Stability theory for dynamic equations on time scales

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 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 11 13 15 19 22 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 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 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 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 Applications 209 where f2 / D e cos ! C 1/! sin C ı /e g2 ; Œt; '/ D C e C ı / e e sin.!.Œt 1C / 1C 2/ 2 '/ sin ! cos ! C 1C 2/ 1C / 1C 2/ sin ' sin.! C '/ 1/! ; sin '/ '/ 1C /C sin.!.Œt C / '/ sin.!Œt 1C / sin !.Œt sin.! sin.!.Œt C 1/ ı / e '/ 1C / '/ sin !.Œt C / : From (5.4.4), (5.4.6), and (5.4.7), we have P12 t/ D D e« ˚ / t; 0/ p/ C e« ˚ / p; 0/ e C /.Œt t/ Œt p p/ C t/ t/ D ! 1/ 2/ p12 t/ p12 t/ ; 2/ 1/ p12 t/ p12 t/ where 1/ p12 t/ D ˛1 C ˇ1 p ! C 1C C e .2/ p12 t/ D C /.Œt t/ 2/ C e C C /.Œt t/ '/ g1 ; Œt; '/ ; ˛1 C ˇ1 p !2 C cos.!t 2/ (5.4.8) '/ sin.!t 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 r1 D ˛1 C ˇ1 / p ! C C 2/ r2 D ˛1 C ˇ1 / p !2 C C 2/ cos ' C e C /.Œt sin ' C e t/ C /.Œt g1 ; Œt t/ g2 ; Œt t; '/ ; t; '/ ; 210 Applications by means of which the abovementioned elements are determined as follows: 1/ 1/ b11 D b22 D C t/ C t/ˇ 2 t/ˇ1 ˇ 2ˇr1 2˛1 ˇ t/.1 C t/ / C C t/ / C t/ 1/ b12 D 2/ 1/ 2/ C t/ 2 C t/˛ 2 t/˛1 ˛ 2˛r1 2˛ˇ1 t/.1 C t/ / C C t/ / C t/ 2/ ; b21 D r2 I b11 D b22 D b12 D ; 2/ b21 D r2 I c11 D c22 D ˛ c12 D c21 D ˛ d11 D d22 D ˛1 /.1 C t/ / C ˇ ˇ1 /.1 C t/ // cos !t; ˛1 /.1 C t/ / C ˇ ˇ1 /.1 C t/ // sin !t; t/˛ˇ.˛1 C ˇ1 / p C t/ /.1 C t/ / ! C C 2/ cos.3!t '/ C e C /.Œt t/ g1 ; Œt C 2t; '/ à  ˇ1 t/ ˛1 t/ cos 3!t; C C t/˛ˇ C t/ C t/ d12 D d21 D t/˛ˇ.˛1 C ˇ1 / p C t/ /.1 C t/ / ! C C 2/ sin.3!t '/ C e C /.Œt t/ g2 ; Œt C 2t; '/ à  ˇ1 t/ ˛1 t/ sin 3!t: C C t/˛ˇ C t/ C t/ Then we find max K.t; S// k1 C k2 p C k1 k2 /2 C d11 C c11 /2 C d12 C c12 /2 / ; D where ˇ p ˇ k1 D 2ˇˇ.ˇ  ˇ1 / 2r1 C t/ C t/.ˇ Ãˇ 2˛1 t/.1 C t/ / ˇˇ ˇ; C t/ ˇ1 / Applications 211 ˇ p ˇ k2 D 2ˇˇ.˛  ˛1 / 2r1 C t/ C t/.˛ ˛1 / Ãˇ 2ˇ1 t/.1 C t/ / ˇˇ ˇ: C t/ Thus, the conditions of structural asymptotic stability of solution x D 0, y D of the system (5.4.1) for all t 0, ˛ f˛1 ; ˛2 g and ˇ fˇ1 ; ˇ2 g are as follows: e 1C 2/ / /.1 C 1 C 1/ p12 t/ 1/ b11 D C t/ 2/ C t/ 2 C t/˛ C p k1 C k2 C k1 2/ C p12 t/ C t/ˇ C b11 D 2 / 2/ 2 t/ˇ1 ˇ 2ˇr1 C t/ ı; 2˛1 ˇ t/.1 C t/ / C t/ / < 0; t/˛1 ˛ 2˛r1 C t/ Ô 1; 21 t/.1 C t/ / C t/ / < 0; (5.4.9) k2 /2 C d11 C c11 /2 C d12 C c12 /2 / < ˚ 1/ 2/ « b11 ; b11 "; where ı and " are preassigned positive numbers and t/ D or Remark 5.4.1 For T D R, i.e., for D and in the absence of structural perturbations, i.e., when the structural set G consists of the only element S D  à 10 , the system (5.4.1) takes the form of the system of ordinary differential ID 01 equations dx ˆ < D x C ˛A.!; t/y; dt dy ˆ : D ˇAT.!; t/x C y; dt I the stability of its solution x D y D 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 Ô Really, in view of the fact that D e C / , ı1 1/ D 0, we find by the formulas for functions g1 and g2 that 212 Applications g1 ; Œt; '/ D g2 ; Œt; '/ D Setting ı Ä 1, from the second inequality (5.4.9), we get an obvious result Ä ı The third condition becomes < ˇ.˛ C ˇ/ C / : ! C C /2 (5.4.10) Similarly, the fourth condition becomes < ˛.˛ C ˇ/ C / : ! C C /2 1/ (5.4.11) 2/ If in the fifth condition we let " D 12 maxf b11 ; b11 g, it will be transformed to an obvious result < " Having designated D , D , ˇ D , ! D , we see that conditions (5.4.10) and (5.4.11) coincide with the conditions of asymptotic stability of solution z1 D z2 D of the system dz ˆ D < dt ˆ : dz2 D dt z1 C ˛e Jt z2 ; z2 C ˛e Jt z1 ; à Fixing the values of parameters as 10 D 199:0, D 2:85, ! D 6:05, we obtain a domain of asymptotic stability of solution x D y D of system (5.4.1) for T D R and in the absence of structural perturbations  where zi R2 , 1; ; ˛; ; R, J D Remark 5.4.2 For T D Z, i.e., when D and in the absence of structural perturbations, the system (5.4.1) takes the form of the system of difference equations x.t C 1/ D C /x.t/ C ˛A.!; t/y.t/; y.t C 1/ D ˇAT.!; t/x.t/ C C /y.t/: The stability of its solution x D y D was studied by Martynuyk and Slyn’ko [3] Let us show that for T D Z conditions (5.4.9) coincide with the stability conditions of this paper, if only ; 2pÔ In view of the fact that D 1C /.1C / and ı.0/ D g1 ; Œt; '/ D C cos.!.t cos.!t ˛.1C /Cˇ.1C // ˛Cˇ cos ! C ! C C /2 cos.!t 1/ '/ C ı.0/ cos.!.t '/ ı.0/ cos !t ; we find '/ 1// C cos.!.t C 1/ '/ Applications 213 ˛ C ˇ/ı.0/ cos.!.t 1// p 2 cos ! C 1/ ! C 1/ p12 t/ D cos !t/ C 2/ : Similarly, g2 ; Œt; '/ D cos ! C C sin.!.t sin.!t 1/ '/ C ı.0/ sin.!.t '/ ı.0/ 2/ p12 t/ D 1/ p12 t/ sin.!t '/ 1// C sin.!.t C 1/ sin !t ; ˛ C ˇ/ı.0/ sin.!.t 1// p 2 cos ! C 1/ ! C 2/ C p12 t/ '/ D ˛.1 C sin !t/ C 2/ : C ˇ.1 C //2 : cos ! C 1/ Now, if ı is considered sufficiently small, the second condition (5.4.9) can be written as j˛.1 C 1/ C ˇ.1 C /j < 1: cos ! C (5.4.12) The third and forth conditions become 1/ b11 D C 1/ 2.1 C 1 /ˇ.˛.1 C 2/ b11 D C 2/ 2.1 C 1/ C ˇ.1 C 1/ C ˇ.1 C // cos !/ // cos !/ cos ! C C ˇ < 0; (5.4.13) /˛.˛.1 C cos ! C C ˛ < 0: (5.4.14) The fifth condition (5.4.9) becomes j˛ˇj j˛.1 C 1/ C ˇ.1 C /j 1/ 2/ Ä minf b11 ; b11 g: cos ! C (5.4.15) It can be easily verified that if the designations from Martynuyk and Slyn’ko [3] are used in (5.4.12) – (5.4.15), these conditions will imply the conditions of stability of 214 Applications solution x D y D of the system of difference equations x C 1/ D x / C ˛A.!; /y /; y C 1/ D ˇAT.!; /x / C C  where A.!; t/ D NC /y à cos !t sin !t and x; y R2 , ˛; ˇ; sin !t cos !t /; 1; 2 R, ! Œ0; /, Fixing the values of parameters, we obtain a domain of asymptotic stability of solution x D y D of the system (5.4.1) for T D Z and in the absence of structural perturbations 5.5 Comments and Bibliography The mathematical analysis on a time scale allows a more complete description of the dynamics of time-continuous and time-discrete systems As is well known, this is one of the classes of hybrid systems extensively developed within the recent decades (see Martynyuk [13]) In this chapter some problems connected with the application of dynamic equations are discussed Section 5.1 The dynamics of neuron networks is a new domain of application of qualitative methods developed in the classical theory of equations Neuron networks (see Hopfield [1]) have been adequately developed on RC and on Z, i.e., for timecontinuous and discrete-time models This section is based on the results from Luk’yanova and Martynyuk [1] Section 5.2 Neuron networks with complex parameters are described in detail in Nitta [1] The main results of the section are taken from Luk’yanova and Martynyuk [2] Section 5.3 Main results of this section are given according to the paper by Babenko and Martynyuk [2] For other applications in mechanics, see Marks II et al [1] Section 5.4 The results of this section are adapted from Babenko [1] and were obtained under the supervision of Martynyuk Some other applications of dynamic equations are due to Gard and Hoffacker [1]; Atici and McMahan [1]; Atici, Biles, and Lebedinsky [1]; and Bohner, Fan, and Zhang [1] For more applications, see the monograph by Martynyuk [11] The paper by Yao [1] deals with almost periodicity of a host-macroparasite model on time scales (see also Ding et al [1], Hang [1], Zhuang and Wen [1], etc.) For linear control systems on time scales, see Bartosiewicz and Pawluszewicz [2] and Marks II et al [1] The conditions of connective stability on time scales were established in the paper by Luk’yanova and Martynyuk [2] References 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circle minus, Circle plus, Comparison functions of the same order of growth, 96 Comparison method, 146 Comparison principle, 180 Converse theorem, 114 Cylinder transformation, Delta derivative of the graph, 178 Delta-derivative, Diamond-alpha-derivative, 19 Diamond-alpha-integral, 20 Dynamic adjacent matrix, 176 Dynamics of the i-th interacting subsystem, 125 Dynamics of the i-th isolated subsystem, 126 Equation with structural perturbations, 127 dynamic Lyapunov, 116 integral dynamic, 53 matrix comparison, 182 matrix differential Lyapunov, 116 quasilinear, 61 quasilinear dynamic, 80 quasistationary Lyapunov, 116 scalar quasilinear, 75 Equilibrium adjacent matrix ; /-stable, 177 asymptotically ; /-stable, 177 globally asymptotically ; /-stable, 177 uniformly ; /-stable, 177 Equilibrium graph ; /-stable, 176 ; /-unstable, 177 asymptotically ; /-stable, 177 globally asymptotically ; /-stable, 177 uniformly ; /-stable, 176 Estimations of solutions, 52 Evolution of the dynamic graph, 178 Exponential stability, 73 Floquet theory, 48 Function auxiliary, 88 matrix-valued, 90 scalar, 88 vector, 89 comparison of K-class, 90 © Springer International Publishing Switzerland 2016 A.A Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Systems & Control: Foundations & Applications, DOI 10.1007/978-3-319-42213-8 221 222 Index delta differentiable, exponential, examples, 10 matrix, 11, 12 properties, Green, 120 positively regressive, scalar, 146 vector, 147 Function v.t; x;  / –increasing, 169 definite –positive, 169 weakly –decreasing, 169 uniformly continuous, 168 Moving set, 159 Moving set A.r/ invariant for solutions, 159 moving set B p/ conditionally invariant with respect to the set A.r/, 159 Nabla derivative, 15 Neuron network complex-valued, 193 Neuron network on a time scale, 185 Neuron system continuous, 186 difference, 186 Nonlinear system r-dynamics, 137 -dynamics, 137 polydynamics, 137 Graininess, Graph dynamic, 175, 200 equilibrium, 176 Parametric and/or external perturbations, 125 Point left-dense, left-scattered, right-dense, right-scattered, Polydynamics on time scales, 136 Induction principle, Inequalities dynamic with a nabla derivative, 29 Gronwall-Bellman, 25 nonlinear, 31 Stakhurskaya’s, 32 Inequality Wazhewsky, 110 Initial value problem, 12, 15, 18, 22 Rd-continuity, Regressive group, 9, 12 Regressivity, 8, 12–14, 18 Jump operators, backward, forward, Set conditionally invariant, 161, 165 uniformly asymptotically stable, 161, 165 Set Kappa operator, 3, 15 Keller’s chain rule, Ld-continuity, 17 Lyapunov function ˛-rhombic derivative , 138 Lyapunov transformation, 43 Matrix-valued function, 91 decreasing, 91 positive (negative) semidefinite, 91 positive definite, 91 radially unbounded, 91 Measure with respect to the measure continuous, 168 invariant, 160 uniformly asymptotically stable, 160 uniformly equi-asymptotically stable, 160 Set B p/ uniformly asymptotically stable, 160 uniformly quasi-asymptotically stable, 160 uniformly stable, 160 Set B p0 ; p/ conditionally invariant, 163 uniformly asymptotically stable, 163 Solution x D unstable, 76, 156 Stability of a dynamic graph, 174 Stability of linear systems, 105 Stability of oscillations, 206 Stability under perturbations, 78 Index Stability under structural perturbations, 206 Stability with respect to two measures, 168 State x D x uniformly asymptotically stable, 188, 189, 192 uniformly exponentially stable, 190 State x D equiasymptotically stable, 153 exponentially stable, 96 stable (asymptotically stable) in the whole, 106 uniformly asymptotically stable, 108 uniformly exponentially stable, 96 uniformly stable, 57 uniformly stable (uniformly asymptotically stable) in the whole, 107 unstable, 103, 104, 109, 155 State z.t/ D ze globally exponentially stable, 195, 197, 198 Structural matrix, 129 structural matrix Ei , 126 structural parameter eij , 126 Structural perturbations, 125 Structure of system, 127 System described by the dynamic equations, 86 heterogeneous linear, 50 linear periodic Lyapunov function, 118 nonatonomous, 37 of difference equations, 86 of ordinary differential equations, 86 periodic linear, 128 time-invariant, 44 System of dynamic equations ; /-equi-stable, 169 ; /-equi-stable, 170 quasi-equi-asymptotically ; /-stable, 169 quasi-uniformly ; /-stable, 169 uniformly ; /-stable, 169, 170 223 uniformly asymptotically ; /-stable, 169, 170 The set Q invariant and uniformly asymptotically stable, 164 Time scale, examples, Volterra model, 199 Time scale T !-periodic, 128 Two-index system of functions, 118 Unperturbed motion, 38, 45 exponentially stable robustly, 45 uniformly, 45 weakly uniformly, 45 asymptotically stable, 38 exponentially stable, 45 exponentially stable, uniformly with respect to t, 40 stable, 38 uniformly stable, 38 Variation of constants, 13, 15 Weighted directed graph, 174 Zero polydynamics, 139 asymptotically stable, 141 uniformly asymptotically stable, 139 uniformly stable, 139 unstable, 139, 142 Zero solution u D equistable, 151 ... 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... 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... 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