynamic Equations

where XW T ! Rn1 n2 , A1 2 Rn1 n1 , and A2 2 Rn2 n2 , and In1 and In2 are unit matrices of orders n1 and n2 , respectively. Lemma 3.5.1 If the matrices A1 and A2 in equation (3.5.14) are regressive, then its solution X.t/ with initial condition X.t0 / D X0 is given by X.t/ D ˆ.t; t0 /X0 ; where the operator ˆ.t; t0 / is defined for all t0 ; t 2 T in the space of n1 n2 -matrices  T 1 ˆ.t; t0 /X D e1 A1 .t; t0 / XeA2 .t; t0 / and e1 Ai .t; t0 /, i D 1; 2, is an exponential function of a system of linear dynamic equations of the ni -th order: x D Ai x. u . Denote by ˛ ˚ ˇ the regressive sum of functions ˛.t/ and Let ~ D  1C.t/u ˇ.t/ .˛.t/ C ˇ.t/ C .t/˛.t/ˇ.t//. The following lemma characterizes the spectrum of the operator ˆ.t; t0 /.

Lemma 3.5.2 For all t  t0 of the scale .ˆ.t; t0 // D fe~.˛j ˚ˇk / .t; t0 / j ˛j 2 .A1 /; ˇk 2 .A2 /g. The following lemma establishes existence conditions for the periodic solution X.t/ of the equation .In1 C .t/AT1 /X  .t/.In2 C .t/AT2 / C AT1 X.t/ C X.t/A2 C .t/AT1 x.t/A2 D F.t/;

(3.5.15)

where F 2 Crd R.T; Rnn / is a -periodic function, A1 and A2 are regressive matrices of orders n1 and n2 , respectively. Denote  D q D p!. Lemma 3.5.3 If at some t0 2 T the inequality e~.˛j ˚ˇk / .t0 C ; t0 / ¤ 1 holds for any ˛j 2 .A1 / and ˇk 2 .A2 /, then equation (3.5.15) has a unique -periodic solution that starts with t0 and is defined by X.t/ D ˆ.t; t0 /.E  ˆ.t0 C ; t0 //

1

tZ 0 C

ˆ.t0 C ; .s//F.s/ s t0

Zt ˆ.t; .s//F.s/ s:

C t0

Lyapunov Theory for Dynamic Equations

131

Before formulating and proving the main theorem, we will introduce some notations: B1 .t; S.t// D 12 .AT11 P11 C P11 A11 C .t/AT11 P11 A11 / C 21 2 .I C .t/AT11 /P12 .t/S2 .t/h2 .t/ C 22 .t/hT2 .t/S2T.t/P22 S2 .t/h2 .t/; B2 .t; S.t// D 12 .t/hT1 .t/S1T.t/P11 S1 .t/h1 .t/ C 21 2 hT1 .t/S1 .t/P12 .t/.I C .t/A22 / C 22 .AT22 P22 C P22 A22 C .t/AT22 P22 A22 /; C.t; S.t// D 12 .I C .t/AT11 /.P11 C PT11 /S1 .t/h1 .t/ C 21 2 ..I C .t/AT11 /.P22 C PT22 /.I C .t/A22 / C AT11 P12 .t/ C P12 .t/A22 C .t/AT11 P12 .t/A22 / C 22 hT2 .t/S2T.t/.P22 C PT22 /.I C .t/A22 /; F.t; S.t// D 

1 .I C .t/AT11 /.P11 C PT11 /S1 .t/h1 .t/ 22

2 T h .t/S2T.t/.P22 C PT22 /.I C .t/A22 /; 21 2  T D.t; S.t// D 21 2 .t/hT2 .t/S2T.t/ P12 S1 .t/h1 .t/; 

K.t; S.t//   kB .t; S.t//  B1 .t; S0 /k 12 kC.t; S.t// C D.t; S.t//k D 1 1 : 2 kC.t; S.t// C D.t; S.t//k kB2 .t; S.t//  B2 .t; S0 /k Let A1 D A11 and A2 D A22 in the expression for ˆ.t; t0 /. Theorem 3.5.1 If there exists a value S0 of the structural matrix S.t/ from the structural set S of system (3.5.9), positive-definite matrices P11 2 Rn1 n1 , P22 2 Rn2 n2 , and real numbers 1 ; 2 ; ı; " > 0 such that the following inequalities hold for all t0 2 T, t  t0 , and S 2 S (1) (2) (3) (4) (1)

e~.˛j ˚ˇk / .t0 C ; t0 / ¤ 1 for all ˛j 2 .A11 /, ˇk 2 .A22 /I 2kP12 .t/k2 C ı  m .P11 / m .P22 /I

M1 D M .B1 .t; S0 // < 0I

M2 D M .B2 .t; S0 // < 0I

M .K.t; S.t/// < maxf M1 ;  M2 g  ",

132

Lyapunov Theory for Dynamic Equations

where P12 .t/ is a -periodic function, P12 .t/ D ˆ.t; t0 /.E  ˆ.t0 C ; t0 //

1

tZ 0 C

ˆ.t0 C ; .t//F. ; S0 / 

t0

Zt ˆ.t; . //F. ; S0 / 

C t0

(its existence is ensured by condition (1) of Theorem 3.5.1 according to Lemma 3.5.3), then the solution x D 0 of system (3.5.12) is asymptotically stable for each S 2 S. Proof Let us show that the function v.t; x; / whose matrices P11 , P22 , and P12 .t/ are selected as prescribed by the theorem satisfies all the conditions of Theorem 4.2 of Bohner and Martynyuk [1], which ensures that the solution x D 0 of the system is asymptotically stable. Condition (2) of Theorem 3.5.1 guarantees that the function v.t; x; / is positive definite for any 1 ; 2 2 RC . Indeed, the following estimate is valid: v.t; x; / D 12 xT1 P11 x1 C 21 2 xT1 P12 .t/x2 C 22 xT2 P22 x2  12 kx1 k2 m .P11 /  21 2 kx1 kkx2 kkP12 .t/k C 12 kx2 k2 m .P22 /: By condition (2) of Theorem 3.5.1, we have r kP12 .t/k  

m .P11 / m .P22 /  ı I 2

therefore, v.t; x; /  12 kx1 k2 m .P11 / p  1 2 kx1 kkx2 k 2. m .P11 / m .P22 /  ı/ C 12 kx2 k2 m .P22 /: The principal minors of the matrix 0 MD@

1 2

 2 .P / q1 m 11

m .P11 / m .P22 /ı 2

1 2

q

1

m .P11 / m .P22 /ı 2 A

22 m .P22 /

of the quadratic form on the right-hand side are positive due to the positive definiteness of the matrix P11 . Therefore, M is a positive-definite matrix according to the Sylvester criterion. Thus, a positive-definite function dependent only on x

Lyapunov Theory for Dynamic Equations

133

and , i. e., a positive-definite function, can be used as a lower-bound estimate for v.t; x; /. Let us find the -derivative of the function v.t; x; / along the solutions of the system (3.5.12). Considering that .fg/ D f  g C f  gD l for any -differentiable functions f and g and using the notation introduced before the statement of the theorem, we obtain the following expression: ˇ v  .t; x; /ˇ(3.5.12) D xT1 B1 .t; S.t//x1 C xT2 B2 .t; S.t//x2 C xT1 .C.t; S.t// C D.t; S.t///x2 ; which can be rearranged into ˇ v  .t; x; /ˇ(3.5.12) D xT1 B1 .t; S0 /x1 C xT2 B2 .t; S0 /x2 C xT1 C.t; S0 /x2 C xT1 .B1 .t; S.t//  B1 .t; S0 //x1 C xT2 .B2 .t; S.t//  B2 .t; S0 //x2 C xT1 .C.t; S.t// C D.t; S.t//  C.t; S0 //x2 : From the expressions for the functions C.t; S.t// and F.t; S.t// and Lemma 3.5.3, we see that the function P12 .t/ is a -periodic solution of the dynamic equation C.t; S0 / D 0 because of which the -derivative of the function v.t; x; / along the solutions of system (3.5.12) becomes ˇ v  .t; x; /ˇ(3.5.12) D xT1 B1 .t; S0 /x1 C xT2 B2 .t; S0 /x2 C xT1 .B1 .t; S.t//  B1 .t; S0 //x1 C xT2 .B2 .t; S.t//  B2 .t; S0 //x2 C xT1 .C.t; S.t// C D.t; S.t///x2 : According to conditions (3)–(5) of Theorem 3.5.1, we have the following upperbound estimate for the right-hand side of the above expression: ˇ v  .t; x; /ˇ(3.5.12)  xT1 B1 .t; S0 /x1 C xT2 B2 .t; S0 /x2 C jxT1 .B1 .t; S.t//  B1 .t; S0 //x1 j C jxT2 .B2 .t; S.t//  B2 .t; S0 //x2 j C jxT1 .C.t; S.t// C D.t; S.t///x2 j  M1 kx1 k2 C M2 kx2 k2 :

134

Lyapunov Theory for Dynamic Equations

Thus, the function v.t; x; / satisfies all the conditions of Theorem 4.2. of Bohner and Martynyuk [1] on asymptotic stability for each S 2 S. Therefore, the solution x D 0 of the system (3.5.12) is asymptotically stable for all S 2 S. The theorem is proved. Setting T D R and T D Z in the condition of Theorem 3.5.1, we can easily establish sufficient stability conditions for linear systems of differential and systems of difference equations with periodic coefficients for all S 2 S. In the former case, the graininess function .t/ D 0 for all t 2 R, the exponential function eA .t; t0 / D T eA.tt0 / , and the operator ˆ.t; t0 / is defined by ˆ.t; t0 /X D eA11 .tt0 / XeA22 .tt0 / and, hence, is dependent only on the difference t  t0 , i. e., ˆ.t; t0 / D ˆ1 .t  t0 /. In the latter case, .t/ D 1 for all t 2 Z, eA .t; t0 / D .I C A/tt0 , and ˆ.t; t0 /X D .I C AT11 /t0 t X.I C A22 /t0 t D ˆ2 .t  t0 /X, where .I C A/r with negative integer exponent denoting the rth power of the matrix .I C A/1 that is inverse of I C A. Thus, we arrive at the following corollaries. Corollary 3.5.1 Let there exist a value S0 of the structural matrix S.t/ from the structural set S of the system of differential equations dx D A.t/x; dt positive-definite matrices P11 2 Rn1 n1 , P22 2 Rn2 n2 , and real numbers 1 , 2 , ı; " > 0 such that the following inequalities hold for all t0 2 R, t  t0 , and S 2 S: 2li for all l 2 Z, ˛j 2 .A11 /, ˇk 2 .A22 /;  2kP12 .t/k2 C ı  m .P11 / m .P22 /;

M1 WD M .B1 .t; S0 // < 0;

M2 WD M .B2 .t; S0 // < 0;

M .K.t; S.t/// < maxf M1 ;  M2 g  ",

(i) ˛j C ˇk ¤ (ii) (iii) (iv) (v)

where P12 .t/ D ˆ1 .t  t0 /.E  ˆ1 .//1 tZ 0 C

eA11 . / F. C t0 ; S0 /eA22 . / d



T

t0

Zt C

eA11 .t / F. ; S0 /eA22 .t / d I T

t0

B1 .t; S.t// D

12 .AT11 P11

C P11 A11 / C 21 2 P12 .t/S2 .t/h2 .t/;

B2 .t; S.t// D 21 2 hT1 .t/S1T.t/P12 .t/ C 22 .AT22 P22 C P22 A22 /;

Lyapunov Theory for Dynamic Equations

135

C.t; S.t// D 12 .P11 C PT11 /.S1 .t/  S10 /h1 .t/ C 22 hT2 .t/.S2 .t/  S20 /T.P22 C PT22 /I 2 T 1 .P11 C PT11 /S1 .t/h1 .t/  h .t/S2T.t/.P22 C PT22 /I 22 21 2 0 1 1 e kC.t; S.t//k C B21 2 kP12 .t/S2 h2 .t/k 2 K.t; S.t// D @ A 1 kC.t; S.t//k 21 2 khT1 .t/e ST1 P12 .t/k 2

F.t; S.t// D 

and e Si D Si .t/  Si0 . Then the solution x D 0 of the system dx D A0 x C S.t/h.t/x; dt

S.t/ 2 S for all t 2 R

is asymptotically stable for all S 2 S. Corollary 3.5.2 Let there exist a value S0 of the structural matrix S.t/ from the structural set S of the system of difference equations x.t/ D A.t/x; positive-definite matrices P11 2 Rn1 n1 , P22 2 Rn2 n2 , and real numbers 1 , 2 , ı; " > 0 such that the following inequalities hold for all t0 2 Z, t  t0 , and S 2 S:   1 C ˛j ˇk (i) ¤ 1 for all ˛j 2 .A11 /, ˇk 2 .A22 /; 1 C ˛j C ˇk C ˛j ˇk (ii) 2kP12 .t/k2 C ı  m .P11 / m .P22 /; (iii) M1 WD M .B1 .t; S0 // < 0; (iv) M2 WD M .B2 .t; S0 // < 0; (v) M .K.t; S.t/// < maxf M1 ;  M2 g  ", where P12 .t/ D ˆ2 .t  t0 /.E  ˆ2 .//1 

1 X

.I C AT11 /  F. C t0 ; S0 /.I C A22 / 

D0

C

t1 X

Dt0

.I C AT11 / t F. ; S0 /.I C A22 / t ;

136

Lyapunov Theory for Dynamic Equations

B1 .t; S.t// D 12 .AT11 P11 C P11 A11 C AT11 P11 A11 / C 21 2 .I C AT11 /P12 .t C 1/S2 .t/h2 .t/ C 22 hT2 .t/S2T.t/P22 S2 .t/h2 .t/; B2 .t; S.t// D 12 hT1 .t/S1T.t/P11 S1 .t/h1 .t/ C 21 2 hT1 .t/S1T.t/P12 .t C 1/.I C A22 / C 22 .AT22 P22 C P22 A22 C AT22 P22 A22 /; C.t; S.t// D 12 .I C AT11 /.P11 C PT11 /.S1 .t/  S10 /h1 .t/ C 22 hT2 .t/.S2 .t/  S20 /T.P22 C PT22 /.I C A22 /; F.t; S.t// D  

1 .I C AT11 /.P11 C PT11 /S1 .t/h1 .t/ 22

2 T h .t/S2T.t/.P22 C PT22 /.I C A22 /; 21 1

D.t; S.t// D 21 2 hT2 .t/S2T.t/PT12 .t C 1/S1 .t/h1 .t/; ˇ2 K.t; S.t// D .Kij .t; S.t///ˇi;jD1 ; and Kii .t; S.t// D kBi .t; S.t//  Bi .t; S0 /k; Kij .t; S.t// D Kji .t; S.t// D

i D 1; 2I

1 kC.t; S.t// C D.t; S.t//k; 2

i ¤ j:

Then the solution x D 0 of the system x.t/ D A0 x C S.t/h.t/x;

S.t/ 2 S for all t 2 T

is asymptotically stable for all S 2 S.

3.6 Polydynamics on Time Scales Combined derivatives on time scales (see Section 1.9) allow to consider complex dynamics of time continuous-discrete systems. In this section a dynamic system with ˙˛ -derivative of the state vector of the system .S/ is considered. On the basis of the alpha-rhombic derivative of the Lyapunov function, the conditions for the stability of some motions of the system .S/ on a time scale are determined.

Lyapunov Theory for Dynamic Equations

137

3.6.1 Problem setting Let the state of some nonlinear system (S) at the instant of time t 2 T be determined by the vector x.t/ 2 Rn , n  1. On the time scale T, we consider the subsets A, B, C, and D. Without loss of generality, we may assume that a 2 A [ D and b 2 B [ D and denote by xP .t/ the Euler derivative, if any, of the vector x with respect to t. Assume that the -dynamics of the nonlinear system (S) on the time scale T is described by the system of equations x .t/ D f .t; x.t//;

x.t0 / D x0 ;

(3.6.1)

where 8 < x..t//  x.t/  .t/ x .t/ D : xP .t/

if t 2 A [ CI at the rest of points;

and x 2 Rn , f W Tk  Rn ! Rn . Assume that the r-dynamics of the nonlinear system (S) on the time scale T is described by the system of equations xr .t/ D g.t; x.t//;

x.t0 / D x0 ;

(3.6.2)

where 8 < x.t/  x..t// r .t/ x .t/ D : xP .t/

if t 2 B [ CI at the rest of points;

and x 2 Rn , gW Tk  Rn ! Rn . The polydynamics of the nonlinear system (S) on the time scale T is described by the system of equations x˙˛ .t/ D F.t; x.t/; ˛/;

x.t0 / D x0 ;

˛ 2 Œ0; 1;

(3.6.3)

where F.t; x.t/; ˛/ D ˛f .t; x/C.1˛/g.t; x/ and x˙˛ .t/ is an ˛-rhombic derivative of the vector x 2 Rn , calculated according to Theorem 1.8.1. In view of x˙˛ .t/ D xP .t/ C O.maxf.t/; .t/g/ at all t 2 R, the system (3.6.3) can be rewritten as xP .t/ D F.t; x.t/; ˛/  O.maxf.t/; .t/g/;

x.t0 / D x0 ;

˛ 2 Œ0; 1:

(3.6.4)

138

Lyapunov Theory for Dynamic Equations

Thus, the problem of polydynamics of the system (S) on a time scale consists in studying the behavior of solutions of the system (3.6.3) or (3.6.4) under different assumptions on the dynamic properties of the systems (3.6.1) and (3.6.2) over the corresponding sets of the time scale T.

3.6.2 Analysis of polydynamics The proposed approach is based on the following generalization of the direct Lyapunov’s method. Namely, the matrix-valued function U.t; x/W T  Rn ! R22 is applied, as well as the function constructed on its basis: v.t; x; / D  TU.t; x/;

(3.6.5)

where  2 R2C is some positive vector. The elements uij .t; / of the matrix-valued function U.t; x/ are constructed as follows: the diagonal elements uii .t; /, i D 1; 2; are related with the subsystems (3.6.1), (3.6.2), respectively, and the extradiagonal element uij .t; /, i ¤ j, is constructed with allowance for the composite dynamics of the system (3.6.3), i.e., this element takes into account the relationship between the “delta” and the “nabla” dynamics of the systems (3.6.1) and (3.6.2). It is assumed that for the function (3.6.5) there exist - and r-derivatives along solutions of the systems (3.6.1) and (3.6.2), which are calculated by the formulas v  .t; x; / D v  .t; x.t/; /j(3.6.1)

at t 2 Tk

(3.6.6)

v r .t; x; / D v r .t; x.t/; /j(3.6.2)

at t 2 Tk :

(3.6.7)

and

Here v  .t; x.t/; / and v r .t; x.t/; / are calculated in conformity with the derivative calculation rules for a complex function on a time scale. The function v ˙˛ .t; x; / D ˛v  .t; x; / C .1  ˛/v r .t; x; /

(3.6.8)

is called an ˛-rhombic derivative of the Lyapunov function v.t; x; / on a time scale if and only if the function (3.6.5) is definite positive and increasing, and v ˙˛ .t; x; /  0 over the set B  Rn at all t 2 Tkk . Thus, the solution of a general problem of polydynamics of the system (3.6.3) on a time scale T is reduced to the construction of an appropriate matrix-valued function and function (3.6.5) and to the adaptation of the direct Lyapunov’s method in terms of the ˛-rhombic derivative (3.6.8) of the Lyapunov function.

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139

3.6.3 Conditions for stability and instability In the subsequent discussion, a comparison function of class K is applied. The function W Œ0; r ! Œ0; 1/ is a comparison function of class K if and only if it is defined, continuous, and strictly increasing over Œ0; r and .0/ D 0. Definition 3.6.1 The zero polydynamics is: (a) uniformly stable on Tkk , if for any " > 0 there exists ı D ı."/ > 0 such that kx.t; t0 ; x0 /k < " at all t 2 Tkk , t > t0 , as soon as kx0 k < ı; (b) uniformly asymptotically stable if it is uniformly stable and there exists a constant > 0 such that lim kx.t; t0 ; x0 /k D 0 at t ! C1 as soon as kx0 k < ; (c) unstable at a fixed t0 , if it is not stable in the sense of Definition 3.6.1(a) at a fixed t0 . The uniform stability conditions for zero polydynamics of the system (3.6.3) are given in the following result. Theorem 3.6.1 Assume that for the systems (3.6.1), (3.6.2) and (3.6.3) a matrixvalued function U.t; x/ and a vector  2 R2C are constructed, such that (1) there exist symmetric constant 2  2-matrices A1 and A2 and vector comparison functions '1 2 K and '2 2 K, for which '1T.kxk/A1 '1 .kxk/  v.t; x; /  '2T.kxk/A2 '2 .kxk/

(3.6.9)

at all .t; x/ 2 Tkk  B; (2) at all .t; x/ 2 Tkk  B the following condition is satisfied: v ˙˛ .t; x; /j(3.6.3)  0

at all ˛ 2 Œ0; 1I

(3) in the inequality (3.6.9), the matrices A1 and A2 are positive definite. Then the zero polydynamics of the system (3.6.3) is uniformly stable. Proof Under the condition (3) of Theorem 3.6.1, the inequality (3.6.9) can be rewritten as

m .A1 /' 1 .kxk/  v.t; x; /  M .A2 /' 2 .kxk/;

(3.6.10)

where m .A1 / > 0 and M .A2 / > 0 are the maximal and the minimal eigenvalues of the matrices A1 and A2 , respectively, the functions ' 1 and ' 2 2 K-class and satisfy the inequalities ' 1 .kxk/  '1T.kxk/'1 .kxk/ at all x 2 D.

and ' 2 .kxk/  '2T.kxk/'2 .kxk/

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From the condition (2) of Theorem 3.6.1, it follows that 8 r ˆ at ˛ D 0I ˆ 0 such that v.t0 ; x0 ; / < M .A2 /' 2 .ı0 /;

(3.6.13)

as soon as kx0 k < ı0 . For any " > 0 choose ı1 > 0 so that

M .A2 /' 2 .ı1 / < m .A1 /' 1 ."/:

(3.6.14)

Calculate ı D minfı0 ; ı1 g and show that if kx0 k < ı, then under the conditions of Theorem 3.6.1, the inequality kx.t; t0 ; x0 /k < " holds at all t  t0 , t 2 Tkk . Let this be not so, then one can find t1  t0 such that kx.t1 ; t0 ; x0 /k D " and kx.t; t0 ; x0 /k < "

(3.6.15)

at all t 2 Œt0 ; t1 /. From the inequalities (3.6.10), (3.6.12)–(3.6.14), we have

m .A1 /' 1 ."/ D m .A1 /' 1 .kx.t1 /k/  v.t; x.t1 /; /  v.t0 ; x0 ; / < M .A2 /' 2 .ı/: The obtained contradiction proves the conclusion of Theorem 3.6.1. Theorem 3.6.2 Assume that (1) for the system (3.6.3) the conditions (1), (3) of Theorem 3.6.1 are satisfied; (2) there exists a symmetric 2  2 matrix D.t/ D D..t/; .t//, the maximal eigenvalue M .t/ of which satisfies the conditions: (a) M .t/ > 0 at all t 2 Tkk ; 0 < .t/ < 1, 0 < .t/ < 1, Rt (b) M .s/˙˛ s ! 1 as t ! 1; t0

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141

(3) there exists a vector comparison function '3 2 K-class, such that v ˙˛ .t; x; /  '3T.kxk/D.t/'3 .kxk/ at all .t; x/ 2 Tkk  B and at all ˛ 2 Œ0; 1. Then the zero polydynamics of the system (3.6.3) is asymptotically stable. Proof From the condition (1) of Theorem 3.6.2, it follows that the function v.t; x; / is definite positive and decreasing. The condition (2)(a) of Theorem 3.6.2 allows the condition (3) of Theorem 3.6.2 to be transformed as v ˙˛ .t; x; /   M .t/' 3 .kxk/;

(3.6.16)

where ' 3 .kxk/  '3T.kxk/'3 .kxk/ at all x 2 B, '3 2 K. Under the conditions of Theorem 3.6.2, all the conditions of Theorem 3.6.1 are satisfied, and therefore the zero polydynamics of the system (3.6.3) is stable. Assume that under the conditions of Theorem 3.6.2 there exists a solution x.t/ D x.t; t0 ; x0 / of the system (3.6.3) such that lim kx.t/k ¤ 0 as t ! C1. There exists a constant a > 0 such that a < kx.t/k < H < C1 at all t 2 Tkk , t  t0 . For the ˛-rhombic derivative of the function (3.6.10), the following fundamental relation holds: Zt v

˙˛

Zt .s; x.s/; /˙˛ s D ˛

t0

v  .s; x.s/; /s

t0

Zt C .1  ˛/

(3.6.17)

v r .s; x.s/; /rs D v.t; x.t/; /  v.t0 ; x0 ; /:

t0

Taking into account the relation (3.6.17), according to the condition (3) of Theorem 3.6.2, we obtain Zt v.t; x.t/; / D v.t0 ; x0 ; / C

v ˙˛ .s; x.s/; /˙˛ s

t0

Zt  v.t0 ; x0 ; / 

M .s/' 3 .kx.s/k/˙˛ s t0

Zt  v.t0 ; x0 ; /  ' 3 .kx0 k/

M .s/˙˛ s: t0

(3.6.18)

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Lyapunov Theory for Dynamic Equations

The function v.t; x; / is positive definite and decreasing, while, according to the estimate (3.6.18), v.t; x.t/; / ! 1 as t ! C1. The obtained contradiction proves that lim kx.t/k D 0 as t ! C1. The theorem is proved. Theorem 3.6.3 Assume that (1) for the system (3.6.3) the conditions (1), (3) of Theorem 3.6.1 are satisfied; (2) there exists a symmetric 2  2 matrix E.t/ D E..t/; .t//, whose maximal eigenvalue M .t/ satisfies the conditions (a) M .t/ > 0 at all t 2 Tkk ; 0 < .t/ < 1, 0 < .t/ < 1; Rt (b) M .s/˙˛ s ! 1 as t ! C1; t0

(3) there exists a vector comparison function '4 2 K-class, such that v ˙˛ .t; x; /  '4T.kxk/E.t/'4 .kxk/

(3.6.19)

at all .t; x/ 2 Tkk  B and at all ˛ 2 Œ0; 1; (4) the point x D 0 belongs to the boundary of the domain G B; (5) the function v.t; x; / D 0 over the set Tkk  .@G \ B" /, where B" D fxW kxk < "g, " > 0  const. Then the zero polydynamics of the system (3.6.3) is unstable. Proof Under the condition (3.6.1) of Theorem 3.6.3, the function v.t; x; / is positive definite and bounded. Therefore for any " > 0, one can find x0 2 @G \ B" and ˛ > 0 such that ˛  v.t0 ; x0 ; / > 0 at all  2 R2C and t0 2 Tkk . From the condition (2)(a), it follows that the inequality (3.6.19) can be represented as v ˙˛ .t; x; /  M .t/' 4 .kxk/

(3.6.20)

at all .t; x/ 2 Tkk B and ˛ 2 Œ0; 1, where ' 4 .kxk/  '4T.kxk/'4 .kxk/, ' 4 2 K-class. Then, while x.t; t0 ; x0 / 2 G; on the basis of (3.6.17), (3.6.20), and the condition 2(b), we get Zt ˛  v.t; x.t/; / D v.t0 ; x0 ; / C

v ˙˛ .s; x.s/; /˙˛ s

t0

Zt  v.t0 ; x0 ; / C ' 4 .kx0 k/

(3.6.21)

M .s/˙˛ s t0

for any .t; x/ 2 Tkk  B and ˛ 2 Œ0; 1. From the inequality (3.6.21), it follows that the solution x.t/ can leave the domain G at some instant of time t 2 Tkk . However, under the condition (5) of Theorem 3.6.1, the solution x.t/ cannot leave

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143

the domain G through the boundary G belonging to B" . Therefore the solution x.t/ will leave the domain G, which means the unstability of the zero polydynamics of the system (3.6.3). The theorem is proved. Example 3.6.1 Consider the following systems of dynamic equations: x .t/ D y.x C y/; y .t/ D x.x C y/

at t 2 A [ C;

(3.6.22)

at t 2 B [ C;

(3.6.23)

and xr .t/ D y.x C y/; yr .t/ D x.x C y/

where x.t0 /; y.t0 / D .x0 ; y0 / 2 R and x C y ¤ 0 for all .x; y/ 2 D  R2 . 2 2 In the domain D  R2 choose the ˇ ˇ function v.x; y/ D x C y and calculate  r ˇ ˇ v .x.t/; y.t// (3.6.22) and v .x.t/; y.t// (3.6.23) . It is easy to show that v  .x.t/; y.t//j(3.6.22) D .t/.x C y/2 .x2 C y2 / > 0 at all .x; y/ 2 D and 0 < .t/ < C1. Hence the zero -dynamics of the system (3.6.22) is unstable. Then we obtain v r .x.t/; y.t//j(3.6.23) D .t/.x C y/2 .x2 C y2 / < 0 at all .x; y/ 2 D and 0 < .t/ < C1. Hence it follows that the zero r-dynamics of the system (3.6.23) is asymptotically stable. Calculating the ˙˛ -derivative of the function v.x; y/, we get v ˙˛ .x.t/; y.t// D .˛.t/  .1  ˛/.t//.x C y/2 .x2 C y2 /: According to Theorem 3.6.2, we find that the zero polydynamics of the systems (3.6.22)–(3.6.23) is asymptotically stable on Tkk if ˛.t/  .1  ˛/.t/ < 0, and unstable if ˛.t/  .1  ˛/.t/  0, at all 0  ˛  1.

3.7 Comments and Bibliography Theorems of the direct Lyapunov method are subject to generalizations and adaptations for many classes of equations of perturbed motion (see Martynyuk [9], Antosiewicz [1], Burton [1, 2], Krasovsky [1], Zubov [1], etc.). In terms of auxiliary matrix-valued functions, a generalization of the direct Lyapunov’s method for time-continuous and discrete-time systems is proposed in the monographs by

144

Lyapunov Theory for Dynamic Equations

Martynuyk [1–3] (see also Djordjevic [1]). Some general theorems of the direct Lyapunov method for dynamic equations are given in Martynyuk-Chernienko [1–3] and Bohner and Martynyuk [1, 2] (see also Hoffacker and Tisdell [1], Peterson and Raffoul [1], Peterson and Tisdell [1] and others.) Section 3.1. Theorems 3.1.1, 3.1.2 and Corollary 3.1.1 are formulated and proved in Bohner and Peterson [1]. Section 3.2. The results discussed in this section are new and similar to the results that are well known for ordinary differential equations (see Martynyuk [1–3]). Section 3.3. Lemma 3.3.1 is given according to the monograph by Martynyuk [11]. Theorems 3.3.1, 3.3.2, and 3.3.4 are given in Martynyuk-Chernienko [1, 2] and Bohner and Martynyuk [1, 2]. Theorem 3.3.3 is taken from Martynyuk [7]. Corollary 3.3.3 is due to Martynyuk-Chernienko and Chernetskaya [1]. Theorems 3.3.5, 3.3.6 are given in Bohner and Martynyuk [1]. Theorems 3.3.7, 3.3.8 are new (cf. DaCunha [1]). Lemma 3.3.2 and Theorems 3.3.9, 3.3.10 are new. Section 3.4. As is well known (see Krasovsky [1], Yoshizawa [1] and Zubov [1]), the converse theorems are of great theoretical importance for the stability theory in the Lyapunov sense, and they are of the same significance for dynamic equations on time scales. Theorem 3.4.1 is formulated and proved in Kloeden and Zmorzynska [1]. In our study an abridged proof of the theorem is only given. The results of Subsections 3.4.2 and 3.4.3 rest on the papers by Martynyuk and Slyn’ko [1], Bohner and Martynyuk [2], and Babenko [1]. Section 3.5. This section is based on the results by Babenko [1] and Martynyuk [10]. Oscillatory systems of (3.5.9) type with invariable structure in continuous case were studied by Lila and Martynyuk [1]. Section 3.6. This section is based on the papers by Martynyuk [4–6]. The key idea of the approach is the application of the diamond ˛-rhombic derivative of the Lyapunov function for a combined dynamic system on a time scale. Like in the classical stability theory, in the theory of stability of dynamic equations, the key element of the investigation technique is the construction of an appropriate Lyapunov function for the equation (class of equations) on a time scale. This problem, in a general case, remains open. For various results of stability of the homogeneous first-order linear dynamic equations on time scales, see Hamza and Oraby [1], Bartosiewicz and Pawluszewicz [1], etc.

Chapter 4

COMPARISON METHOD

4.0 Introduction The comparison method in the dynamics of continuous and discrete systems is developed quite well, and the main results are described in a number of generalizing monographs. As is well known, the main idea of the comparison method is the substitution of the initial system for a lower order system such that the dynamic properties of the comparison equation (system) imply the corresponding dynamic properties of the initial system. Dynamic equations on time scales compose a new class of systems of equations, for which the development of the comparison method is both of theoretical and practical interest. In this chapter the stability of dynamic equations is studied in terms of multicomponent functions and the corresponding comparison equations (systems). The results of this chapter are given as follows. Section 4.1 contains the basic theorems of the comparison method for scalar and vector Lyapunov functions constructed on the basis of a matrix-valued auxiliary function. In Section 4.2 a method for obtaining the stability conditions on the basis of the comparison method is developed. Here the conditions for stability, uniform stability, and uniform asymptotic stability of motion are set out. Section 4.3 presents the results of stability analysis of conditionally invariant sets for dynamic equations. Besides, it is the first time when a two-component Lyapunov function is used; the function components account for the different dynamic properties of system solutions in different domains of the phase space. Section 4.4 provides the analysis of the stability with respect to two measures. In terms of the existence of matrix-valued functions and special majorants of total -derivative of a Lyapunov function, the sufficient conditions for the stability on time scales are established. In the final section, Section 4.5, the stability of a dynamic graph is studied by the method of matrix Lyapunov functions and comparison principle. © Springer International Publishing Switzerland 2016 A.A. Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42213-8_4

145

146

4.1

Comparison Method

Theorems of the Comparison Method

Consider the system of dynamic equations x .t/ D f .t; x.t//;

x.t0 / D x0 ;

(4.1.1)

where F.t/ D f .t; x.t// 2 Crd .T/ and x .t/ is the -derivative of the state vector x.t/ 2 Rn of the system (4.1.1) at the instant of time t 2 T. We shall formulate a theorem of the comparison principle for the class of scalar auxiliary functions described in Section 3.2. Theorem 4.1.1 For the system (4.1.1) assume that there exists a scalar function with respect to x at v.t; x/ 2 Crd .T  Rn ; RC /, v.t; x/ is locally Lipschitzian ˇ any t 2 T, and for the total -derivative v  .t; x.t//ˇ(4.1.1) , there exists a majorant g.t; v.t; x//, g.t; u/ 2 Crd .T  RC ; R/ satisfying the conditions .1/ g.t; u1 /  g.t; u2 / at all t 2 T as soon as u1  u2 ; .2/ v  .t; x.t//  g.t; v.t; x.t/// at all t 2 T n ft0 g. Now assume that the maximal solution r.t/ D r.tI t0 ; u0 / of the comparison inequality u .t/  g.t; u.t//;

u.t0 / D u0  0

(4.1.2)

exists at all t  t0 . Then along solutions of the system (4.1.1), the following estimate holds: v.t; x.t// < r.t/

at all t  t0

(4.1.3)

as soon as v.t0 ; x0 / < u0 : Proof We apply the induction principle on a time scale T (see Theorem 1.1.1). Let the statement S.t/ be defined as follows: S.t/W v.t; x.t// D m.t/ < r.t/

at all t 2 Œt0 ; 1/:

I. It is clear that S.t0 / holds, since, according to the condition of the theorem, v.t0 ; x0 / < u0 and r.t0 ; t0 ; u0 / D u0 . II. Let the values t  t0 be right-scattered, and let the statement S.t/ hold. From the condition (2) of Theorem 4.1.1, we obtain m .t/  g.t; m.t//  g.t; u.t//  u .t/; since m.t/ < u.t/ implies g.t; m.t//  g.t; u.t// in view of the condition (1) of Theorem 4.1.1.

Comparison Method

147

Then m..t// D m.t/ C .t/m .t/ < r.t/ C .t/r .t/ D r..t// and therefore the statement S..t// holds. III. Let the values t  t0 be left-dense and let the statement S. / hold at all 2 Œt0 ; t/. Then we have m. / < r. / at all t0  < t and, by continuity, m.t/ < r.t/. Hence .r  m/  0 on the interval Œt0 ; t, and the difference m  r does not decrease over Œt0 ; t. Therefore .r  m/.t/  .r  m/.t0 / > 0, which leads to the conclusion on the correctness of the statement S.t/. IV. Let the values t  t0 be right-dense and let the statement S.t/ hold. Since m.t/ < r.t/, it follows by continuity that there exists a neighborhood W such that m. / < r. / at all t 2 W and therefore S. / holds at all 2 W \ .t; 1/. Theorem 4.1.1 is proved. We shall now formulate a result of the comparison principle with a vector auxiliary function. Theorem 4.1.2 For the system (4.1.1) assume that there exists a vector function V.t; x/ 2 Crd .T  Rn ; Rm C /, V.t; x/ is locally Lipschitz with respect to x, and a m majorant G.t; V/ 2 Crd .TRm C ; R / of the total -derivative of the function V.t; x/ along solutions of the system (4.1.1), such that: .1/ ui C .t/Gi .t; u/  wi C .t/Gi .t; w/ as soon as u  w, ui D wi and i 2 Œ1; m; .2/ V  .t; x.t//  G.t; V.t; x.t/// at all t 2 T n ft0 g, t0 2 T; .3/ the maximal solution R.t/ D R.t; t0 ; W0 / of the system of dynamic inequalities W  .t/  G.t; W.t//;

W.t0 / D W0  0

exists at all t  t0 . Then along solutions of the system (4.1.1), the following estimate is true: V.t; x.t// < R.t/

at all t  t0

(4.1.4)

as soon as V.t0 ; x0 / < W0 . The estimate (4.1.4) holds componentwise. Proof Denote M.t/ D V.t; x.t//, MW T ! Rm C and apply Theorem 1.1.1 for the statement S.t/W M.t/ < R.t/

at all t 2 Œt0 ; 1/:

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Comparison Method

I. The statement S.t/ holds at the point t D t0 , by the condition of Theorem 4.1.2, therefore M.t0 / < R.t0 /. II. Let t  t0 be right-scattered and let the statement S.t/ hold. According to the condition (1) of Theorem 4.1.2, we have G.t; M.t//  G.t; W.t//

(4.1.5)

as soon as M.t/ < W.t/. From this fact and the conditions (2)–(3) of Theorem 4.1.2, it follows that M  .t/  G.t; M.t//  W  .t/; since G.t; / is a vector function, quasimonotone in its second argument. Taking into account the inequality (4.1.5), we obtain M..t// D M.t/ C .t/M  .t/ D M.t/ C .t/G.t; M.t//  W.t/ C .t/W  .t/ < R.t/ C .t/G.t; R.t// D R..t//; where 0 < .t/ < C1. Hence it follows that the statement S..t// holds. III. Let the points t  t0 be right-dense, and let the statement S.t/ hold at these points. Since M.t/ < R.t/, from the condition of continuity, it follows that there exists a neighborhood Q of the point t, such that M.r/ < R.r/ at all r 2 Q, and then S.r/ holds at all r 2 Q \ .t; 1/. IV. Let the points t  t0 be left-dense. Assume that S.r/ holds at all r 2 Œt0 ; t/. Then we get M.r/ < R.r/

at all t0  r < t

and, by continuity, M.t/ < R.t/. Hence it follows that over the interval Œt0 ; t .R  M/  0; this shows that the difference R  M does not diminish over Œt0 ; t. Hence we have .R  M/.t/  .R  M/.t0 / > 0: Therefore, the statement S.t/ holds. Corollary 4.1.1 If for the system (4.1.1) there exists a vector function VW TRn ! T Rm C and for the function v.t; x; ˛/ D ˛ V.t; x/ all the conditions of Theorem 4.1.1 are satisfied, then the conclusion of the theorem remains valid. Remark 4.1.1 The condition v.t0 ; x0 / < u0 in Theorem 4.1.1 and the condition V.t0 ; x0 / < W0 in Theorem 4.1.2 and, consequently, the estimates v.t; x.t// < r.t/

Comparison Method

149

and V.t; x.t// < R.t/ in Theorems 4.1.1 and 4.1.2 can be replaced by the nonstrict ones with the sign “,” and moreover, the condition (1) in Theorem 4.1.1 is replaced by the following: .1/

the function u C .t/g.t; u/ is nondecreasing with respect to u at all t 2 T;

and the condition (1) in Theorem 4.1.2 is replaced by the following: .1/

the vector function W C .t/G.t; W/ is quasimonotone nondecreasing with respect to W at all t 2 T.

Theorem 4.1.3 Assume that for the system (4.1.1) there exists a vector function V.t; x/ 2 Crd .T  Rn ; Rm C /, V.t; x/ is locally Lipschitz with respect to x, and a m minorant H.t; V.t; x// 2 Crd .T  Rm C ; R / for the total -derivative of the function V.t; x/ along solutions of the system (4.1.1), such that: .1/ u C .t/H.t; u/  w C .t/H.t; w/ as soon as u  w, ui D wi at i 2 Œ1; m; .2/ V  .t; x.t//  H.t; V.t; x.t/// at all .t; x/ 2 T  Rn ; .3/ the minimal solution Y.t/ D Y.t; t0 ; Y0 / of the system of dynamic equations Y  .t/ D H.t; Y.t//;

Y.t0 / D Y0

exists at all t 2 T. Then along any solution x.t/ of the system (4.1.1), the following estimate holds: V.t; x.t//  Y.t/

at all t 2 T;

(4.1.6)

as soon as V.t; x0 /  Y0 . The estimate (4.1.6) holds componentwise. The proof of this theorem is similar to that of Theorem 4.1.2. Example 4.1.1 Consider the system of dynamic equations 1 4 x i .t/ D  xi C ui .t; x/ D wi .t; xi ; x/; 2 where xi 2 Rni , ui 2 Crd .T  Rn ; Rni /, n D

m P

i D 1; 2; : : : ; m;

(4.1.7)

ni . With the independent dynamic

iD1

subsystems 1 x i .t/ D  xi ; 2

xi .t0 / D xi0 ;

i D 1; 2; : : : ; m; of the system (4.1.7), we associate the vector function V.x/ D .xT1 x1 ; : : : ; xTm xm /T

(4.1.8)

150

Comparison Method

with the components vi .xi /, i D 1; 2; : : : ; m. In this case, for the -derivative of the functions vi .xi /, we have vi .xi .t// D xTi xi C 2xTi .t/kui .t; x/ C wi .t; xi ; x/k2

(4.1.9)

at all i D 1; 2; : : : ; m, .t; xi / 2 T  Rni . Now assume that (S1 )

there exists an open domain D Rn such that xTi ui .t; x/  0;

(S2 )

i D 1; 2; : : : ; mI

(4.1.10)

there exists a vector function G 2 Crd .T  RC ; Rm / such that ui C .t/ gi .t; ui /  wi C .t/ gi .t; wi /; as soon as ui  wi and i D 1; 2; : : : ; m and kwi .t; xi ; x/k2  gi .t; xT1 x1 ; : : : ; xTi1 xi1 ; xTiC1 xiC1 ; : : : ; xTm xm /

(4.1.11)

at all t 2 T n ft0 g, t0 2 T and xi 2 Rni . Under the condition (S1 ), the relation (4.1.9) implies the estimates vi .xi .t//  vi .xi .t// C .t/ gi .t; v1 .x1 .t//; : : : ; vi1 .xi1 .t//; viC1 .xiC1 .t//; : : : ; vm .xm .t///

(4.1.12)

at all i D 1; 2; : : : ; m and .t; xi / 2 T  Rni . Along with the inequality (4.1.12), consider the dynamic system of equations u i .t/ D ui .t/ C .t/gi .t; u1 ; : : : ; ui1 ; uiC1 ; : : : ; um /:

(4.1.13)

If the right-hand parts of the system (4.1.13) satisfy the condition (1) of Theorem 4.1.2, then the system of equation (4.1.13) is a comparison system for the systems (4.1.7).

4.2 Stability Theorems The key element of the comparison method is the construction of an appropriate scalar, vector, or matrix-valued function and a corresponding comparison system. Assume that for the system (4.1.1) a matrix-valued auxiliary function is constructed by some method: U.t; x/ D Œvij .t; x/;

i; j D 1; 2; : : : ; m;

(4.2.1)

Comparison Method

151

where vii 2 Crd .T  Rn ; RC / at all i D 1; 2; : : : ; m and vij 2 Crd .T  Rn ; R/ at all .i ¤ j/ 2 Œ1; m. Then the function v.t; x; / D  TU.t; x/;

 2 Rm C;

(4.2.2)

is applied to obtain stability conditions for the state x D 0 of the system (4.1.1) within the framework of the comparison method. We shall make the following assumptions about the system (4.1.1) and the comparison equation (4.1.2): H1 : H2 : H3 : H4 :

The state x D 0 is a unique equilibrium state of the system (4.1.1). The solution x.t/ D x.t; t0 ; x0 / of the system (4.1.1) exists at all t  t0 . The equation (4.1.2) has a solution u D 0, i.e., g.t; 0/ D 0 at all t 2 T. The maximal solution r.t; t0 ; u0 / of the equation (4.1.2) exists at all t  t0 .

Recall that the definitions of stability of the zero solution of the comparison equation (4.1.2) are formulated in the same way as the definitions of stability of the state x D 0 of the system (4.1.1) with the difference that the solution u.t/ of the equation (4.1.2) satisfies the condition u.t/ 2 RC at all t 2 T. For example, the zero solution u D 0 of the equation (4.1.2) is equi-stable, if for any " > 0 and t0 2 T there exists a function ı D ı.t0 ; "/ continuous at the right-dense points t0 for any " > 0, such that u.t; t0 ; u0 / < " at all t  t0 , as soon as u0 < ı. We shall next discuss stability result for the state x D 0 of the system of dynamic equation (4.1.1). Theorem 4.2.1 Assume that for the system (4.1.1) there exist a matrix-valued function (4.2.1) and a majorant g.t; u/ of the total -derivative of the function (4.2.2), such that .1/ the functions v.t; x; / 2 Crd .TRn Rm C ; RC /, v.t; x; / is locally Lipschitzian with respect to x and the following two-sided estimate holds: '1T.kxk/A'1 .kxk/  v.t; x; / 

T 1 .kxk/B

1 .kxk/

(4.2.3)

at all .t; x/ 2 T  Rn , where A and B are constant symmetric positive definite matrices, the vector functions '1 ; 1 2 KR-class componentwise; .2/ the function .t/g.t; u/ C u, where g 2 Crd .T  RC ; R/, is nondecreasing with respect to u at all t 2 T and v  .t; x.t/; /  g.t; v.t; x.t/; //I

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Comparison Method

.3/ the zero solution of the dynamic comparison equation u .t/ D g.t; u.t//;

u.t0 / D u0  0

(4.2.4)

has a certain type of stability. Then the state x D 0 of the system (4.1.1) has the same type of stability. Proof We shall first transform the estimate (4.2.3) as

m .A/'.kxk/  v.t; x; /  M .B/ .kxk/;

(4.2.5)

where m ./; M ./ is the minimal and the maximal eigenvalues of the matrices A and B, '; 2 KR-class, such that '.kxk/  '1T.kxk/'1 .kxk/;

.kxk/ 

T 1 .kxk/ 1 .kxk/

at all x 2 Rn . Let " > 0 be given, and for m .A/'."/ and t0 2 T let there exist ı1 D ı1 .t0 ; "/ > 0 such that r.t/ < m .A/'."/ at all t  t0 ;

(4.2.6)

as soon as u0 < ı1 , where u.t/ D u.t; t0 ; u0 / is any solution of the comparison equation (4.2.4). Now choose ı D ı.t0 ; "/ > 0 so that

M .B/ .ı/ < ı1 ;

ı1 D ı1 .t0 ; "/ > 0:

Show that if kx0 k < ı, then kx.t/k < " at all t 2 T, where x.t/ D x.t; t0 ; x0 / is any solution of the system of dynamic equation (4.1.1). If this is not true, then there should exist a value t1 2 T, t1 > t0 such that the solution x.t/ of the dynamic equation (4.1.1) at the instant of time t1 leaves the "-neighborhood of the state x D 0, i.e., kx.t/k < ";

at t0  t < t1

and kx.t1 /k  ":

Denote m.t/ D v.t; x.t/; / and, in view of Theorem 4.1.1, obtain the estimate m.t/  r.t; t0 ; u0 /;

t0  t  t1 ;

(4.2.7)

where r.t; t0 ; u0 / is the maximal solution of the comparison equation (4.2.4) with the estimate m.t0 /  u0 . From the estimate (4.2.5) and the inequalities (4.2.6) and

Comparison Method

153

(4.2.7) for the point t1 2 T, we find that

m .A/'."/  m .A/'.kx.t1 /k/  v.t1 ; x.t1 /; /  r.t1 ; t0 ; u0 / < m .A/'."/

(4.2.8)

under the conditions u0 D v.t0 ; x0 ; / < M .B/ .kx0 k/ < M .B/ .ı/ < ı1 : The obtained contradiction (4.2.8) shows that t1 … T and therefore kx.t/k < " at all t 2 T as soon as kx0 k < ı. Theorem 4.2.1 is proved. Corollary 4.2.1 If all the conditions of Theorem 4.2.1 hold with the function g.t; u/ D p.t/g.u/, where p.t/  0 and is rd-continuous, and with the comparison equation u .t/ D p.t/g.u/;

u.t0 / D u0 > v.t0 ; x0 ; /;

then the conclusion of the theorem remains valid. Corollary 4.2.2 If in the conditions of Theorem 4.2.1 the majorant g.t; u/  0 at all t 2 T, then the uniform stability of solutions of the equation (4.2.4) implies the uniform stability of the state x D 0 of the system of dynamic equation (4.1.1). Corollary 4.2.3 If in the conditions of Theorem 4.2.1 the majorant g.t; u/ D c.u/, where c 2 KR-class at all t 2 T, then the uniform asymptotic stability of the zero solution of the comparison equation u .t/ D c.u.t// implies the uniform asymptotic stability of the state x D 0 of the system of dynamic equation (4.1.1). Consider next the case when the function v.t; x; / is not decreasing, i.e., in the estimate (4.2.3) the m  m-matrix B  0. Theorem 4.2.2 Assume that for the system (4.1.1) there exist a matrix-valued function (4.2.1) and function v.t; x; / 2 Crd .T Rn  Rm C ; RC / and vector comparison functions ' i .kxk/ 2 KR-class at all i D 1; 2; : : : ; m such that .1/ ' T.kxk/A'.kxk/  v.t; x; / at all .t; x/ 2 T  Rn ; where the m  m-matrix A is symmetric and positive definite; .2/ there exist vector comparison functions ai .kxk/ 2 KR-class at all i 2 Œ1; m and an m  m-matrix C which is symmetric and negative definite, such that v  .t; x.t/; /  aT.kxk/Ca.kxk/

at all .t; x/ 2 T  Rn I

.3/ there exists a constant K > 0 such that k f .t; x/k  K at all .t; x/ 2 T  Rn . Then the solution x D 0 of the system (4.1.1) is equiasymptotically stable.

154

Comparison Method

Proof Rewrite the condition (1) of Theorem 4.2.2 as

m .A/'.kxk/  v.t; x; /; where m .A/ and './ are the same as in the estimate (4.2.5). Let " > 0 and t0 2 T be given. From the fact that v.t; x; / is rd-continuous on T and v.t; 0; /  0, it follows that there exists ı D ı.t0 ; "/ > 0 such that v.t0 ; x0 ; / < m .A/'."/ as soon as kx0 k < ı. Show that under the conditions of Theorem 4.2.2 for the solutions x.t/ of the system (4.1.1), the estimate kx.t/k < " holds at all t 2 T, as soon as kx0 k < ı. Let lim inf kx.t/k ¤ 0. For the given  > 0 one can find > 0 such that kx.t/k > t!1  at all t > . From the condition (2) of Theorem 4.2.2, we obtain Zt v.t; x.t/; /  v.t0 ; x0 ; /  M .C/

g.kx.s/k/s;

t  ;



where g./ 2 KR-class and aT.kxk/a.kxk/  g.kxk/ at all x 2 Rn . Hence follows the estimate 0  v.t0 ; x0 ; /  M .C/g./.t  /; which at large values of t leads to a contradiction with the condition (1) of Theorem 4.2.2. Therefore lim inf kx.t/k D 0 as t ! C1. Let lim sup kx.t/k ¤ 0 as t ! C1. In this case, given " > 0, there exists a divergent sequence ftk g such that kx.tk /k > 0 at tk 2 T. The value tk 2 T can belong to one of the subsets A; B; C; D (see Section 3.1) on a time scale T. Let on the subset B there exist a subsequence fti g 2 B in the divergent sequence ftk g 2 B. Then the condition (2) of Theorem 4.2.2 implies v..ti /; x..ti //; /  v.ti ; x.ti /; /  .t/v  .t; x.t/; /  v.t0 ; x0 ; /  M .C/

i X

.tj /g.kx.tj /k/

jD1

(4.2.9)

 v.t0 ; x0 ; /  M .C/ig."/: As i ! 1, the inequality (4.2.9) leads to a contradiction with the condition (1) of Theorem 4.2.2, since the graininess .tj / > 0 at any j. Over the sets A; C; D, one can indicate other divergent sequences fti g such that ti < ti or ti < tiC1 for which the following relations hold: kx.ti /k D "; kx.ti /k D

1 1 " and " < kx.t/k < " at all t 2 .ti ; ti /; 2 2

Comparison Method

155

or kx.ti /k D

1 1 "; kx.tiC1 /k D " and " < kx.t/k < " 2 2 at all t 2 .ti ; tiC1 /:

Taking into account the condition (3) of Theorem 4.2.2, it is easy to find that ti  ti >

" 2k

and tiC1  ti >

" ; 2k

k > 0:

In this case, we have the estimate 0  v.ti ; x.ti /; /  v.t0 ; x0 ; /  M .C/g

" " i; 2 2k

which at large values of i leads to a contradiction with the condition (1) of Theorem 4.2.2. Therefore lim sup kx.t/k D 0 at t ! 1. Theorem 4.2.2 is proved. The instability of the zero solution of the dynamic equation (4.1.1) is understood as follows. The zero solution x D 0 of the dynamic equation (4.1.1) is unstable on T, if there exist a 2 T, a  t0 , and a constant b < 1 such that for any pair of numbers .ı; "/ independent of each other ("  b, ı < "), one can find a number > a and values of x0 satisfying the condition kx0 k  ı at t D a and yielding the relation kx. /k D ". Theorem 4.2.3 If the function v.t; x; / is increasing and its -derivative (3.2.2) by virtue of the equation (4.1.1) satisfies the inequality v  .t; x; / 

.kxk/D.t/ .kxk/

T

(4.2.10)

2 K-class and the m  m-matrix in the codomain .t; x/ 2 T  S, S Rn , where D.t/ is such that the minimal eigenvalue m of the matrix .DT.t/ C D.t//=2 satisfies the condition Zt

m .s/s

at t ! 1;

t0

then the state x D 0 of the dynamic equation (4.1.1) is unstable. Proof The instability of the state x D 0 of the equation (4.1.1) will be proved if we show that for an arbitrarily small value of ı, one can find a solution x.t/ which starts in the domain kx0 k < ı and within a finite period of time reaches the boundary of the surface of the sphere kxk D ". Let such a value ı be given. Choose the initial values x.t0 / D x0 so that at t D t0 the inequality kx0 k < ı holds and v.t0 ; x0 ; / > 0. Then choose a constant H1 so that v.t; x; / < v.t0 ; x0 ; / within the sphere kxk D H1 . The solution x.t/ D x.tI t0 ; x0 / of the dynamic equation (4.1.1) with these initial

156

Comparison Method

values will either reach the surface of the sphere kxk D " (" < H2 < 1) within a finite period of time or remain in the ring H1  kxk  H2 at all t  t0 . Let us show that an arbitrarily long stay of the solution x.t/ in the ring is impossible. Indeed, from the inequality (4.1.10), it follows that v  .t; x; /  

.kxk/D.t/ .kxk/

T

1 2

.kxk/.DT.t/ C D.t// .t/  m .t/ .kxk/;

T

(4.2.11)

where .kxk/  T.kxk/ .kxk/ at all x 2 fxW H1  kxk  H2 g and the function 2 K-class. From the inequality (4.2.11), we find that Zt v.t; x; /  v.t0 ; x0 ; / D

Zt



v .s; x.s/; /s 

.kx0 k/

t0

m .s/s:

(4.2.12)

t0

Hence it follows that v.t; x; / is unboundedly increasing with time, which contradicts the property of the decreasing function v.t; x; /. Theorem 4.2.3 is proved. Theorem 4.2.4 If the function v.t; x; / is bounded and its -derivative 3.2.2 by virtue of the equation (4.1.1) is reducible to the form v  .t; x; / D .t/v.t; x; / C w.t; x; /;

(4.2.13)

where .t/ 2 R and lim inf .t/ > 0, and w.t; x; / either is identically equal to t!1 zero or in the neighborhood S of the state x D 0 satisfies the condition lim

v.t;x; /!0

w.t; x; / D0 v.t; x; /

(4.2.14)

uniformly with respect to t 2 T, then the solution x D 0 of the dynamic equation (4.1.1) is unstable on T. Proof Under the condition lim inf .t/ > 0 and the condition (4.2.14), one can find t!1

 > 0 and t1 2 Œt0 ; 1/ such that .t/   at all t 2 Œt1 ; 1/, and if 0 < " < , there exists ı1 > 0 such that jw.t; x; /j  "v.t; x; / at all t 2 Œt0 ; 1/, if only jv.t; x; /j < ı1 . Taking this into account, we reduce the condition (4.2.13) to the form v  .t; x; /  .t/v.t; x; / C "v.t; x; /  ..t/  "/v.t; x; /  .  "/v.t; x; /

(4.2.15)

at all t 2 Œt1 ; 1/ \ T. Since the function v.t; x; / is bounded by the condition of the theorem, and the function .t/ is positively regressive, the inequality (4.2.15)

Comparison Method

157

implies the estimate v.t; x; /  e" .t; t1 /v.t1 ; x1 ; /

(4.2.16)

at all t 2 Œt1 ; 1/ \ T. The inequality (4.2.16) leads to a contradiction with the assumptions of Theorem 4.2.4 on the functions v.t; x; / and w.t; x; /, and consequently, for any arbitrarily small x1 such that v.t1 ; x1 ; / > 0, there will be an instant t  t1 when the solution x.tI t1 ; x1 / reaches the boundary of the sphere kxk D ". If w.t; x; /  0, then the inequality (4.2.15) becomes an equality, and v.t; x; / D e.t/ .t; t1 /v.t1 ; x1 ; /; which, as above, leads to a contradiction. Theorem 4.2.4 is proved. Note that Theorems 4.2.3 and 4.2.4 are similar to the first and second Lyapunov’s theorems on the instability of dynamic equations on time scales. These theorems were obtained by means of the function v.t; x; / defined on the right-dense subsets of the time scale T. We shall consider some examples. Example 4.2.2 Let x .t/ D y.x C y/; y .t/ D x.x C y/:

(4.2.17)

If T D R, then .t/ D 0 at all t 2 R and the system (4.2.17) becomes dx D y.x C y/; dt dy D x.x C y/: dt

(4.2.18)

For the function v.x; y/ D x2 C y2 , we have ˇ v.x.t/; P y.t//ˇ(4.2.18) D 0

at all t 2 R:

From the first Lyapunov’s theorem on the instability of motion, it follows that the zero solution of the system (4.2.18) is stable. At the same time, we obtain on T ˇ v  .x.t/y.t//ˇ(4.2.17) D .t/.x C y/2 .x2 C y2 /:

(4.2.19)

Since 0 < .t/ < 1 for any time scale T, it follows from Theorem 4.2.3 that the state x D y D 0 of the system (4.2.17) is unstable on T.

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Comparison Method

Example 4.2.3 Let the following system of dynamic equations be specified on a time scale T: x .t/ D p.t; x; y/y  q.t/x.x2 C y2 /;

(4.2.20)

y .t/ D p.t; x; y/x  q.t/y.x2 C y2 /; where q.t/  0 at all t 2 T, p 2 Crd and lim

.x2 Cy2 /!0

p.t; x; y/ D 0: x2 C y2

Along with the systemˇ (4.2.20), consider the function v.x; y/ D x2 C y2 and its total -derivative v  .x; y/ˇ./ , for which it is easy to obtain the following expression: ˇ v  .x; y/ˇ(4.2.20) D 2q.t/.x2 C y2 /2 C  C .t/ p2 .t; x; y/.x2 C y2 / C q2 .t/.x2 C y2 /2 ;

t 2 T:

(4.2.21)

Note that if T D R, then .t/ D 0 and dv .x; y/  2q.t/.x2 C y2 /2 : dt

(4.2.22)

Hence it follows that the solution x D y D 0 of the system (4.2.20) on R is asymptotically stable. At the same time, on a time scale T, the dynamics of the system (4.2.20) is more diverse. Namely, if sup q.t/ < 1, then x D 0, y D 0 of t2T

the system (4.2.20) is asymptotically stable. If this condition is not fulfilled, then we can get stability and/or instability. Example 4.2.4 Consider the linear system of dynamic equations x .t/ D x C y y .t/ D x C y:

(4.2.23)

For the function v.x; y/ D xy on R, we have dv D 2xy C .x2 C y2 / D 2v C .x2 C y2 / dt

(4.2.24)

Comparison Method

159

and, according to Lyapunov’s theorem, the solution x D y D 0 of the system (4.2.23) is unstable. On the time scale T, we obtain v  .x; t/ D .xy/ D xy C x Œy C .t/y  D 2xy C .x2 C y2 / C .t/.x C y/2 D 2v C .x2 C y2 / C .t/.x2 C y2 /: Since 0 < .t/ < C1, the solution x D y D 0 of the system (4.2.23) is unstable at any graininess of the time scale.

4.3 Stability of Conditionally Invariant Sets Consider the system of dynamic equations x .t/ D f .t; x.t/; ˛/;

x.t0 / D x0 ;

(4.3.1)

where x.t/ 2 Rn , t 2 T, ˛ is an inaccuracy parameter for the coefficients in the right-hand part of the system of equation (4.3.1), and the -derivative of the vector x is determined by the formula ( 

x .t/ D

x. .t//x.t/ ; .t/

if t 2 A [ C;

xP .t/

at the rest of points of T:

For the system (4.3.1), we assume that assumptions H1 –H4 (see Section 3.1) are fulfilled. Let a function r.˛/ > 0 be given such that r.˛/ ! r0 (r0 D const) as k˛k ! 0 and r.˛/ ! C1 as k˛k ! C1. In a normalized space .Rn ; k  k/, we consider the moving sets A.r/ D f x 2 Rn W kxk  r.˛/ g; B. p/ D f x 2 Rn W kxk  p.˛/ g; where 0 < r.˛/ < p.˛/ at all ˛ 2 S  Rd . Definition 4.3.1 A moving set A.r/ is invariant for solutions of the system of dynamic equation (4.3.1), if from the condition x0 2 A.r/ it follows that x.t; ˛/ 2 A.r/ at all t  t0 , t0 2 T. Definition 4.3.2 A moving set B. p/ is conditionally invariant with respect to the set A.r/ for solutions of the system of dynamic equation (4.3.1), if from the condition x0 2 A.r/ it follows that x.t; ˛/ 2 B. p/ at all t  t0 , t0 2 T.

160

Comparison Method

From now on x.t; ˛/ D x.t; t0 ; x0 ; ˛/ is any solution of the system (4.3.1) with the initial conditions .t0 ; x0 / 2 T  Rn at any ˛ 2 S. Now consider the sets S.A; ı/ D f x 2 Rn W kxk  r.˛/ C ı g; S.B; "/ D f x 2 Rn W kxk  p.˛/ C " g; where " > 0 and ı D ı."/ > 0. Definition 4.3.3 The set B. p/, being conditionally invariant with respect to the set A.r/, is uniformly asymptotically stable if: .a/ it is uniformly stable, i.e., given " > 0, at any t0 2 T there exists ı D ı."/ such that from the condition x0 2 S.A; ı/, it follows that x.t; ˛/ 2 S.B; "/ at all t  t0 , where the m  m-matrix A is symmetric and positive definite; (b) it is uniformly quasi-asymptotically stable, i.e., given 0 <  < ", at any t0 2 T there exist ı0 D ı0 ./, 0 < ı0 < ı, and D ./ > t0 such that the condition x0 2 S.A; ı0 / implies x.t; ˛/ 2 S.B; / at all t  t0 C ./. Along with the system of equation (4.3.1), we shall consider the scalar dynamic equation u .t/ D g.t; u; .˛//;

u.t0 / D u0  0;

(4.3.2)

where g 2 Crd .T  RC  Rd ; R/, g.t; 0; .˛// D 0 at all t 2 T. Assume that the maximal solution u.t; / D u.t; t0 ; u0 ; / of the equation (4.3.2) exists at all t  t0 , t0 2 T. Then for the equation (4.3.2), we consider the set  D f u 2 RC W u  r0 g;

r0 > 0:

(4.3.3)

Definition 4.3.4 A set  is: .1/ invariant for solutions of the equation (4.3.2), if from the condition u0  r0 it follows that u.t; /  r0 at all t  t0 , t0 2 T; .2/ uniformly asymptotically stable if .a/ it is uniformly stable, i.e., for a given " > 0 and any t0 2 T, there exists ı  D ı  ." / > 0 such that the condition u0 < r0 C ı  implies u.t; / < r0 C " at all t  t0 ; (b) it is uniformly equiasymptotically stable, i.e., for a given  > 0 and any t0 2 T, there exist ı0 > 0 and  D  . / > 0 such that from the condition u0 < r0 C ı0 , it follows that u.t; / < r0 C  at all t  t0 C

 . /. We shall now apply the function (4.2.1) and determine the conditions for the uniform asymptotic stability of the set B. p/.

Comparison Method

161

Theorem 4.3.1 Assume that for the system (4.3.1) there exist a matrix-valued function (4.2.1) and a majorant g.t; u; / of the total -derivative of the function (4.2.2) such that .1/ the function v.t; x; / 2 Crd .T  Rn  Rm C ; RC /, v.t; x; / is locally Lipschitzian with respect to x and for it the two-sided estimate '1T.kxk/A'1 .kxk/  v.t; x; / 

T 1 .kxk/B

1 .kxk/

holds at all .t; x/ 2 T  Rn , where the m  m-matrices A and B are symmetric and positive definite and the vector functions '1 and 1 2 KR-class componentwise; .2/ the function .t/g.t; u; /Cu, where g 2 Crd .TRC Rd ; R/, is nondecreasing with respect to u at all t 2 T such that v  .t; x.t; ˛/; /  g.t; v.t; x.t; ˛/; /; /I .3/ the set  with the constant r0 D M .B/ .r/, r D max r.˛/ is invariant and ˛2S

uniformly asymptotically stable; .4/ for any value .˛/ 2 Rd , ˛ 2 S there exist r D r. / > 0 and p D p. / > 0 such that m .A/ .r/ D M .B/'. p/; where r D r. / ! 0, p D p. / ! 0 as k k ! 0. Then the set B is conditionally invariant with respect to the set A and uniformly asymptotically stable. Proof Like in the proof of Theorem 4.2.1, we reduce the condition (1) of Theorem 4.3.1 to

m .A/'.kxk/  v.t; x; /  M .B/ .kxk/:

(4.3.4)

We shall first show that the set B is conditionally invariant with respect to the set A. From the condition (3) of Theorem 4.3.1, it follows that if u0 < M .B/ .r/, then at all t  t0 the following estimate holds: u.t; /  M .B/ .r/;

(4.3.5)

where u.t; / is any solution of the dynamic comparison equation (4.3.2). Show that if x0 2 A.r/, then x.t; ˛/ 2 B. p/ at all t  t0 . Let this be not so. Then one can find a value t2 2 T, t2 > t0 such that x0 2 A.r/, x.t; ˛/ 2 B. p/ at t0 < t < t2 , but x.t2 ; ˛/ 2 B. p/. According to comparison Theorem 4.1.1, we have the estimate v.t; x.t/; /  u.t; /

at all t0  t  t2 ;

(4.3.6)

162

Comparison Method

where u.t; / is the maximal solution of the comparison equation (4.3.2). Let u0 D v.t0 ; x0 ; /, then u0 D v.t0 ; x0 ; /  M .B/ .kx0 k/  M .B/ .r/: Taking into account the estimate (4.3.6) and the condition (4) of Theorem 4.3.1, we obtain the inequality

m .A/'.r/ < m .A/'.kx.t2 /k/  v.t2 ; x.t2 /; /  u.t; /  M .B/ .r/ D m .A/'.r/; which is a contradiction. Consequently, the set B is conditionally invariant with respect to the set A. Now prove the second assertion of Theorem 4.3.1. Let " > 0 and t0 2 T be given. Choose ı1 D ı1 ."/ > 0 so that the following inequality will hold:

M .B/ .r C ı1 / < m .A/'. p C "/: From the condition (3) of Theorem 4.3.1, it follows that under the condition u0
0. From the condition (3) of Theorem 4.3.1, it follows that for the given 0 <  < "0 and any t0 2 T, there exist ı0 > 0 and D ./ > 0 such that the condition u0 < M .B/ .r.˛/ C ı0 / implies the estimate u.t; /
0 and any t0 2 T, the following conditions are satisfied: .a/ there exists a ı D ı."/ > 0 such that the condition r0 .˛/ ı  kx0 k  r.˛/Cı implies the estimate p0 .˛/  "  kx.t; ˛/k  p.˛/ C " at all t  t0 , t0 2 T; (b) there exist ı0 > 0 and D ."/ > 0 such that the condition r0 .˛/  ı0  kx0 k  r.˛/ C ı0 implies the estimate p0 .˛/  "  kx.t; ˛/k  p.˛/ C " at all t  t0 C ."/, t0 2 T.

164

Comparison Method

The estimates of total -derivatives of the components vi .t; x; b/ of the vector function (4.3.8) by virtue of the system of dynamic equation (4.3.1) result in some comparison system u .t/ D G.t; u.t/; .˛//;

u.t0 / D u0  0;

(4.3.9)

where u 2 R2C , G 2 Crd .T  R2C  Rd ; R2 /. For the system (4.3.1), the function (4.3.8) and the comparison system (4.3.9) assume that the estimate from the comparison Theorem 4.1.3 holds. Let u.t; / be any solution of the system (4.3.9). Definition 4.3.7 The set Q D f u 2 R2C W k0  u  k g is invariant and uniformly asymptotically stable for the solutions u.t; / of the system (4.3.1) if: .1/ there exist constants k0 ; k, .k0  k/ such that the condition k0  ku0 k  k implies the estimate k0  ku.t; /k  k at all t  t0 , t0 2 T; .2/ for the given " > 0 and any t0 2 T .a/ there exists ı  D ı  ."/ > 0 such that the condition k0 ı   ku0 k  kCı  implies the estimate k0  "  ku.t; /k  k C " at all t  t0 , t0 2 T; (b) there exist ı0 > 0 and D ." / > 0 such that the condition k0  ı0  ku0 k  k C ı0 implies the estimate k0  "  ku.t; /k  k C " at all t  t0 , t0 2 T. We shall now prove the following result. Theorem 4.3.2 Assume that for the system (4.3.1) there exist a matrix-valued function (4.3.7), a 2  2-matrix ˆ, and a constant vector b 2 R2C such that for the components vi .t; x; b/ of the function V.t; x; b/ D ˆU.t; x/b, vi .t; x; b/ 2 Crd .T  Rn  R2C ; R/, the two-sided estimates hold: (1) bT1 .kxk/A1 b1 .kxk/  v1 .t; x; b/  aT1 .kxk/jB1 a1 .kxk/ at all .t; x/ 2 T  .r/, where A1 ; B1 are m  m-constant matrices and .b1 ; a1 / 2 KR-class; (2) bT2 .kxk/A2 b2 .kxk/  v2 .t; x; b/  aT2 .kxk/jB2 a2 .kxk/ at all .t; x/ 2 T  .r0 /, where A2 and B2 are m  m-constant matrices and .b2 ; a2 / 2 KR-class; (3) at x 2 .r/ the following inequality is true v1 .t; x; b/  g.t; v1 .t; x; b/; .˛// and u C .t/g.t; u; .˛// is a function nondecreasing with respect to u; (4) at all x 2 .r0 / the following inequality holds v2 .t; x; b/  h.t; v2 .t; x; b/; .˛// and w C .t/h.t; w; .˛// is a function nonincreasing with respect to u; (5) for any value r0 .˛/  r.˛/ at all ˛ 2 S, there exist constants k0  k such that k D M .B1 /a1 .r/ D m .A1 /b1 . p/ and k0 D m .A2 /b2 .r0 / D M .B2 /a2 . p0 / and, besides, k ! 0 as r.˛/ ! 0 and k0 ! 1 as r0 .˛/ ! 1;

Comparison Method

165

(6) the set  D f u 2 R2C W kuk  r0 g is invariant and uniformly asymptotically stable for solutions of the comparison system (4.3.9). Then, if the matrices Ai ; Bi , i D 1; 2 are symmetric and positive definite, then the set B is conditionally invariant with respect to the set A and uniformly asymptotically stable for solutions of the system (4.3.1). Proof We shall first transform the estimates of the functions vi .t; x; b/, i D 1; 2 from the conditions (1) and (2) of Theorem 4.3.2. Let m .Ai / be the minimal eigenvalues of the matrices Ai , i D 1; 2 and M .Bi / be the maximal eigenvalues of the matrices Bi , i D 1; 2. Then the estimates of the functions vi .t; x; b/, i D 1; 2 are reducible to the form .1/0

m .A1 /b1 .kxk/  v1 .t; x; b/  M .B1 /a1 .kxk/;

(4.3.10)

where a1 ; b1 2 KR-class and b1 .kxk/  bT1 .kxk/b1 .kxk/ and a1 .kxk/  aT1 .kxk/a1 .kxk/ at all x 2 Rn ; .2/0

m .A2 /b2 .kxk/  v2 .t; x; b/  M .B2 /a2 .kxk/;

(4.3.11)

where a2 ; b2 2 KR-class and b2 .kxk/  bT2 .kxk/b2 .kxk/ and a2 .kxk/  aT2 .kxk/a2 .kxk/ at all x 2 Rn . We shall now show that the set B is conditionally invariant with respect to the set A for solutions of the system (4.3.1). Let this be not so. Then at the initial values .t0 ; x0 /W t0 2 T and r0 .˛/  kx0 k  r.˛/, one can find a solution x.t; t0 ; x0 ; ˛/ D x.t; ˛/ and a value t2 2 T such that .a/ kx.t2 ; ˛/k > p.˛/ and r0 .˛/  kx.t; ˛/k at t 2 Œt0 ; t2  \ T; .b/ kx.t2 ; ˛/k < p0 .˛/ and r.˛/  kx.t; ˛/k at t 2 Œt0 ; t2  \ T. Under the conditions (3) and (4) of Theorem 4.3.2 for the functions vi .t; x.t/; b/, i D 1; 2, we have the estimates v1 .t; x.t; ˛/; b/  r.t; t0 ; v1 .t0 ; x0 ; b//;

(4.3.12)

v2 .t; x.t; ˛/; b/  r.t; t0 ; v2 .t0 ; x0 ; b//

(4.3.13)

at all t 2 Œt0 ; t2 \T. Here r.t; / and r.t; / are the maximal and the minimal solutions of the comparison equations u 1 .t/ D g.t; u1 .t/; .˛//;

u1 .t0 / D u0  0I

(4.3.14)

u 2 .t/

u2 .t0 / D u0  0

(4.3.15)

D h.t; u2 .t/; .˛//;

166

Comparison Method

respectively. From the estimates (4.3.10) and (4.3.12) and the condition (5) of Theorem 4.3.2, we obtain in the case (a)

m .A1 /b1 .p/ < M .A1 /b1 .kx.t2 ; ˛/k/  v1 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; v1 .t0 ; x0 ; b//  r.t2 ; t0 ; M .B1 /a1 .kx0 k//  r.t2 ; t0 ; M .B1 /a1 .r//  M .B1 /a1 .r/ D m .A1 /b1 . p/; or in the case (b)

M .B2 /a2 . p0 / > M .B2 /a2 .kx.t2 ; ˛/k/  v2 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; v2 .t0 ; x0 ; b//  r.t2 ; t0 ; m .A2 /b2 .kx0 k//  r.t2 ; t0 ; m .A2 /b2 . p0 //  m .A2 /b2 .r0 / D M .B2 /a2 . p0 /: The obtained contradictions prove that the set B is conditionally invariant with respect to the set A for solutions of the system (4.3.1). Let us prove the second part of the assertion of the theorem. Let 0 < " < p0 and t0 2 T be satisfied. Under the conditions (5) and (6) of Theorem 4.3.2 for the given

M .B2 /a2 . p0  "/, m .A1 /b1 . p C "/, one can find "1 > 0, ı1 > 0 and ı > 0 such that k C ı1 D M .B1 /a1 .r C ı/ < m .A1 /b1 . p C "/ D k C "1 ; k0  " D M .B2 /a2 . p0  "/ < m .A2 /b2 .r0  ı/ D k0  ı1 : The condition k0  ı1 < ku0 k < k C ı1 implies the estimate k0  "1 < ku.t; /k < k C "1 at all t  t0 , t0 2 T. We shall now show that for the chosen ı > 0, the set B is uniformly stable with respect to the set A for solutions of the system (4.3.1), i.e., the condition r0 .˛/  ı < kx0 k < r.˛/ C ı

(4.3.16)

implies the estimate p0 .˛/  " < kx.t; ˛/k < p.˛/ C " at all t  t0 ; t0 2 T:

(4.3.17)

Let this be not so and let there exist a solution x.t; ˛/ and an instant of time t2 2 T, t2 > t0 such that one of the following inequalities holds: .a/ kx.t2 ; ˛/k  p.˛/ C " and kx.t; ˛/k  r0 .˛/ at t 2 Œt0 ; t2  \ T; .b/ kx.t2 ; ˛/k  p0 .˛/  " and kx.t; ˛/k  r.˛/ at t 2 Œt0 ; t2  \ T.

Comparison Method

167

Consider the case (a). From the estimate (4.3.12), we get

m .A1 /b1 .p.˛/ C "/  M .A1 /b1 .kx.t2 ; ˛/k/  v1 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; v1 .t0 ; x0 ; b//  r.t2 ; t0 ; M .B1 /a1 .kx0 k//  r.t2 ; t0 ; M .B1 /a1 .r.˛/ C ı// < m .A1 /b1 . p.˛/ C "/: In the case (b) from the estimate (4.3.13) and the inequality (4.3.11), we have

M .A2 /a2 .p0 .˛/  "/ > M .A2 /a2 .kx.t2 ; ˛/k/  v2 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; v2 .t0 ; x0 ; b//  r.t2 ; t0 ; m .A2 /b2 .kx0 k//  r.t2 ; t0 ; m .A2 /b2 .r0 .˛/  ı//  m .A2 /b2 .r0 .˛// D M .A2 /a2 . p0 .˛/  "/: The obtained contradictions prove that the set B is uniformly stable with respect to the set A for solutions of the system (4.3.1). Now prove that the set B is uniformly asymptotically stable with respect to the set A . Let " D p0 D max p0 .˛/ and choose ı0 D ı. p0 / > 0 so that the condition ˛2S

r0 .˛/  ı0 < kx0 k < r.˛/ C ı0 implies the estimate 0 < kx.t; ˛/k < p.˛/ C " at all t  t0 ; t0 2 T: Let 0 < " < p0 and t0 2 T. According to the condition (b) of Theorem 4.3.2, the set  is uniformly asymptotically stable, and therefore for the given

M .B2 /a2 . p0 .˛/"/ and m .A1 /b1 . p.˛/C"/, there exists D ."/ > 0, t0 C 2 T such that the condition

m .A2 /b2 .r0 .˛/  ı0 / < ku0 k < M .B1 /a1 .r.˛/ C ı0 / implies the estimate

M .B2 /a2 . p0 .˛/  "/ < ku.t; /k < m .A1 /b1 . p.˛/ C "/ at all t  t0 C . We shall show that under the initial condition (4.3.16), two-sided estimate holds: p0 .˛/  " < kx.t; ˛/k < p.˛/ C "

(4.3.18)

at all t  t0 C ."/, t0 2 T. Let this be not so, then there should exist a solution x.t; ˛/ and a point of time t2 2 T such that .a/ kx.t2 ; ˛/k  p.˛/ C " at t2  t0 C , t0 2 T; .b/ kx.t2 ; ˛/k  p0 .˛/  " at t2  t0 C , t0 2 T.

168

Comparison Method

Taking into account the estimates (4.3.10), (4.3.12), and (4.3.13), we obtain

m .A1 /b1 . p.˛/ C "/  v1 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; M .B1 /a1 .r.˛/ C ı0 // < m .A1 /b1 . p.˛/ C "/;

M .B2 /a2 . p0 .˛/  "/  v2 .t2 ; x.t2 ; ˛/; b/  r.t2 ; t0 ; m .A2 /b2 .r0 .˛/  ı0 // > M .B2 /a2 . p0 .˛/  "/: The obtained contradictions show that the set B is uniformly asymptotically stable with respect to the set A . Theorem 4.3.2 is proved.

4.4 Stability with Respect to Two Measures Originally the notion of stability with respect to two measures appeared in the continuum mechanics as a tool for description of the dynamic behavior of elastic constructions. For dynamic equations on time scales, the stability with respect to two measures was studied in several papers. The purpose of this section is to establish criteria for the stability of motion with respect to two measures on the basis of the generalized Lyapunov method. Consider the system x .t/ D f .t; x.t//;

x.t0 / D x0 ;

(4.4.1)

where F.t/ D f .t; x.t// 2 Crd .T/ and f .t; 0/ ¤ 0 for all t 2 T. The qualitative analysis of solutions of the system (4.4.1) will be conducted on the basis of the two measures .t; x.t// and 0 .t0 ; x.t0 // which take on values from the set M D f  2 C.T  Rn ; RC / W

inf

.t;x/2TRn

.t; x/ D 0 g:

Definition 4.4.1 Let the measures 0 ;  2 M. Assume that .a/ the measure  is continuous with respect to the measure 0 if there exist a constant ˇ > 0 and a comparison function ' 2 C.T; R/, s ! '.t; s/, ' 2 KRclass for each t 2 T and .t; x/ < '.t; 0 .t; x// as soon as 0 .t; x/ < ˇ; .b/ the measure  is uniformly continuous with respect to the measure 0 if the function '.t; s/ in Definition 4.4.1(a) does not depend on t 2 T. Together with the system (4.4.1), consider the auxiliary function v.t; x; / D  TU.t; x/;

 2 Rm C;

(4.4.2)

Comparison Method

169

where U 2 Crd .TRn ; Rmm /, U.t; x/ is locally Lipschitzian with respect to x 2 Rn . The sign definiteness of the function (4.4.2) with respect to the measure .t; x/ is defined below. Definition 4.4.2 Let the function v.t; x; / 2 Crd .T  Rn   Rm C ; R/. Assume that the function v.t; x; / is: (a) definite -positive, if there exist a constant > 0 and a comparison function a 2 KR-class, such that a..t; x//  v.t; x; / at all .t; x/ 2 T  Rn under the condition that .t; x/ < ; (b) -decreasing on T  Rn , if there exist a constant  > 0 and a comparison function b 2 KR-class, such that under the condition .t; x/ <  , the estimate v.t; x; /  b..t; x// holds at all .t; x/ 2 T  Rn ; .c/ weakly -decreasing if in Definition 4.4.2(b) there exist the comparison functions b 2 CK-class. Now the notion of stability with respect to two measures is formulated as follows. Definition 4.4.3 The system of dynamic equation (4.4.1) is said to be: .a/ .0 ; /-equi-stable if for each " > 0 and t0 2 T, there exists ı D ı.t0 ; "/ > 0, and the function ı.t0 ; "/ is rd-continuous with respect to t0 for each " > 0, such that .t; x.t// < " at all t  t0 , t0 2 T as soon as 0 .t0 ; x0 / < ı; .b/ uniformly .0 ; /-stable, if in Definition 4.4.3(a) the function ı does not depend on t0 2 T; .c/ quasi-equiasymptotically .0 ; /-stable if for each " > 0 and t0 2 T, there exist positive numbers ı0 D ı0 .t0 / and D .t0 ; "/ such that .t; x.t// < " at all t  t0 C , t0 C 2 T as soon as 0 .t0 ; x0 / < ı0 ; .d/ quasi-uniformly .0 ; /-stable if the conditions of Definition 4.4.3(c) are satisfied and the values ı0 and do not depend on t0 2 T; .e/ uniformly asymptotically .0 ; /-stable if the conditions of Definitions 4.4.3(a) and (d) are satisfied at the same time. Remark 4.4.1 The notion of stability with respect to two measures embraces a number of known definitions of stability depending on the method of choosing the measures 0 and . Here are some methods of choosing the two measures: .1/ if in the system (4.4.1) f .t; x.t// D 0 at x D 0, then the measures 0 .t; x/ D .t; x/ D kxk are applied in the study of the stability of its zero solution; .2/ the measures 0 .t; x/ D kxk and .t; x/ D kxks , 1  s  n are applied in the study of the stability with respect to a part of variables of the system (4.4.1); .3/ the measures 0 .t; x/ D .t; x/ D kx  x0 .t//k are applied in the study of the stability of the prescribed motion x0 .t/ of the system (4.4.1); .4/ the measures 0 .t; x/ D .t; x/ D kxk C .t/, where .t/ 2 L-class, are applied in the study of the stability of the asymptotically invariant set f0g; .5/ the measures 0 .t; x/ D .t; x/ D d.x; A/, where d.x; A/ is the distance from the point x to the set A Rn , are applied in the study of the stability of the invariant set A;

170

Comparison Method

.6/ the measures 0 .t; x/ D d.x; A/ and .t; x/ D d.x; B/ are applied in the study of the stability of the conditionally invariant set B with respect to the set A, A B Rn . We shall next consider the conditions for the stability with respect to two measures of the system (4.4.1). Introduce the set S.; H/ D f .t; x/ 2 T  Rn W .t; x/ < H; H > 0 g: Theorem 4.4.1 Assume that for the system (4.4.1) there exist a matrix-valued function U.t; x/ 2 Crd .T  Rn ; Rmm / and a vector  2 Rm C such that: .1/ for the function v.t; x; / D  TU.t; x/, the following estimates hold: (a) '1T..t; x//A'1 ..t; x//  v.t; x; / at all .t; x/ 2 T  Rn , where A is an m  m-constant symmetric matrix and '1 ..t; x// is a vector comparison function with the components '1i ..t; x//, i D 1; 2; : : : ; m, '1i 2 KRclass; (b) '2T.t; .t; x//B'2 .t; .t; x//  v.t; x; / at all .t; x/ 2 T Rn , where B is an m  m-constant symmetric matrix, and '2 .t; .t; x// is a vector comparison function with the components '2i .t; .t; x//, i D 1; 2; : : : ; m, '2i 2 CKclass; (c) '3T..t; x//B '3 ..t; x//  v.t; x; / at all .t; x/ 2 T  Rn , where B is an m  m-constant symmetric matrix and '3 ..t; x// is a vector comparison function with the components '3i ..t; x//, i D 1; 2; : : : ; m, '3i 2 KR-class; .2/ for the function v  .t; x; /, the following inequality holds: v  .t; x.t/; / 

.0 .t; x//C .0 .t; x//

T

at all .t; x/ 2 T  S.; H/, where C is an m  m-constant symmetric matrix and .0 .t; x// is a vector comparison function with the components i .0 .t; x//, i D 1; 2; : : : ; m, i 2 K-class. Then: (A) if the measure  2 M is continuous with respect to the measure 0 and the matrices A and B are positive definite and the matrix C is negative semidefinite, then the system (4.4.1) is .0 ; /-equi-stable; (B) if the measure  2 M is uniformly continuous with respect to the measure 0 and the matrices A and B are positive definite and the matrix C is negative semidefinite, then the system (4.4.1) is uniformly .0 ; /-stable; (C) if the measures 0 ;  2 M and the measure  is uniformly continuous with respect to the measure 0 and the matrices A and B are positive definite and matrix C is negative definite, then the system (4.4.1) is uniformly asymptotically .0 ; /-stable.

Comparison Method

171

Proof We shall first transform the condition (1) of Theorem 4.4.1 to the form .a /

m .A/b..t; x//  v.t; x; /

at all .t; x/ 2 T  Rn , where m .A/ is the minimal eigenvalue of the matrix A, b./ 2 KR-class such that '1T..t; x//'1 ..t; x//  b..t; x//I .b /

M .B/a.t; .t; x//  v.t; x; /

at all .t; x/ 2 T  Rn , where M .B/ is the maximal eigenvalue of the matrix B, a./ 2 CK-class such that '2T.t; .t; x//'2 .t; .t; x//  a.t; .t; x//I .c /

M .B /Qa..t; x//  v.t; x; /

at all .t; x/ 2 T  Rn , where M .B / is the maximal eigenvalue of the matrix B , aQ ./ 2 KR-class such that '3T..t; x//'3 ..t; x//  aQ ..t; x//: Reduce the condition (2) of Theorem 4.4.1 in the codomain .t; x/ 2 S.; H/ to the form v  .t; x.t/; /  M .C/c.0 .t; x//;

(4.4.3)

where M .C/ is the maximal eigenvalue of the matrix C and the comparison function c./ 2 K-class. We shall now prove the assertion (A) of Theorem 4.4.1. If the matrices A and B are positive definite, then the function v.t; x; / is weakly 0 -decreasing and positive definite. Therefore for t0 2 T and x0 2 Rn , there exist a constant H0 > 0 and a function a./ 2 CK-class, such that v.t0 ; x0 ; /  M .B/a.t0 ; 0 .t0 ; x0 //;

(4.4.4)

as soon as 0 .t0 ; x0 / < H0 . In addition, one can find a constant H1 > 0 such that

m .A/b..t; x//  v.t; x; /;

(4.4.5)

172

Comparison Method

as soon as .t; x/ < H1 . Since the measure  is continuous with respect to the measure 0 , there exist a constant H0 > 0 and a comparison function ' 2 CK-class, such that .t0 ; x0 /  '.t0 ; 0 .t0 ; x0 //;

(4.4.6)

as soon as 0 .t0 ; x0 / < H0 , where H0 is chosen so that '.t0 ; H0 / < H1 . Let t0 2 T and " 2 .0; H0 / be given. Since the function a./ 2 CK-class, there exists a function ı.t0 ; "/ > 0, rd-continuous with respect to t0 , such that

M .B/a.t0 ; ı.t0 ; "// < m .A/b."/:

(4.4.7)

The estimates (4.4.4)–(4.4.6) imply

m .A/b..t0 ; x0 //  v.t0 ; x0 ; /  M .B/a.t0 ; 0 .t0 ; x0 // < m .A/b."/ at 0 .t0 ; x0 / < ı. Hence it follows that .t0 ; x0 / < ". We shall next show that for any solution x.t/ D x.t; t0 ; x0 / of the system (4.4.1) with the initial conditions 0 .t0 ; x0 / < ı, the following inequality holds: .t; x.t// < "

at all t  t0 ;

t0 2 T:

(4.4.8)

If this is not so, then for the solution x.t/, there exists a value t1 2 T, t1 > t0 such that .t1 ; x.t1 //  " and .t; x.t// < " at all t 2 Œt0 ; t1 /. Since the matrix C is negative semidefinite, we get from the estimate (4.4.3) that v  .t; x.t/; /  0

at .t; x/ 2 S.; H/

and therefore v.t; x.t/; /  v.t0 ; x0 ; /

at t0  t  t1 :

Hence we find that

m .A/b."/  m .A/b..t1 ; x.t1 //  v.t1 ; x.t1 /; /  v.t0 ; x0 ; / < M .B/a.t0 ; ı/ < m .A/b."/: The obtained contradiction proves that the system (4.4.1) is .0 ; /-equi-stable. The proof of the assumption (B) is carried out in the same way, taking into account the fact that the estimate .c/ is applied and the function v.t; x; / is 0 decreasing, and the measure  is uniformly continuous with respect to the measure 0 . The value ı is chosen irrespective of t0 2 T.

Comparison Method

173

We shall now prove the assumption (C). From the estimate .a/ it follows that there exist a constant 0 < H0 < H and functions aQ ; b 2 KR-class such that

m .A/b..t; x//  v.t; x; /

at all .t; x/ 2 S.; H/

(4.4.9)

and v.t; x; /  M .B /Qa.0 .t; x//;

(4.4.10)

as soon as 0 .t; x/ < H0 . According to the assumption (B), the system (4.4.1) is uniformly .0 ; /-stable. Let " D H0 and ı0 D ı.H0 /, then .t; x.t// < H0

at all t  t0 ;

t0 2 T;

as soon as 0 .t0 ; x0 / < ı0 . Now assume that 0 < " < H0 and ı D ı."/. Let  /Q a.ı0 / 0 .t0 ; x0 / < ı0 and D ."/ D

MM.B.C/c.ı/ C 1, 2 T. The uniform asymptotic .0 ; /-stability occurs if there exists t 2 Œt0 ; t0 C  such that .t ; x.t // < ı. Assume that 0 .t; x.t//  ı at all t 2 Œt0 ; t0 C , t0 2 T at the initial values 0 .t0 ; x0 / < ı0 . The inequality (4.4.3) implies the estimate tZ 0 C

v.t0 C ; x.t0 C /; /  v.t0 ; x0 ; /  M .C/

c.0 .s; x.s///s t0

from which we find that Z

M .C/

t0 C

c.0 .s; x.s/// s  v.t0 ; x0 ; /  M .B /Qa.0 .t0 ; x0 //

t0

(4.4.11)



 M .B /Qa.ı0 /: On the other hand, tZ 0 C

M .C/

c.0 .s; x.s/// s  M .C/c.ı/ > M .B /Qa.ı0 /:

(4.4.12)

t0

The inequality (4.4.12) contradicts the inequality (4.4.11) and thus the assumption . 0 .t; x.t//  ı / is not true. Theorem 4.4.1 is proved.

174

Comparison Method

4.5 Stability of a Dynamic Graph The purpose of this section is to study the stability of a dynamic graph by the method of matrix Lyapunov functions.

4.5.1 Description of a dynamic graph We shall consider a weighted directed graph (later referred to as a graph) D D .V; E/ which is an ordered pair where V is a nonempty finite set of N vertices and E is a set of the edges of the graph. The vertices .v1 ; v2 ; : : : ; vN / connect the graph edges .vj ; vi / so that each edge is oriented from vj to vi at all .i; j/ 2 N D f1; 2; : : : ; Ng. Each edge .vj ; vi / is associated with a weight eij , if the edge .vj ; vi / 2 D while eij D 0 if .vj ; vi / ¤ D. The concept of isomorphism of N  N matrix E D .eij / is associated with the digraph D. In what follows we will use this concept of isomorphism and the permutation of D and E as applied to the case under consideration. Now define a space of graphs D with fixed number N of vertices, as a linear space over the field F of real numbers. For any D1 ; D2 2 D, there exists a unique graph D1 C D2 2 D

(4.5.1)

which is called the sum of graphs D1 and D2 , and for any D 2 D and an arbitrary number ˛ 2 F , there exists a unique graph ˛D 2 D:

(4.5.2)

If in the formula (4.5.2) we assume ˛ D 0, then ˛D D 0, which corresponds to the zero graph D D 0 2 D. This graph consists of N disconnected vertices, and therefore the matrix E is empty. The above procedure that specifies D as a linear space can be interpreted in the context of the linear space C of adjacent matrices. For two N  N matrices E1 D .e1ij / and E2 D .e2ij / the sum is .e1ij / C .e2ij / D .e1ij C e2ij / 2 C

(4.5.3)

and for any N  N matrix E D .eij / 2 C and a scalar ˛ 2 F , we obtain ˛eij D .˛eij / 2 C: Note that the zero element of the space C is the N  N matrix E D 0 2 C.

(4.5.4)

Comparison Method

175

Now, in order to introduce the notion of graph motion and its stability in the space D, we shall introduce a norm of the graph .D/ with the following properties: (a) .D/ > 0 at all D 2 D .D ¤ 0/I (b) .˛D/ D j˛j.D/ at all D 2 D and ˛ 2 F I

(4.5.5)

(c) .D1 C D2 /  .D1 / C .D2 / at all .D1 ; D2 / 2 D: For the space of adjacent matrices C isomorphic to the space D, consider the matrix norm W RNN ! RC in the space RNN with the properties: (a) .E/ > 0 at all E 2 RNN .E ¤ 0/I (b) .˛E/ D j˛j.E/ at all E 2 RNN and at all ˛ 2 F I

(4.5.6)

(c) .E1 C E2 /  .E1 / C .E2 / at all .E1 ; E2 / 2 RNN : Using these norms, we introduce a metric in the space D by the formula .D1 ; D2 / D .D1  D2 / at all .D1 ; D2 / 2 D

(4.5.7)

and in the matrix space D by the formula .E1 ; E2 / D .E1  E2 / at all .E1 ; E2 / 2 C:

(4.5.8)

Taking into account some results of the monograph by Siljak [3], we shall consider an axiomatic definition of a dynamic graph as a mapping of the abstract space D into itself. Let a family of mappings ˆ.t; D/ in the space D for any D 2 D and an arbitrary t 2 R be associated with some graph ˆ 2 D. Definition 4.5.1 A dynamic graph D is a one-parameter mapping ˆW R  D ! D of the space D into itself, which satisfies the following axioms: (a) ˆ.t0 ; D0 / D D0 at all t0 2 R and at all D0 2 D; (b) ˆ.t; D/ is continuous at all t 2 R and at all D 2 D; (c) ˆ.t2 ; ˆ.t1 ; D// D ˆ.t1 C t2 ; D/ at all .t1 ; t2 / 2 R and at all D 2 D. The axiom (a) establishes the fact of the existence of an initial graph D.t0 / D D0 . The axiom (b) prescribes continuity of the mapping ˆ.t; D/ with respect to all t and all D, including t0 and D0 . The axiom (b) determines that a dynamic graph is a oneparameter group of transformations of the space D into itself. In applications of the theory of dynamic graphs, the notion of an adjacent matrix plays a key role; therefore the introduction of such a notion is justified.

176

Comparison Method

Definition 4.5.2 A dynamic adjacent matrix E is a one-parameter mapping ‰W R  RNN ! RNN of the space RNN into itself, satisfying the following axioms: (a) ‰.t0 ; E0 / D E0 at all t0 2 R and at all E0 2 RNN ; (b) the mapping ‰.t; E/ is continuous at all t 2 R and at all E 2 RNN ; (c) ‰.t2 ; ‰.t1 ; E// D ‰.t1 C t2 ; E/ at all .t1 ; t2 / 2 R and at all E 2 RNN . In the analysis of the dynamic graph ˆ.t; D/, the mapping is called the motion of the dynamic graph D, while ‰.t; E/ is called the motion of the adjacent matrix E. The graph of stationary motion determined by the formula ˆ.t; De / D De

at all t 2 R

(4.5.9)

is of interest. The graph De will also be called the equilibrium graph. Analogously, an adjacent equilibrium matrix is determined by the formula ‰.t; Ee / D Ee

at all t 2 R:

(4.5.10)

We shall now consider the notion of stability (instability) of a dynamic graph, if a graph of stationary motion (equilibrium) is specified.

4.5.2 Problem of stability The analysis of the shape and character of the graph motions in the neighborhood of an equilibrium graph or an equilibrium adjacent matrix is of interest, since this analysis allows to determine the persistency in time conditions for a particular structure of a complex system described by a given graph. We shall introduce some definitions, taking into account the notion of stability in the sense of Lyapunov and two metrics: 0 .; De / and .; De / in order to characterize the initial and current state of the dynamic graph. Definition 4.5.3 The equilibrium graph De is called (a) .0 ; /-stable if for any " > 0 and t0 2 R, there exists  D .t0 ; "/ > 0 such that the inequality 0 .D0 ; De / < 

(4.5.11)

.D.t; D0 /; De / < "

(4.5.12)

implies the estimate

at all t  t0 ; (b) uniformly .0 ; /-stable, if in the conditions of Definition 4.5.3, (a) the value  does not depend on t0 2 R;

Comparison Method

177

(c) asymptotically .0 ; /-stable, if it is .0 ; /-stable and for any t0 2 R, there exists  > 0 such that at 0 .D0 ; De / < 

(4.5.13)

lim .D.t; D0 /; De / D 0I

(4.5.14)

the following relation holds: t!1

(d) globally asymptotically .0 ; /-stable if the conditions of Definition 4.5.3 (c) are satisfied at an arbitrarily large  and at all D 2 D; (e) .0 ; /-unstable if the conditions of Definition 4.5.3 (a) are not satisfied. In the case when the properties of a dynamic graph are studied in terms of an adjacent equilibrium matrix, there is a good reason to consider the following definition. Definition 4.5.4 An equilibrium adjacent matrix Ee 2 RNN is said to be: (a) .0 ; /-stable if for any " > 0 and t0 2 R, there exists  D .t0 ; "/ > 0 such that the inequality 0 .E0 ; Ee / < 

(4.5.15)

.E.t; E0 /; Ee / < "

(4.5.16)

implies the estimate

at all t  t0 ; (b) uniformly .0 ; /-stable if all the conditions of Definition 4.5.4 (a) are satisfied with  independent of t0 2 R; (c) asymptotically .0 ; /-stable if it is .0 ; /-stable and for any t0 2 R there exists  > 0 such that at 0 .E0 ; Ee / < 

(4.5.17)

the following relation holds: lim .E.t; E0 /; Ee / D 0I

t!1

(d) globally asymptotically .0 ; /-stable if the conditions of Definition 4.5.4 (c) are satisfied for an arbitrary fixed  and for any matrix E0 2 RNN .

178

Comparison Method

Remark 4.5.2 As there are several possibilities of choosing the two measures, Definitions 4.5.3 and 4.5.4 may be interpreted in different ways. Let us dwell on some of them: (1) let De D 0 and 0 .t; / D .t; / D kDk, where k  k is an Euclidean norm. Then Definition 4.5.3 specifies the stability of a dynamic graph with respect to the zero graph; (2) let Ee D 0 and 0 .t; / D .t; / D kEk. Then Definition 4.5.4 specifies the stability of a dynamic adjacent matrix with respect to the zero adjacent matrix E D 0 2 C.

4.5.3 Evolution of a dynamic graph Let a time scale T with a graininess function .t/ D .t/  t, where .t/ D inffs 2 T; s > tg be given. The function .t/ determines a forward jump operator W T ! T. Define Tk by the formula T=fMg if T has a right-scattered maximum M, and by Tk D T in the other cases (see Chapter 1). Definition 4.5.5 Fix t 2 Tk and let DW T ! D. Determine some matrix D .t/ (if any) with the following properties: for any " > 0 there exists a neighborhood W of a point t for which kŒD..t//  D.s/  D .t/Œ.t/  sk  j.t/  sj at all s 2 W. In this case we will say that D .t/ is a delta derivative of the graph D.t/ at a point t. The evolution of the dynamic graph on a time scale T is described by the matrix equation D .t/ D G.t; D/;

D.t0 / D D0 2 D;

(4.5.18)

where GW T  D ! D. In terms of the dynamic adjacency matrix E.t/, the equation (4.5.18) becomes E .t/ D F.t; E/;

E.t0 / D E0 2 RNN ;

(4.5.19)

where FW T  RNN ! RNN . If T D R, then .t/ D 0 and E D dE dt , and the initial problem (4.5.19) reduces to the initial problem for the ordinary differential matrix equation dE D F.t; E/; dt

E.t0 / D E0 2 RNN :

(4.5.20)

Comparison Method

179

If T D Z, then .t/ D 1 and E D E.t/ D E.t C 1/  E.t/, and the initial problem (4.5.19) reduces to the initial problem for the difference matrix equation E.t C 1/  E.t/ D F.t; E.t//;

E.t0 / D E0 2 RNN :

(4.5.21)

The objective of the qualitative analysis of a dynamic graph is to study the solutions of the matrix system of dynamic equation (4.5.19).

4.5.4 Matrix-valued functions and their applications Now, we associate with the system (4.5.19) the matrix-valued function V.t; E/W T  RNN ! RNN and its total dynamic derivative along the solutions of the system (4.5.19) U  .t; E/ D Ut .t; E..t/// Z 1   P E t; E.t/ C h.t/E .t/ dh E .t/ U C 0

D

(4.5.22)

Ut .t; E..t/// Z

1

C 0

  P E t; E.t/ C h.t/F.t; E.t// dh F.t; E.t//; U

where Ut is calculated as a -derivative of the matrix-valued function U.t; E/ P E is a partial derivative of the with respect to t according to Definition 4.5.5, and U matrix-valued function U.t; E/ with respect to the matrix argument E 2 RNN . Assume that for the expression (4.5.22), there exists a matrix-valued function G.t; V.t; E// such that U  .t; E/j(4.5.19)  G.t; U.t; E//:

(4.5.23)

Along with the matrix inequality (4.5.23), consider the matrix equation M  .t/ D G.t; M.t//;

M.t0 / D M0 2 RNN ;

(4.5.24)

where M.t/ D U.t; E.t//, E.t/ D E.tI t0 ; E0 / at all t 2 T. Further, we shall introduce some notions and definitions for the dynamic equations (4.5.19) and (4.5.24). Assume that for the system (4.5.19), a time scale T with the graininess function .t/ is chosen. Let X1 D RNN and A1 X1 be a space of initial data E0 , such that E.t0 I t0 ; E0 / D E0 2 A1 . Denote by SE a family of motions of the dynamic graph on the time scale T.

180

Comparison Method

Then the sequence of sets and spaces fT; X1 ; A1 ; I; SE g determines the evolution of the dynamic graph on a time scale. Analogously, for the system (4.5.24), take a time scale T with the same graininess function .t/ and denote X2 D RNN , A2 X2 is a space of initial values M0 such that M.t0 I t0 ; M0 / D M0 2 A2 . Let SM be a family of motions of the matrix system (4.5.24). Then the sequence fT; X2 ; A2 ; I; SM g determines the evolution of the matrix dynamic equation (4.5.24) on a time scale. Let the sets N1 X1 and N2 X2 be invariant with respect to the families of motions SE and SM , respectively. The matrix mapping UW T  X1 ! X2 relates the sets N2 and N1 as N2 D U.T  N1 / D fM 2 X2 W M D U.t ; E1 / for some E1 N1 and t 2 Tg:

(4.5.25)

The family of motions SM of the system (4.5.24) and the family of motions SE of the dynamic graph (4.5.19) are related as Sm D M.SE /;

(4.5.26)

where M.SE / D fM.I t0 ; B/ W M.tI t0 ; B/ D U.t; E.tI t0 ; A// for any E.tI t0 ; A/ 2 SE , B D U.t0 ; A/, A 2 A1 and t0 2 Tg. It seems of interest to obtain conditions under which the dynamic properties of the pairs .SM ; N2 / and .SE ; N1 / are equivalent. Note that the systems (4.5.19) and (4.5.24) are determined in the same space of variables RNN , but the system (4.5.24), in view of its construction according to the inequality (4.5.23), can prove to be more traceable compared with the initial system (4.5.19).

4.5.5 A variant of comparison principle Before proceeding to the stability conditions for the system of dynamic equation (4.5.24), we shall formulate a result on the relationship between the dynamic properties of the pairs .SM ; N2 / and .SE ; N1 /. Let 1 .E; N1 / be a metric in the space X1 and 2 .U.t; E/; N2 / be a metric in the space X2 . The function W Œ0; r1  ! RC (respectively W Œ0; 1 ! RC ) is of the Hahn class if .0/ D 0 and .r/ is strictly increasing over Œ0; r1  (on RC ). Functions of this class serve as comparison functions in the theory of stability of motion. Lemma 4.5.1 Assume that evolutions of the systems (4.5.19) and (4.5.24) are determined and there exists a matrix-valued function UW T  X1 ! X2 , such that: (a) the sets of motions SM and SE are related by (4.5.26);

Comparison Method

181

(b) the sets V1 and N2 are closed and related by (4.5.25); (c) there exist comparison functions 1 ; 2 2 K-class, such that 1 .1 .E; N1 //

 2 .U.t; E/; N2 / 

2 .1 .E; N1 //

at all t 2 T and E 2 RNN . Then the following statements hold: (a) the invariance of the pair .SE ; N1 / implies the invariance of the pair .SM ; N2 /; (b) certain type of stability of the pair .SM ; N2 / implies the same type of stability of the pair .SE ; N1 /; (c) the exponential stability of the pair .SM ; N2 / implies the exponential stability of the pair .SE ; N1 / if the comparison functions are of the form i .r/ D ai rb0 , ai > 0, b0 > 0, i D 1; 2. Proof We consider the statement (b) and assume that the pair .SM ; N2 / is stable. Besides, for any "2 > 0 and any t0 2 T, one can find 2 D 2 ."2 ; t0 / > 0 such that 2 .M.tI t0 ; B/; N2 / < "2 at all M.I t0 ; B/ 2 SM and at all t 2 T.B; t0 / T as soon as 2 .B; N2 / < 2 . To prove the stability of the pair .SE ; N1 / for an arbitrary " > 0 and t0 2 T choose "2 D 1 ."/ and  D 21 .2 /. If 1 .A; N1 / < , then, according to the condition (c) of Lemma 4.5.1, we obtain 2 .B; N2 /  2 .1 .A; N1 // < 2 ./ D 2 . It means that for any solution M.t; t0 ; B/ 2 SM the estimate 2 .M.tI t0 ; B// < "2 is true at all t 2 T.B; t0 /. From the conditions (a) and (b) of Lemma 1, we have that E.I t0 ; A/ 2 N1 at all t 2 T.A; t0 / D T.B; t0 /, where B D U.t0 ; A/. From the condition (c) of Lemma 1, it follows that 1 .E.tI t0 ; A/; N1 / 

1

D

1

.U.t; E.tI t0 ; A//; N2 / .2 .M.tI t0 ; B/; N2 / 

1

."2 / D ";

at all t 2 T.A; t0 / D T.B; t0 / as soon as 1 .A; N1 / < . The statement (b) is proved. Other statements of the comparison principle are proved in a similar way. To obtain the sufficient stability conditions for a dynamic graph based on the analysis of the system (4.5.24), we shall fix upon the choice of the matrix-valued function U.t; E/ and the matrix of the function G.t; U/ in the inequality (4.5.23). Let U.t; E/ D EET and G.t; U/ D AU; where A is an N  N-constant matrix, and E 2 RNN . Taking into account the relation E..t// D E.t/ C .t/E .t/

(4.5.27)

182

Comparison Method

on a time scale T with the graininess .t/, we obtain U  .E.t// D EF T .t; E/ C F.t; E/ET C .t/F.t; E/F T .t; E/:

(4.5.28)

In view of (4.5.28), the inequality (4.5.23) becomes U  .E.t//j(4.5.19)  AU.E.t//

(4.5.29)

at all t 2 T, and the matrix comparison equation (4.5.24) M  .t/ D AM.t/;

M.t0 / D M0 2 RNN

(4.5.30)

is linear. Dynamic graphs on time scales can be used as a flexible modeling concept for studying structural changes in a broad range of large-scale systems as diverse as population communities and arms race, compartmental systems and multi-agent formations, chemical processes, and other real-world phenomena. In general, the comparison principle on time scales can be used in a suitable way to explore conditions for a breakdown of large-scale systems due to failures of their components or subsystems. In our approach we have formulated stability conditions for dynamic equations in terms of matrix Lyapunov functions and the comparison principle. The direct Lyapunov method together with the comparison principle proves to be an ideal tool for the stability analysis of dynamic equations.

4.6 Comments and Bibliography In the theory of ordinary differential equations, the comparison method with both scalar and vector functions has been adequately developed in many studies (see, e.g., Aleksandrov and Platonov [1], Corduneanu [1], Borne, Dambrine, Perruquetti, Richard [1], Grujiˇc, Martynyuk, and Ribbens-Pavella [1], Lakshmikantham, Leela, and Martynyuk [1, 2], Matrosov [1], Šiljak [2], Yoshizawa [1], and others.) Meanwhile, in the theory of dynamic equations, the development of the comparison method is at early stage, with a number of problems remaining still open, among them the construction of appropriate Lyapunov functions and adequate comparison equations (systems) as well as the establishment of stability criteria for dynamic comparison equations. This chapter provides a description of a general methodology of stability analysis in terms of the comparison principle and specifies a method for constructing dynamic comparison equations (systems) by means of the simplest Lyapunov functions. Section 4.1. The comparison Theorem 4.1.1 is based on some results from the monographs by Bohner and Peterson [1] and Lakshmikantham, Sivasundaram, and Kaymakçalan [1]. Theorems 4.1.2 and 4.1.3 are new in the context with the vector Lyapunov function.

Comparison Method

183

Section 4.2. The stability Theorems 4.2.1–4.2.2 and their corollaries are new. For their statement, some results of the monograph by Martynyuk [1, 11] were used. The instability Theorems 4.2.3–4.2.4 are taken from Martynyuk [8]. Section 4.3. Theorem 4.3.1 involves a new class of scalar Lyapunov functions on a time scale constructed in terms of matrix-valued functions (see MartynyukChernienko [1, 2]). Theorem 4.3.2 employs the vector Lyapunov function defined on a time scale. The results of this section are new. Section 4.4 contains the results on stability of a system of dynamic equations with respect to two measures and under assumption on the existence of an appropriate matrix-valued Lyapunov function (cf. Lakshmikantham, Leela, and Martynyuk [1]). Theorem 4.4.1 is new. Section 4.5. Dynamic graphs on time scales are defined in linear space as a one-parameter group of transformations of the graph space into itself. Stability of dynamic graphs is formulated in terms of two different measures in the nonnegative orthant of the graph spaces. The conditions for stability are established in terms of the comparison principle and the concept of matrix Lyapunov functions. All the results of Section 4.5 are new. To obtain them, some of the results from Šiljak [3] were used.

Chapter 5

APPLICATIONS

5.0 Introduction Time-continuous and time-discrete dynamic systems as a whole (hybrid systems) are of undoubted interest for applications. The mathematical analysis developed on time scales allows to consider the real-world phenomena in a more accurate description. The purpose of this chapter is the application of some general results of Chapters 1–4 to the study of linear and nonlinear dynamic equations on time scales, which represent a new model of real processes. The chapter is organized as follows. Section 5.1 contains the sufficient conditions for the uniform asymptotic and uniform exponential stability of the equilibrium state of a neural network on a time scale. The model treated in this section is a generalization of the mathematical model of a neuron system proposed by Hopfield [1]. Section 5.2 is concerned with the study of dynamic processes in the case when parameters of a neuron network model are presented in a complex form. Section 5.3 deals with the analysis of stability of a class of Volterra models on dynamic graph. Section 5.4 is devoted to the analysis of stability of oscillations in linear systems with structural perturbations.

5.1 Stability of Neuron Network Consider a neuron network on a time scale, whose dynamics is described by equations of the form x .t/ D Bx.t/ C Ts.x.t// C J;

t 2 Œ0; C1/ \ T:

(5.1.1)

© Springer International Publishing Switzerland 2016 A.A. Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42213-8_5

185

186

Applications

The solution x.tI t0 ; x0 / at t D t0 takes the value x0 , i.e., x.t0 I t0 ; x0 / D x0 ;

t0 2 Œ0; C1/ \ T;

x 0 2 Rn ;

(5.1.2)

where t 2 T, T is an arbitrary time scale, 0 2 T, sup T D C1. In the system (5.1.1), x .t/ is a -derivative on the time scale T; the vector x 2 Rn characterizes the state of the neurons, T D ftij g 2 Rnn ; the components tij describe the interaction between the i-th and the j-th neurons, s W Rn ! Rn , s.x/ D .s1 .x1 /; s2 .x2 /; : : : ; sn .xn //T; the function si describes the response of the i-th neuron, B 2 Rnn , B D diagfbi g, i D 1; 2; : : : ; n; J 2 Rn is a constant vector of the primary input. If T D R, then x D d=dt and at bi > 0 the initial problem (5.1.1)–(5.1.2) is equivalent to the initial problem for a continuous neuron system of Hopefield type: dx.t/ D Bx.t/ C Ts.x.t// C J; dt x.t0 I t0 ; x0 / D x0 ;

t0  0;

t  0;

x 0 2 Rn :

If T D N0 , then x .k/ D x.k C 1/  x.k/ D x.k/ and at j1  bi j < 1 the initial problem (5.1.1)–(5.1.2) is equivalent to the following difference neuron system x.k/ D Bx.k/ C Ts.x.k// C J; x.k0 I k0 ; x0 / D x0 ;

k 0 2 N0 ;

t 2 N0 ;

(5.1.3)

x0 2 R :

(5.1.4)

n

Regarding the system (5.1.1), we make the following assumptions: S1 : S2 : S3 : S4 :

The vector function f .x/ D Bx C Ts.x/ C J is regressive, i.e., the operator I C .t/f .t; / at all t 2 Tk is invertible. Here IW Rn ! Rn is a unit vector. There exist positive constants Mi > 0, i D 1; 2; : : : ; n, such that jsi .u/j  Mi at all u 2 R. There exist positive constants Li > 0, i D 1; 2; : : : ; n, such that jsi .u/  si .v/j  Li ju  vj at all u; v 2 R. The graininess function of the time scale 0 < .t/ 2 M at all t 2 Œ0; C1/\ T, where M R is a compact set.

By Theorem 8.24 from the monograph by Bohner and Peterson [1], it follows that if for any .t0 ; x0 / 2 Œ0; C1/ \ T  Rn the conditions S1  S3 are satisfied, then the problem (5.1.1)–(5.1.2) has exactly one solution over the interval Œt0 ; C1/ \ T. We introduce the notation ƒ D diag fLi g 2 R

nn

;

rD

X n X n iD1

b D minfbi g; i

jD1

b D maxfbi g; i

Mj jTij j C jJi j

2

L D maxfLi g i

=b2i

1=2

;

Applications

187

in the same manner as in the proof of Theorem 3.1 from Wang and Zou [1] and Theorem 1 from Zhang [1]. Lemma 5.1.1 If for the system (5.1.1) the conditions S1  S3 are satisfied, then there exists an equilibrium state x D x of the system (5.1.1), and kx k  r. If, in addition, the matrix Bƒ1  jTj is an M-matrix, then this equilibrium state is unique. The regressivity of the function f .x/ D Bx C Ts.x/ C J is one of the conditions for the existence of a unique solution of the problem (5.1.1)–(5.1.2). Below is given a sufficient condition for the regressivity of the function f .x/. Lemma 5.1.2 Let the assumption S3 hold. If at each fixed t 2 T the matrix .I  .t/B/ƒ1  .t/jTj is an M-matrix, then the function f .x/ D Bx C Ts.x/ C J is regressive. Proof Fix the point t 2 T and consider the mapping RW Rn ! Rn specified by the formula R.x/ D x C .t/f .t; x/ D .I  .t/B/x C .t/Ts.x/ C .t/J: Denote e B D .I  .t/B/, e T D .t/T, e J D .t/J and obtain R.x/ D e Bx C e Ts.x/ C e J: Tj is an M-matrix, the mapping RW Rn ! Rn is a Since the matrix e Bƒ1  je homeomorphism (see Zhang [1]). Hence follows the invertibility of the mapping R.x/, which is equivalent to the invertibility of the mapping I C .t/f ./W Rn ! Rn . Lemma 5.1.2 is proved. Denote the equilibrium state of the system (5.1.1) by x . Perform the substitution of the variable y.t/ D x.t/  x and rewrite the initial problem (5.1.1)–(5.1.2) as y4 .t/ D By.t/ C Tg.y.t//; y.t0 I t0 ; y0 / D y0 ;

t 2 Œ0; C1/ \ T;

t0 2 Œ0; C1/ \ T;

y 0 2 Rn ;

(5.1.5) (5.1.6)

where y 2 Rn , gW Rn ! Rn , g.y/ D .g1 .y1 /; g2 .y2 /; : : : ; gn .yn //T , g.y/ D s.y C x /  s.x /. If for the system (5.1.1) the assumptions S1 –S3 hold true, then for the system (5.1.5) the following statements are true: G1 : G2 : G3 :

The vector function gQ 1 .y/ D By C Tg.y/ is regressive. jgi .u/j  2Mi at all u 2 R and i D 1; 2; : : : ; n. jgi .u/  gi .v/j  Li ju  vj at all u; v 2 R and i D 1; 2; : : : ; n.

Note that if the conditions G1 –G3 are satisfied, then the problem (5.1.5)–(5.1.6) has exactly one solution over the interval Œt0 ; C1/ \ T at any initial data .t0 ; x0 / 2 Œ0; C1/ \ T  Rn .

188

Applications

Theorem 5.1.1 Assume that for the system (5.1.1) on a time scale T the assumptions S1 –S4 hold and there exists a constant  2 M such that .t/   at all t 2 Œ0; C1/ \ T. If the inequality 2b  2LkTk   .b C LkTk/2 > 0 holds, then the equilibrium state x D x of the system (5.1.1) is uniformly asymptotically stable. Proof It is clear that the behavior of the solution x.t/ of the system (5.1.1) in the neighborhood of the equilibrium state x is equivalent to the behavior of the solution y.t/ of the system (5.1.5) in the neighborhood of zero. To prove this, we shall use the Lyapunov function v.y/ D yT y. If y.t/ is -differentiable at the point t 2 Tk , then for the  -derivative of the function v.y.t// along solutions of the system (5.1.5), we have the expression v  .y.t//j(5.1.5) D yT .t/ y .t/ C Œ yT .t/ y..t// D yT .t/ y .t/ C Œ yT .t/ Œ y.t/ C .t/y .t/ D 2yT .t/ŒBy.t/ C Tg.y.t// C .t/k  By.t/ C Tg.y.t//k2  2ƒm .B/ky.t/k2 C 2ky.t/k kTk kg.y.t//k C  .kBk ky.t/k C kTk kg.y.t//k/2 D 2b ky.t/k2 C 2kTk ky.t/k ky.t/k C  .b ky.t/k C kTk kg.y.t//k/2 : Taking into account the estimate kg.y.t//k  L ky.t/k, we get the expression v  .y.t//j(5.1.5)  2 b ky.t/k2 C 2LkTk ky.t/k2  2 C  b ky.t/k C LkTk ky.t/k   D  2b  2LkTk   .b C LkTk/2 ky.t/k2 : Thus, all the conditions of Corollary 4.2 from Bohner and Martynyuk [1] are satisfied. Hence, the zero equilibrium condition y D 0 of the system (5.1.5) is uniformly asymptotically stable. The theorem is proved. Theorem 5.1.2 Let the following conditions be satisfied: (1/ for the system (5.1.1) on a time scale T, the assumptions S1 –S4 hold true; (2/ the functions si 2 C2 .R/ and there exist constants Ki > 0 such that js00i .u/j  Ki at all u 2 R, i D 1; 2; : : : ; n; (3/ there exists a constant  > 0 such that .t/   at all t 2 Œ0; C1/ \ T;

Applications

189

(4/ there exists a positive-definite symmetric matrix P 2 Rnn such that the following inequality is true:

M .PB1 C BT1 P/ C  kPkkB1 k2 < 0; where B1 D B C TG, G D diagfs0i .0/g 2 Rnn . Then the equilibrium state x D x of the system (5.1.1) is uniformly asymptotically stable. Proof Using the expansion of the functions gi .yi / into the Maclaurin series, one can easily find that the function g.y/ can be represented in the form g.y/ D Hy C gQ 2 .y/, where H D diagfg0i .0/g 2 Rnn , gQ 2 W Rn ! Rn and the following estimate holds true: kQg2 .y/k  Kkyk2 ;

K D maxfKi g=2: i

(5.1.7)

For the -derivative of the function v.y/ D yT Py along solutions of the system (5.1.5), we obtain v  .y.t//j(5.1.5) D yT .t/Py .t/ C Œ yT .t/ Py..t// D yT .t/Py .t/ C Œ yT .t/ Py.t/ C .t/Œy .t/T Py .t/   T D yT .t/P B1 y.t/ C T gQ 2 .y.t// C B1 y.t/ C T gQ 2 .y.t// Py.t/ T   C .t/ B1 y.t/ C T gQ 2 .y.t// P B1 y.t/ C T gQ 2 .y.t//   yT .t/ PB1 C BT1 P y.t/ C 2yT .t/PT gQ 2 .y.t//

(5.1.8)

C .t/kPk kB1 y.t/ C T gQ 2 .y.t//k2    ƒM .PB1 C BT1 P/ C C.t/kPk kB1 k2 ky.t/k2 C 2kPk kTk kQg2 .y.t//k ky.t/k C .t/kPk kQg2 .y.t//k2 kTk2 C 2.t/kPk kB1 k kTk kQg2 .y.t//k ky.t/k: Taking into account the inequality (5.1.7) and the condition (3) of Theorem 5.1.2, we arrive at the inequality   v  .y.t//j(5.1.5)  ƒM .PB1 C BT1 P/ C  kPk kB1 k2 ky.t/k2 C 2KkPk kTk ky.t/k3 C 2 KkPk kB1 k kTk ky.t/k3 C  K 2 kPk kTk2 ky.t/k4 :

190

Applications

Denote .kyk/ D akyk2 ;

  a D  ƒM .PB1 C BT1 P/ C  kB1 kkPk2 > 0;   m. / D 2a3=2 KkPkkTk 1 C  kB1 k 3=2 C  a2 K 2 kPkkTk2

2

;

and for the -derivative of the function V along solutions of the system (5.1.5), we obtain the estimate v  .y.t//j(5.1.5)   .kyk/ C m. .kyk//: Since

2 K-class and lim m. / D 0, all the conditions of Corollary 4.2 from !0

Bohner and Martynyuk [1] are satisfied, and therefore the zero equilibrium state y.t/ D 0 of the system (5.1.5) is uniformly asymptotically stable. The theorem is proved. Theorem 5.1.3 Let the following conditions be satisfied: (1/ for the system (5.1.1), the assumptions S1 –S3 hold; (2/ the functions si 2 C2 .R/ and there exist constants Ki > 0 such that js00i .u/j  Ki at all u 2 R, i D 1; 2; : : : ; n; (3/ there exists a positive- definite symmetric matrix P 2 Rnn and there exists a constant M > 0 such that j1 C .t/A.t/j  M at all t 2 Œt0 ; C1/ \ T, where A.t/ D M .PB1 C BT1 P/ C .t/kPkkB1 k2 and B1 D B C TG, G D diag fs0i .0/g 2 Rnn . Then: .i/ if lim sup ˇA . / D q < 0, then the equilibrium state x D x of the t!1

system (5.1.1) is exponentially stable; .ii/ if supfˇA .t/W t 2 Œ0; C1/g D q < 0, then the equilibrium state x D x of the system (5.1.1) is uniformly exponentially stable. Proof Using the expression (5.1.8) for the -derivative of the function v.y/ D yT Py along solutions of the system (5.1.5), we obtain the estimates   v  .y.t//j(5.1.5)  ƒM .PB1 C BT1 P/ C .t/kPk kB1 k2 ky.t/k2 C 2kPk kTk kQg2 .y.t//k ky.t/k C 2.t/kPk kB1 k kTk kQg2 .y.t//k ky.t/k  C .t/kPk kQg2 .y.t//k2 kTk2  ƒM .PB1 C BT1 P/   C .t/kPk kB1 k2 ky.t/k2 C 2KkPk kTk ky.t/k  C 2.t/KkPk kB1 k kTk ky.t/k C .t/K 2 kPk kTk2 jy.t/k2 ky.t/k2 D A.t/ky.t/k2 C ˆ.t; v.y//;

Applications

191

where  p ˆ.t; v/ D 2KkPk kTk.1 C .t/kB1 k/ v C .t/K 2 kPk kTk2 v v: Now consider the set T D ft 2 Œ0; C1/ \ TW .t/ ¤ 0g. If there exists sup T < C1, then there exists t1 2 Œ0; C1/\T such that .t/ D 0 at all t 2 Œt1 ; C1/\T. If the set T is unbounded, then from the inequality lim sup ˇA .t/ D q < 0, it t!1

follows that there exists a sufficiently large t2 2 Œ0; C1/ \ T such that at all t 2 Œt2 ; C1/ \ T the inequality ˇA .t/ < 0 holds. Then at all t 2 Œt2 ; C1/T \ T ˇ ˇ logˇ1 C .t/.ƒM .PB1 C BT1 P/ C .t/kPkkB1 k2 /ˇ < 0; or .t/.ƒM .PB1 C BT1 P/ C .t/kPk kB1 k2 /  1 < 1; kPk kB1 k2 2 .t/ C ƒM .PB1 C BT1 P/.t/  2  0: Since D D ƒM .PB1 CBT1 P/2 C8kPkkB1 k2  0, we obtain the inequality .t/  1 for all t 2 Œt2 ; C1/ \ T , where 1 D .ƒM .PB1 C BT1 P/ C

p D/=2kPkkB1 k2  0:

Consequently, .t/  1 at all t 2 Œt3 ; C1/ \ T, t3 D maxft1 ; t2 g. If t 2 Œ0; .t3 / \ T, then .t/  t3 . Hence it follows that the estimate .t/   D maxf1 ; t3 g holds at all t 2 Œ0; C1/T . Then p ˆ.t; v/ D 2KkPk kTk.1 C .t/kB1 k/ v C .t/K 2 kPk kTk2 v v p  2KkPk kTk.1 C  kB1 k/ v C  K 2 kPk kTk2 v; and ˆ.t; v/=v ! 0 as v ! 0 uniformly with respect to t. All the conditions of Theorem 2 from Martynyuk [7] are satisfied, therefore the equilibrium state y D 0 of the system (5.1.5) is exponentially stable, which is equivalent to the exponential stability of the equilibrium state x D x of the system (5.1.1). Now prove the second part of the theorem. From the condition supfˇA W t 2 Œ0; C1/ \ Tg D q < 0 for t 2 T , it follows that logj1 C .t/.ƒM .PB1 C BT1 P/ C .t/kPkkB1 k2 /j  .t/q < 0 at all t 2 T ; therefore ƒM .PB1 C BT1 P/ C .t/  2kPkkB1 k2

p D

D  ;   0; t 2 T :

192

Applications

That is .t/   for any t 2 Œ0; C1/ \ T. Hence, as above, we have that ˆ.t; v/=v ! 0 as v ! 0 uniformly with respect to t and the equilibrium state x D x of the system (5.1.1) is exponentially stable. The theorem is proved. Remark 5.1.1 Consider the scale T D N0 (.t/  1). In this case the initial problem (5.1.1)–(5.1.2) will be equivalent to the problem (5.1.3)–(5.1.4), and the condition for the uniform asymptotic stability of the equilibrium state of the system (5.1.1) obtained in Theorem 5.1.1, at  D 1 will become 2b  2LkTk  .b C LkTk/2 > 0: This result fully agrees with the following result for the discrete system (5.1.3). Theorem 5.1.4 Let for the neuron discrete system (5.1.3) the assumptions S2 , S3 hold. Then the equilibrium state x D x of the system (5.1.3) will be uniformly asymptotically stable provided that 2b  2LkTk  .b C LkTk/2 > 0: Proof Perform the substitution of variables y.k/ D x.k/  x and write the equation (5.1.3) as y.k C 1/ D .B C I/y.k/ C Tg.x.k//;

k 2 N0 :

(5.1.9)

For the first difference of the function v.y/ D yT y, the following estimates hold: v.y.k//j(5.1.9) D yT .k C 1/y.k C 1/  yT .k/y.k/ D Œ.B C I/y.k/ C Tg.y.k//T Œ.B C I/y.k/ C Tg.y.k//  yT .k/y.k/ D yT .k/BT By.k/  2yT .k/BT y.k/  2y.k/T BTg.y.k// C 2yT .k/Tg.y.k// C GT .y.k//T T Tg.y.k//  kBk2 ky.k/k2  2ƒm .B/ky.k/k2 C 2LkBk kTk ky.k/k2 C 2LkTk ky.k/k2 C kTk2 kg.y.k//k2  2  b  2b C 2LbkTk C 2LkTk C kTk2 L2 k.y.k//k2  D  2b  2LkTk  .b C LkTk/2 k.y.k//k2 : The proof of the theorem follows immediately.

Applications

193

Example 5.1.1 On the time scale, P1; D

1 [

Πj.1 C /; j.1 C / C 1 ;

> 0;

jD0

consider the bicomponent neuron network x 1 D b1 x1 C t11 s.x2 / C t12 s.x2 / C u1 ; x 2 D b2 x1 C t21 s.x1 / C t22 s.x2 / C u2 ; where x1 , x2 2 R, b1 D b2 D 1, T D the graininess function ( .t/ D

0; t 2 ;

t2

(5.1.10)

 0;1 0;5  0;5 0;1 , s.r/ D th r. For the time scale P1;

S1 jD0

Πj.1 C /; j.1 C / C 1/ ;

jD0

fj.1 C / C 1g :

S1

Choosing the matrix P D diagf0; 5I 0; 5g, we obtain the function

ˇA .t/ D

8 1 ˆ ˆ < log j1 C .0; 9 C 0; 53 /j; ˆ ˆ :0; 9 C 0; 53 ;

t2

1 S

t2

1 S

fj.1 C / C 1g ;

jD0

Πj.1C /; j.1 C / C 1/ ;

jD0

and the regressivity condition in the form of inequalities 

1  1; 1 > 0; .1  1; 1 /2  0; 25 2 > 0:

At 0 < < 0; 625 all the conditions of Lemmas 5.1.1 and 5.1.2 and Theorem 5.1.3 are satisfied. The system (5.1.10) has a unique equilibrium state at any u1 , u2 2 R, and this equilibrium state is uniformly exponentially stable.

5.2 Stability of a Complex-Valued Neuron Network Let C denote a set of complex numbers. A function z .t/ D .Re z.t// Ci .Im z.t// will be called the -derivative of the complex-valued function z D Re z C i Im zW T ! Cn on a time scale T. The function f W T  Cn ! Cn is called regressive if the operator I C .t/f .t; / at all t 2 Tk is invertible. Here IW Cn ! Cn is a unit operator. From now on the following notation will be used: j!j is the module of the complex number ! and !N is a number complex conjugate to !. If z D

194

Applications

.z1 ; z2 ; : : : ; zn /T is a vector with complex components, then z D .Nz1 ; zN2 ; : : : ; zNn /, kzk2 D z z D jz1 j2 C jz2 j2 C    C jzn j2 . If A D faij g is a matrix with complex components, then jAj D fjaij jg, AT D faji g, A D fNaji g, ƒm .A/, ƒM .A/ are the smallest and the largest eigenvalues of the matrix A, respectively, kAk D .ƒM .A A//1=2 . Consider a complex-valued neuron network whose dynamics is described by equations of the form z .t/ D Cz.t/ C Ts.x.t// C J;

t 2 Π; C1/ \ T:

(5.2.1)

The solution z.tI t0 ; z0 / at t D t0 takes the value z0 , i.e., z.t0 I t0 ; z0 / D z0 ;

t0 2 Π; C1/ \ T;

z0 2 Rn ;

(5.2.2)

where 2 T, T is an arbitrary time scale for which sup T D C1. In the system (5.2.1) ; z .t/ is a -derivative on a time scale T; the vector z 2 Cn characterizes the state of the neurons, T D ftij g 2 Cnn ; the components tij describe the connections between the i-th and the j-th neurons, sW Cn ! Cn , s.z/ D .s1 .z1 /; s2 .z2 /; : : : ; sn .zn //T; the function si describes the response of the i-th neuron, C 2 Cnn , C D diag fc1 ; c2 ; : : : ; cn g, i D 1; 2; : : : ; n; J 2 Cn is a constant vector of the primary input. If T D R, then z D dz=dt, and under the condition ci < 0, the initial problem (5.2.1)–(5.2.2) is equivalent to the initial problem for a continuous complex-valued neuron system of Hopefield type (see Nitta [1]): dz.t/ D Cz.t/ C Ts.z.t// C J; dt z.t0 I t0 ; z0 / D z0 ;

t0  ;

t  ; z0 2 Cn :

If T D Z, then z .k/ D z.k C 1/  z.k/ D z.k/, Œ ; C1/ \ T D f ; C 1; C 2; : : : g and at jci C 1j < 1 the initial problem (5.2.1)–(5.2.2) is equivalent to the following (see Nitta [1]) z.k C 1/ D Cz.k/ C Ts.x.k// C J; x.k0 I k0 ; z0 / D z0 ;

t 2 f ; C 1; C 2; : : : g;

k0 2 f ; C 1; C 2; : : : g;

z0 2 Cn :

With regard to the system (5.2.1), we introduce the following assumptions: Assumption 5.2.1 The vector function f .z/ D CzC Ts.z/ C J is regressive. Assumption 5.2.2 There exist positive constants li > 0, i D 1; 2, : : : ; n, such that at all %; ! 2 C the following inequalities hold: jsi .%/  si .!/j  li j%  !j;

i D 1; 2; : : : ; n:

Applications

195

If Assumptions 5.2.1 and 5.2.2 hold, then at any initial data .t0 ; z0 / 2 Œ ; C1/ \ T  Cn the problem (5.2.1)–(5.2.2) has exactly one solution over an infinite interval Œt0 ; C1/ \ T. The conditions for the existence of a unique equilibrium state of the system (5.2.1) are given in Bohner, Rao, and Sanyal [1]. Lemma 5.2.1 Assume that the function sW Cn ! Cn is such that s.0/ D 0 and there exists a positive constant L > 0 such that js.z/  s.&/j  Ljz  &j at all z; & 2 Cn . Determine the constant ˇ D kI C Ck C LkTk. If ˇ 2 .0; 1/, then the system (5.2.1) has a single equilibrium state. Denote .t/ D .t/.kCk C LkTk/, L D maxfl1 ; l2 : : : ; ln g and prove the following statement. Lemma 5.2.2 Let Assumption 5.2.2 hold and s.0/ D 0. If .t/ < 1 at all t 2 T, then the function f .z/ D Cz C Ts.z/ C J is regressive for any J 2 Cn . Proof To prove Lemma 5.2.2, it is necessary to show that the mapping FW Cn ! Cn specified by the formula F.z/ D z C .t/.Cz C Ts.z/ C J/ is invertible at each fixed t 2 T, i.e., that at any fixed t 2 T the equation F.z/ D w has a unique solution at all w 2 Cn . Fix t 2 T. If .t/ D 0, then the equation F.z/ D w has a unique solution z D w. Now let .t/ ¤ 0. In this case the equation F.z/ D w has a unique solution if and only if the mapping GW Cn ! Cn specified by the formula G.z/ D w  .t/.Cz C Ts.z/ C J/ has a single fixed point. For any z 2 Sd .0/, where d > .t/kJk=.1  /, the following inequalities hold: kG.z/k D .t/kCz C T.s.z// C Jk  .t/.kCk C LkTk/kzk C .t/kJk  d C .t/kJk < d; which means that G.Sd .0// Sd .0/. Show that the mapping G is contracting. At any z1 ; z2 2 Cn the following inequalities hold kG.z1 /  G.z2 /k D .t/kC.z2  z1 / C T.s.z2 /  s.z1 /k  .t/.kCk C LkTk/kz2  z1 k  .t/kz2  z1 k: Since .t/ < 1, then, according to the principle of contracting mappings, the mapping G has a single fixed point. Lemma 5.2.2 is proved. Assume that the system (5.2.1) has an equilibrium state ze . Definition 5.2.1 The equilibrium state z D ze of the system (5.2.1) is said to be globally exponentially stable if there exists a function p 2 RC such that ep .t; t0 / ! 0 at t ! 1 and there exist constants N > 0 and ˛ > 0 such that kz.tI t0 ; z0 /  ze k < N.ep .t; t0 //˛ at all z0 2 Cn , t 2 Œt0 ; C1/ \ T and t0 2 Œ ; C1/ \ T.

196

Applications

Perform the substitution of variables .t/ D z.t/  ze and rewrite the initial problem (5.2.1)–(5.2.2) as  .t/ D C .t/ C Tg. .t//; .t0 I t0 ; 0 / D 0 ;

t 2 Π; C1/ \ T;

t0 2 Π; C1/ \ T;

0 2 Cn ;

(5.2.3)

where g. / D s. C ze /  s.ze /. It is clear that the behavior of the solution z.t/ of the system (5.2.1) in the neighborhood of the equilibrium state ze is equivalent to the behavior of the solution .t/ of the system (5.2.3) in the neighborhood of zero. Denote !.t/ D .j 1 .t/j; j 2 .t/j; : : : ; j n .t/j/T, ƒ D diagfl1 ; l2 ; : : : ; ln g and prove the following auxiliary statement. Lemma 5.2.3 If Q 2 Cnn , then for the -derivative of the function v.t/ D  .t/Q .t/ along the solution of the system (5.2.3), the following inequality holds: v  .t/j(5.2.3)   .t/A.t/ .t/ C !.t/TB.t/!.t/; where A.t/ D C Q C QC C .t/C QC;



 B.t/ D ƒjTjTjQj C jQjjTjƒ C .t/ ƒjTjTjQjjCj C jCjTjQjjTjƒ C .t/ƒjTjTjQjjTjƒ:

(5.2.4)

(5.2.5)

In particular, if Q D diag fq1 ; q2 ; : : : ; qn g is a diagonal matrix, then the following inequality holds: v  .t/j(5.2.3)  !.t/TD.t/!.t/; where D.t/ D A.t/ C B.t/:

(5.2.6)

Proof Denote !.t/ D !, .t/ D , .t/ D , g. .t// D g, and since .  Q / D .  / Q  C  Q  D .  / Q  C  Q  D .  / Q. C   / C  Q  D .  / Q C  Q  C .  / Q  ; we obtain the inequalities v  .t/j(5.2.3) D .C C Tg/ Q C  Q.C C Tg/ C .B C Tg/ Q.B C Tg/

Applications

197

D  C Q C  QC C   C QC C g T  Q C g T  QC C  QTg C   C QTg C g T  QTg h i D  C Q C QC C C QC C 2Ref  QTgg C 2Ref  C QTgg C g T  QTg: Consider the second summand separately. Ref  QTgg  j  QTgj D j 

X

X ;k;j

N qk tkj gj j 

X

j  jjqk jjtkj jjgj j

;k;j

j  jjqk jjtkj jlj j j j D ! TjQjjTjƒ!:

;k;j

Estimating in the same way the third and the fourth summands, for the -derivative of the function v.t/ D  .t/Q .t/ along the solution of the system (5.2.3), we obtain the estimate h i v  .t/j(5.2.3)   C Q C QC C .t/C QC C 2! TjQjjTjƒ! C 2! TjCjTjQjjTjƒ! C ! TƒjTjTjQjjTj!   A.t/ C ! TB.t/!: This completes the proof. If the matrix Q is diagonal, then the matrix A.t/ is diagonal too, and v  .t/j(5.2.3)   A.t/ C ! TB.t/! D A.t/  C ! TB.t/! D A.t/! T! C ! TB.t/! D ! TA.t/! C ! TB.t/! D ! TD.t/!: The second part of the lemma is proved. Now formulate the sufficient conditions for the exponential stability. Theorem 5.2.1 Let the conditions of Assumptions 5.2.1 and 5.2.2 be satisfied. If there exists a positive-definite Hermitian matrix Q 2 Cnn for which the function p.t/ D

ƒM .A.t// C ƒM .B.t// ; ƒm .Q/

where the matrices A.t/, B.t/ are specified by the formulas (5.2.4) and (5.2.5), is such that p 2 RC and limt!1 ep .t; t0 / D 0, then the equilibrium state z D ze of the system (5.2.1) is globally exponentially stable.

198

Applications

Proof We denote by .t/ an arbitrary solution of the system (5.2.3) and consider the function v. / D  Q . For the -derivative of the function v. .t// along the solution of the system (5.2.3) in view of Lemma 5.2.3, the following inequalities hold: v  . .t//j(5.2.3)   .t/A.t/ .t/ C j!.t/jTB.t/j!.t/j  ƒM .A.t//  .t/ .t/ C ƒM .B.t//! T! h i D ƒM .A.t// C ƒM .B.t// k .t/k2 

ƒM .A.t// C ƒM .B.t// v. .t// D p.t/v. .t//: ƒm .Q/

Hence, according to Theorem 6.1 from Bohner and Peterson [1], we obtain the estimate v. .t//  v. 0 /ep .t; t0 /; or  .t/ .t/  v. 0 /ƒ1 m .Q/ep .t; t0 /: Hence follows the global exponential stability of the zero equilibrium solution of the system (5.2.3). Theorem 5.2.2 Let the conditions of Assumptions 5.2.1 and 5.2.2 be satisfied. If there exists a positive-definite Hermitian matrix Q D diag fq1 ; q2 ; : : : ; qn g 2 Cnn for which the function p.t/ D ƒM .D.t//ƒm .Q/1 , where the matrix D.t/ is specified by the formula (5.2.6), is such that p 2 RC and lim ep .t; t0 / D 0, then t!1 the equilibrium state z D ze of the system (5.2.1) is globally exponentially stable. Example 5.2.1 On an arbitrary time scale T, consider a bicomponent complexvalued neuron network of the form z 1 D c1 z1 C t11 s.z1 / C t12 s.z2 /; z 2 D c2 z1 C t21 s.z1 / C t22 s.z2 /;

(5.2.7)

where z1 , z2 2 C, c1 D c2 D 0:1, the function s.u/ satisfies Assumption 5.2.2 with the constants l1 D l2 D 1,   0:02083 C 0:02083 i 0:04166 C 0:02083 i TD : 0:06250  0:04166 i 0:02083 C 0:02083 i In view of the fact that L D 1, kTk D 0:09139, .t/ D 0:19139 .t/, the condition for the regressivity of the function f .z/ becomes .t/ < 5:22475.

Applications

199

Choose a matrix Q D I and calculate the matrices jTj D

  0:02946 0:04658 ; 0:07511 0:02946

  0:2 C 0:01 .t/ 0 A.t/ D ; 0 0:2 C 0:01 .t/   0:05892 C 0:01370 .t/ 0:12170 C 0:12600 .t/ B.t/ D : 0:12170 C 0:12600 .t/ 0:05892 C 0:00953 .t/ For the time scale T D R, the graininess function .t/  0 and  D.t/ D

 0:12930 0:12170 : 0:12170 0:12930

Since ƒM .D.t// D 0:00759, we obtain the function p.t/  0:00759 and ep .t; t0 / D ep.tt0 / D e0:00759.tt0 / ! 0. Since all the conditions of Theorem 5.2.2 are satisfied, the zero equilibrium state of the system (5.2.7) is globally exponentially stable. For the time scale T D Z, the graininess function .t/  1 and A.t/ D

  0:19 0 ; 0 0:19

B.t/ D

  0:05892 0:12170 : 0:12170 0:05892

Since ƒM .A.t//CƒM .B.t// D 0:00937, we obtain the function p.t/  0:00937 and ep .t; t0 / D .1 C p/tt0 ! 0. All the conditions of Theorem 5.2.1 are satisfied and therefore the equilibrium state of the system (5.2.7) is globally exponentially stable. Remark 5.2.1 It is impossible to apply Theorem 4.4 from Bohner, Rao, and Sanyal [1] because D 0:2 C ˇ 1 C 0:09139ˇ > 0 at all ˇ > 0.

5.3 Volterra Model on Time Scale From the analysis of the nature of complex systems in mathematical biology, it becomes clear that complex systems with time-varying interaction between subsystems have not been researched. Indeed, in the literature complex systems are

200

Applications

described by the system of differential equations (see Šiljak [1]): dxi D gi .t; xi / C hi .t; ei1 x1 ; ei2 x2 ; : : : ; eiN xN /; dt

i D 1; 2; : : : ; N;

(5.3.1)

where equations dxi D gi .t; xi /; dt

i D 1; 2; : : : ; N;

describe the motion of the disconnected subsystems. Functions hi describe action of all subsystems of the complex system on the i-th subsystem. Parameter eik replies for the action of the k-th subsystem on the i-th one and eik is constant. So, the actual problem is to construct the mathematical model and research the complex systems with time-varying interconnection between their subsystems.

5.3.1 Generalization of Volterra model Since interconnection matrix E D Œeij Ni;jD1 in the complex system (5.3.1) may be considered as an adjacent matrix of some graph G D .V; E/, where V D fV1 ; V2 ; : : : ; VN g is a nonempty finite set of N nodes and E D f.Vi ; Vj /jVi ; Vj 2 V; i; j D 1; Ng is a set of ribs, then the earlier mentioned problem is to construct the example of complex systems, in which interconnections between subsystems would assign some time-varying or, perhaps, dynamic graph (see Šiljak [3]). Following the setting problem in Section 4.5, consider the generalization of the well known in mathematical biology and ecology Volterra model of the community of n species. The generalized system is described by the system of dynamic equations on some time scale T: n  X Ni .t/ D Ni "i  ij Nj ;

i D 1; 2; : : : ; n;

(5.3.2)

jD1

where Ni .t/ is a number of individuals of the i-th species at the moment t 2 T, Ni .t/ is a delta derivative of the function Ni .t/ at a point t 2 T. In the case when T D R (when the number of the species changes quickly enough, such i scales are considered; communities of bacteria are an example), Ni .t/ D dN . If dt T D hZ, h > 0 (when the number of the species changes over long periods of time such scales are considered; communities of higher animals are an example), then Ni .t/ D Ni .t/ D Ni .t C h/  Ni .t/. When the number of species changes with the different intensities on the different time intervals, the scale with inconstant graininess function .t/ (.t/  0, when T D R, and .t/  h, when T D hZ), can be applied to such species dynamics modeling. The intensity can be affected, for example, by habitat conditions (climate, geography, forage base, etc.).

Applications

201

In addition, in (5.3.2) "i denotes a rate of natural growth or mortality of the i-th species in the absence of other species. The sign and the absolute value of ij .i ¤ j/ represent the nature and intensity of influence of the j-th species to i-th; ii is an indicator of infraspecific competition. We assume now that n species whose dynamics are described by the system (5.3.2) are the preys and identify interconnections in a community of m species, where the individuals are predators, feeding on individuals of preys. Denote by Sk .k D 1; 2; : : : ; m/ the set of those n species of the prey’s community, which form the forage base of the k-th species of the predator community. Also define N.Sk / by the formula: N.Sk / D

X

Ni ;

i2Sk

that is, N.Sk / is equal to the volume of the k-th predator’s forage base. Predator’s community dynamics can be described by the system (5.3.2): m  X Mi .t/ D Mi ˛i  ˇij Mj ;

i D 1; 2; : : : ; m;

(5.3.3)

jD1

where Mi .t/ is the number of individuals of the i-th species at the moment t 2 T and Mi .t/ is a delta derivative of the function Mi .t/. Also in (5.3.3) ˛i denotes a rate of natural growth or mortality of the i-th species in the absence of other species, and ˇij represent the nature and intensity of influence of the j-th species to the i-th. In this case, it seems natural to assume that the effect of the j-th to the i-th is dependent on the percentage of the species, forming the mutual forage base, in the j-th species forage base. That is: ˇij D ˇij

 N.S \ S / i j : N.Sj /

N.S \S /

i j The more large the ratio N.S is (the interval Œ0; 1 is the range of the ratio), the j/ larger the j-th species makes bids for the mutual with the i-th species forage base, thereby affecting on the i-th species of the community of predators. So, we have constructed an example of the complex system that is described by the system of equations

m   N.S \ S / X i j Mi .t/ D Mi ˛i  Mj ; ˇij N.S / j jD1

i D 1; 2; : : : ; m;

(5.3.4)

and the interconnections between the subsystems are described by the system of equation (5.3.2).

202

Applications

So, adjacent matrix E.t/ D Œeij m i;jD1 of some dynamic graph D is constructed. The matrix satisfies the following system of equations: E.t/ D B Ni .t/

 N.S \ S / i j ; N.Sj /

n  X D Ni "i  ij Nj ;

(5.3.5) i D 1; 2; : : : ; n:

jD1

5.3.2 Stability analysis Let us consider the particular case when the functions ˇij are linear: ˇij

 N.S \ S / N.Si \ Sj / i j D Qij : N.Sj / N.Sj /

Let the community of preys consist of the three species z1 ; z2 ; z3 and the community of predators consist of two species. Suppose that the forage base S1 of the first species of the predators is fz1 ; z2 g and the forage base S2 of the second species of the predators is fz2 ; z2 g. Then the interconnection parameters ˇij satisfy the following relations: ˇ11 D Q11 ; ˇ21 D Q21

ˇ12 D Q12

N2 ; N1 C N2

N2 ; N2 C N3

ˇ22 D Q22 ;

3  X Ni .t/ D Ni "i  ij Nj ;

(5.3.6)

i D 1; 2; 3:

jD1

The equation (5.3.6) describes the evolution of a dynamic graph, consisting of two preys. The value ˇij .t/, as it was mentioned, denotes the weight of the edge .Vi ; Vj /. For the dynamic graph D , which is represented by equation (5.3.6), consider the problem of existence of the adjacent equilibrium matrix and of its stability in terms of Definition 4.5.2. As we see from the formula (5.3.6), the value of the adjacent equilibrium matrix Ee is assigned by the equilibrium state of the system of dynamic equation (5.3.2) on the time scale. That is, adjacent equilibrium matrix Ee equals   e Q11 ˇ12 E D e ˇ21 Q22 e

Applications

203

if and only if components Nie (i D 1; 2; 3) of the equilibrium vector of the system (5.3.2) satisfy the system of equations: 3  X ij Nj D 0; Ni "i 

i D 1; 2; 3;

jD1

Q12 N2 e D ˇ12 ; N2 C N3

(5.3.7)

Q21 N2 e D ˇ21 : N1 C N2 Suppose now that the adjacent matrix equals to Ee D E and let N  D .N1 ; N2 ; N3 /T be a corresponding state vector of the system (5.3.2) (i.e., the solution of the system (5.3.7)). Establish the stability conditions of the state N  . It is easy to see that stability conditions of the state N  of the system (5.3.2) are also stability conditions of the equilibrium matrix Ee D E . In the system (5.3.2), replace the value Ni to xi by the formula: xi D Ni  Ni ;

i D 1; 2; 3;

(5.3.8)

to obtain stability conditions. We obtain the system of dynamic equations 3  X    x .t/ D N .t/ D .x C N / "  .x C N / D i i ij j i i i j jD1 3 3 3   X X X D xi " i  ij Nj  Ni ij xj  ij xi xj ; jD1

jD1

(5.3.9) i D 1; 2; 3;

jD1

and x 1 .t/

3  X D "1  1j Nj  N1 11 x1  N1 12 x2  N1 13 x3 jD1



3 X

1j x1 xj ;

jD1

x 2 .t/

D

N2 21 x1

3  X C "2  2j Nj  N2 22 x2  N2 23 x3 jD1



3 X jD1

2j x2 xj ;

(5.3.10)

204

Applications 3  X   x 3j Nj  N3 33 x3 3 .t/ D N3 31 x1  N3 32 x2 C "3  jD1



3 X

3j x3 xj :

jD1

Denoting x D .x1 ; x2 ; x3 /T ;

0

1 P1 N1 12 N1 13 A D @N2 21 P2 N2 23 A ; N3 31 N3 32 P3 F.x/ D .F1 .x/; F2 .x/; F3 .x//T ;

Fi .x/ D 

3 X

ij xi xj ;

jD1

where Pi D "i 

3 P jD1

ij Nj  Ni ii , i D 1; 2; 3, we obtain the vector form of the

system (5.3.10): x .t/ D Ax C F.x/;

(5.3.11)

lim kF.x/k D 0:

(5.3.12)

with the conditions kxk!0

Now the stability conditions of the equilibrium state N  of the system (5.3.2) are the stability conditions of the trivial equilibrium of the system (5.3.11), which can be obtained by the generalized Lyapunov direct method. According to the method, consider the positive-definite function v.x/ D xT x D x21 C x22 C x23 ;

x 2 R3 ;

and compute the total -derivative of v.x/ with respect to the solutions of the system (5.3.11). Using the product rule, we find ˇ ˇ v  .x/ˇ

(5.3.11)

ˇ  T ˇ D x x C xT x ˇ

(5.3.11)

ˇ  T  ˇ x C .t/x C xT x ˇ D x

(5.3.11)

(5.3.13)

Applications

205

ˇ ˇ D .Ax C F.x//T .x C .t/.Ax C F.x/// C xT .Ax C F.x//ˇ

(5.3.11)

D x .A C A C .t/A A/x C ‰..t/; x/ T

T

T

D xT .AT ˚ A/x C ‰..t/; x/; where ‰..t/; x/ D F T .x/x C xT F.x/ C .t/.xT AT F.x/ C F T .x/Ax C F T .x/F.x//: Here we have used a symbol of regressive sum: AT ˚ A D AT C A C .t/AT A. Now if there exists negative-definite matrix B 2 R33 such that inequality xT .AT ˚ A/x  xT Bx;

8t 2 T;

8x 2 D  R3 ;

(5.3.14)

holds, then the equilibrium state x D 0 is stable by Theorem 3.3.2 from Chapter 3. Indeed, conditions (1), (2a), and (2b) for the function v.x/ hold. From (5.3.13) and (5.3.14), we obtain ˇ ˇ v  .x/ˇ  xT Bx C ‰..t/; x/; (5.3.11)

where the function ‰..t/; x/ satisfies the inequality k‰..t/; x/k  2kF.x/kkxk.1 C .t/kAk/: Using the equality (5.3.12), we compute lim

kxk!0

k‰..t/; x/k  lim 2kF.x/k.1 C .t/kAk/ D 0: kxk kxk!0

That is, conditions (2b) and (2c) of Theorem 3.3.2 hold; therefore by Theorem 3.3.2 the equilibrium state x D 0 of the system (5.3.11) is asymptotically stable which implies the asymptotical stability of the state N D N  of the system (5.3.2). So, in the case when the system (5.3.7) can be solved with respect to N1 , N2 , and N3 , there exists the equilibrium matrix Ee D

  e Q11 ˇ12 ; e ˇ21 Q22

which is asymptotically stable, when (5.3.14) holds.

206

Applications

5.4 Stability of Oscillations The examples of systems modeled by the system of second-order dynamic equations are LCL, CLC, and LC-chains with switches, and the motion of a spring-suspended body of mass m which certain additional mass m is attached to or separated from at some instants of time (see Bohner and Peterson [1, 2], Marks II et al. [1], etc.).

5.4.1 Statement of the problem We consider a system of fourth-order linear dynamic equations which can describe two connected systems with switches of given type (

x .t/ D 1 x C ˛A.!; t/y; y .t/ D ˇAT.!; t/x C 2 y;

where x; y 2 R2 , t 2 T D

S

(5.4.1)

Œk; k C , 2 .0; 1/, 1 ; 2 2 R, ! 2 Œ0; 2/,

k2Z

 A.!; t/ D

 cos !t sin !t :  sin !t cos !t

With regard to the system (5.4.1), we also suppose that there exist p; q 2 N (. p; q/ D 1) such that T D 2 D pq , i.e.,  D Tq D p. ! Assume that system (5.4.1) is subject to structural perturbations in the sense of Šiljak [1]. The parameters ˛ and ˇ, responsible for structural perturbations, take their values from the sets f˛1 ; ˛2 g and fˇ1 ; ˇ2 g, respectively. Since the matrices ˛1 A.!; t/ and ˛2 A.!; t/ differ by the scalar multiplier only, instead of S1 .t/h1 .t/ we shall write S1 .t/h1 A.!; t/, where S1 .t/ D .s11 ; s12 /, h1 D .˛1 ; ˛2 /T. Similarly, instead of S2 .t/h2 .t/ we shall write S2 .t/h2 AT.!; t/, where S2 .t/ D .s21 ; s22 /, h1 D .ˇ1 ; ˇ2 /T. The structural parameter sij W T ! f0; 1g is the .i; j/-th element of the structural matrix S.t/ of system (5.4.1) such that the correlation sij D 1 implies sik D 0 for all k ¤ j and skj D 1 for all k. We state the problem on the stability of solution x D 0, y D 0 of the system (5.4.1) for all S 2 G.

5.4.2 Stability under structural perturbations To solve this problem, we need the following result.

Applications

207

Theorem 5.4.1 If there exists a value S0 of the structural matrix S.t/ from the structural set G of the system x .t/ D A.t/x;

(5.4.2)

positive-definite matrices P11 2 Rn1 n1 , P22 2 Rn2 n2 and real numbers 1 ; 2 ; ı; " > 0 such that for all t0 2 T, t  t0 and S 2 G the inequalities (1) (2) (3) (4) (5)

e.˛j ˚ˇk / .t0 C ; t0 / ¤ 1, for all ˛j 2 .A11 /, ˇk 2 .A22 /; kP12 .t/k2 C ı  m .P11 / m .P22 /I

M1 D M .B1 .t; S0 // < 0;

M2 D M .B2 .t; S0 // < 0;

M .K.t; S.t/// < minf M1 ;  M2 g  "

are satisfied, where P12 .t/ denotes the -periodic function tZ 0 C

P12 .t/ D

G.t; . //F. ; S0 / ; t0

then the solution x D 0 of the system x .t/ D A0 x C S.t/h.t/x;

S.t/ 2 G;

8 t 2 T;

(5.4.3)

is asymptotically stable for every S 2 G. Following the conditions of this theorem, we find expressions for functions P12 .t/, B1 .t; S.t//, B2 .t; S.t//, C.t; S.t//,F.t;S.t//, D.t; S.t// assuming that 1 D 10 . As A11 D 1 I and A22 D 2 I, so 2 D 1, P11 D I, P22 D I and S0 D 01 eA11 .t; t0 / D e1 .t; t0 /I, eA22 .t; t0 / D e2 .t; t0 /I and then ˆ.t; t0 / D e.1 ˚2 / .t; t0 /I. Let t0 D 0 and t ¤ k, then e.1 ˚2 / .t; 0/ D e.1 C2 /.tŒt/ Œt ; where

D

e.1 C2 / : .1 C .1  /1 /.1 C .1  /2 /

(5.4.4)

The expression obtained for e.1 ˚2 / .t; 0/ is also correct in the case t D k what can be verified immediately. Further F.t; S.t// D .S1 .t/h1 .1 C .t/1 /  S2 .t/h2 .1 C .t/2 //A.!; t/:

(5.4.5)

208

Applications

Using (5.4.4) and (5.4.5), we get Zt 0

 '1 .t/ '2 .t/ D .t/; ˆ.t; . //F. ; S0 / D '2 .t/ '1 .t/ 

where '1 .t/ is found by the formula  ˛1 C ˇ1 '1 .t/ D  p cos.!t  '/ ! 2 C .1 C 2 /2   C e.1 C2 /.Œtt/ g1 . ; Œt; '/ C Œt f1 . / ;

(5.4.6)

in which f1 . / D

 . C /  2 1 e 1 2 cos.!  '/  2 cos ! C 1     cos..  1/!  '/  2 cos '  cos.! C '/   C ı. /e.1 C2 / 2 cos !  cos.  1/! ;

2

g1 . ; Œt; '/ D

 1  cos.!Œt  '/

2  2 cos ! C 1

C e.1 C2 / cos.!.Œt  1 C /  '/

 C ı. / e.1 C2 / cos !.Œt  1 C / C cos.!.Œt C 1/  '/

  e.1 C2 / cos.!.Œt C /  '/  ı. / e.1 C2 / cos !.Œt C / ;

where ı. / D

p ! 2 C .1 C 2 /2

.1 /.˛1 .1C.1 /1 /Cˇ1 .1C.1 /2 // ˛1 Cˇ1

and ' denotes an auxiliary angle for which cos ' D p p ! . ! 2 C.1 C2 /2

1 C2

! 2 C.1 C2 /2

, sin ' D

Similarly, we find '2 .t/ by the formula  ˛1 C ˇ1 '2 .t/ D  p sin.!t  '/ 2 2 ! C .1 C 2 /   C e.1 C2 /.Œtt/ g2 . ; Œt; '/ C Œt f2 . / ;

(5.4.7)

Applications

209

where f2 . / D

 . C /  2 1 e 1 2 sin.!  '/

2  2 cos ! C 1     sin..  1/!  '/  2 sin '  sin.! C '/   C ı. /e.1 C2 / 2 sin !  sin.  1/! ;

g2 . ; Œt; '/ D

2

 1  sin.!Œt  '/  2 cos ! C 1

C e.1 C2 / sin.!.Œt  1 C /  '/

 C ı. / e.1 C2 / sin !.Œt  1 C / C sin.!.Œt C 1/  '/

  e.1 C2 / sin.!.Œt C /  '/  ı. / e.1 C2 / sin !.Œt C / :

From (5.4.4), (5.4.6), and (5.4.7), we have P12 .t/ D

e.1 ˚2 / .t; 0/ . p/ C .t/ 1  e.1 ˚2 / . p; 0/

! .1/ .2/ e.1 C2 /.Œtt/ Œt p12 .t/ p12 .t/ D . p/ C .t/ D ; .2/ .1/ 1  p p12 .t/ p12 .t/ where  ˛1 C ˇ1 .1/ cos.!t  '/ p12 .t/ D  p 2 2 ! C .1 C 2 /  C e.1 C2 /.Œtt/ g1 . ; Œt; '/ ; .2/ p12 .t/

˛1 C ˇ1



(5.4.8)

D p sin.!t  '/ ! 2 C .1 C 2 /2  C e.1 C2 /.Œtt/ g2 . ; Œt; '/ :

Applying formula (5.4.8) we compute the elements of matrices B1 .t; S.t//, B2 .t; S.t//, C.t; S.t//, D.t; S.t//. Introduce the designations   .˛1 C ˇ1 / r1 D  p cos ' C e.1 C2 /.Œtt/ g1 . ; Œt  t; '/ ; 2 2 ! C .1 C 2 /   .˛1 C ˇ1 / r2 D  p  sin ' C e.1 C2 /.Œtt/ g2 . ; Œt  t; '/ ; ! 2 C .1 C 2 /2

210

Applications

by means of which the abovementioned elements are determined as follows: .1/

.1/

b11 D b22 D 21 C .t/12 C .t/ˇ 2  2.t/ˇ1 ˇ  .1/

2ˇr1 2˛1 ˇ.t/.1 C .t/1 / C ; .1 C .t/2 / 1 C .t/2

.1/

b12 D b21 D r2 I .2/

.2/

b11 D b22 D 22 C .t/22 C .t/˛ 2  2.t/˛1 ˛  .2/

2˛r1 2˛ˇ1 .t/.1 C .t/2 / C ; .1 C .t/1 / 1 C .t/1

.2/

b12 D b21 D r2 I c11 D c22 D 2..˛  ˛1 /.1 C .t/1 / C .ˇ  ˇ1 /.1 C .t/2 // cos !t; c12 D c21 D 2..˛  ˛1 /.1 C .t/1 / C .ˇ  ˇ1 /.1 C .t/2 // sin !t; 2.t/˛ˇ.˛1 C ˇ1 / p .1 C .t/1 /.1 C .t/2 / ! 2 C .1 C 2 /2    cos.3!t  '/ C e.1 C2 /.Œtt/ g1 . ; Œt C 2t; '/   ˇ1 .t/ ˛1 .t/ cos 3!t; C C 2.t/˛ˇ 1 C .t/2 1 C .t/1

d11 D d22 D 

2.t/˛ˇ.˛1 C ˇ1 / p .1 C .t/1 /.1 C .t/2 / ! 2 C .1 C 2 /2    sin.3!t  '/ C e.1 C2 /.Œtt/ g2 . ; Œt C 2t; '/   ˇ1 .t/ ˛1 .t/ sin 3!t: C C 2.t/˛ˇ 1 C .t/2 1 C .t/1

d12 D d21 D 

Then we find 1 k1 C k2 2 p  C .k1  k2 /2 C 2..d11 C c11 /2 C .d12 C c12 /2 / ;

max .K.t; S// D

where ˇ  p ˇ 2r1 k1 D 2ˇˇ.ˇ  ˇ1 / C .t/.ˇ  ˇ1 / 1 C .t/2 ˇ 2˛1 .t/.1 C .t/1 / ˇˇ  ˇ; 1 C .t/2

Applications

211

ˇ  p ˇ 2r1 ˇ k2 D 2ˇ.˛  ˛1 / C .t/.˛  ˛1 / 1 C .t/1 ˇ 2ˇ1 .t/.1 C .t/2 / ˇˇ  ˇ: 1 C .t/1 Thus, the conditions of structural asymptotic stability of solution x D 0, y D 0 of the system (5.4.1) for all t  0, ˛ 2 f˛1 ; ˛2 g and ˇ 2 fˇ1 ; ˇ2 g are as follows: e.1 C2 / ¤ 1; .1 C .1  /1 /.1 C .1  /2 /  .1/ 2  .2/ 2 p12 .t/ C p12 .t/  1  ı; .1/

b11 D 21 C .t/12 C .t/ˇ 2  2.t/ˇ1 ˇ  C

2ˇr1 < 0; 1 C .t/2

.2/

b11 D 22 C .t/22 C .t/˛ 2  2.t/˛1 ˛  C

2˛1 ˇ.t/.1 C .t/1 / .1 C .t/2 /

2˛ˇ1 .t/.1 C .t/2 / .1 C .t/1 /

2˛r1 < 0; 1 C .t/1

p  1 k1 C k2 C .k1  k2 /2 C 2..d11 C c11 /2 C .d12 C c12 /2 / 2 ˚ .1/ .2/

< min  b11 ; b11  ";

(5.4.9)

where ı and " are preassigned positive numbers and .t/ D 0 or 1  . Remark 5.4.1 For T D R, i.e., for D 1 and in the absence of structural perturbations,   i.e., when the structural set G consists of the only element S D 10 ID , the system (5.4.1) takes the form of the system of ordinary differential 01 equations 8 dx ˆ < D 1 x C ˛A.!; t/y; dt dy ˆ : D ˇAT.!; t/x C  y; 2 dt

I

the stability of its solution x D y D 0 was studied in the paper by Lila [1]. It can be easily shown that for T D R conditions (5.4.9) coincide with the conditions of stability of this paper if and only if 2 ¤ 1 . Really, in view of the fact that

D e.1 C2 / , ı1 .1/ D 0, we find by the formulas for functions g1 and g2 that

212

Applications

g1 . ; Œt; '/ D g2 . ; Œt; '/ D 0. Setting ı  1, from the second inequality (5.4.9), we get an obvious result 0  1  ı. The third condition becomes 1