Explicit Stability Conditions for Continuous Systems: A Functional Analytic Approach (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

314

Michael I. Gil’

Explicit Stability Conditions for Continuous Systems A Functional Analytic Approach

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Author Prof. Dr. Michael I. Gil’ Ben Gurion University of Negev Department of Mathematics P.O. Box 653 84 105 Beer Sheva Israel

ISSN 0170-8643 ISBN 3-540-23984-7 Springer Berlin Heidelberg New York Library of Congress Control Number: 2005920065 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, 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. Typesetting: Data conversion by the authors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141Yu - 5 4 3 2 1 0

Preface

This book deals with nonautonomous linear and nonlinear continuous finite dimensional systems. Explicit conditions for the asymptotic, absolute, inputto-state and orbital stabilities are discussed. The problem of stability analysis of various systems continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems of control theory, because of the absence of its complete solution. The problem of the synthesis of a stable system is closely connected with the problem of stability analysis. Any progress in the problem of analysis implies success in the problem of synthesis of stable systems. The basic method for the stability analysis of nonlinear systems is the Lyapunov functions one. By this method many very strong results are obtained, but finding Lyapunov’s functions is often connected with serious mathematical difficulties, especially in regard to nonstationary systems. The stability conditions presented in this book are mainly formulated in terms of the eigenvalues of auxiliary matrices. This fact allows us to apply the well-known stability criteria for polynomials and matrices (for example the Hurwitz criterion) to the stability analysis of time-varying (linear and nonlinear) systems. The main methodology presented in this publication is based on a combined use of recent norm estimates for matrix-valued functions with the following well-known methods and results: a) the method of characteristic exponents (the first Lyapunov method); b) the multiplicative representations of solutions; c) the freezing method; d) the positivity of the Green (impulse) functions. Here we do not consider the Lyapunov functions method because several excellent books cover this topic. A significant part of this book is devoted to a solution to the problem connected with the Aizerman conjecture. Recall that in 1949 M. A. Aizerman conjectured that a single input-single output system is absolutely stable in the Hurwitz angle. This hypothesis caused great interest among the specialists. With the help of counter examples it was shown that the conjecture is not true, in general. The problem of finding a class of systems that satisfy Aizerman’s hypothesis arose. One of the most powerful results in this direction was obtained in 1966 by N. Truchan who showed that Aizerman’s hypothesis is satisfied by systems having linear parts in the form of single loop

VI

Preface

circuits with up to five stable aperiodic links connected in tandem. In 1981 the author showed that any system satisfies the Aizerman hypothesis if its Green function is non-negative. That result includes the Truchan one. Similar results are derived for multivariable systems. They are also presented here. The aim of this book is to provide new tools for specialists in control system theory and stability theory of ordinary differential equations. We suggest a systematic exposition of the approach to stability analysis which is based on estimates for matrix-valued functions. That approach allows us to investigate various classes of systems from the unified viewpoint. The book is intended not only for specialists in stability theory, but for anyone interested in various applications who has had at least a first year graduate level course in analysis. I was very fortunate to have fruitful discussions with Professors M.A. Aizerman, V.M. Alekseev, M.A. Krasnosel’skii, A. Pokrovskii and A.A. Voronov, to whom I am very grateful for their interest in my investigations. Acknowledgment. This work was supported by the Kamea Fund of the Israel. Michael I. Gil’

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definitions of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Eigenvalues of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Matrix-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Norms of Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Upper Bounds for Lyapunov’s Equation . . . . . . . . . . . . . . . . . . . 1.10 Lower Bounds for Lyapunov’s Equation . . . . . . . . . . . . . . . . . . . 1.11 Estimates for Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Absolute Values of Matrix Functions . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Statement of the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.2 Proof of Theorem 1.12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 8 9 10 13 15 17 18 21 22 23 23 25 26

2.

Perturbations of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Evolution Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Perturbations of Evolution Operators . . . . . . . . . . . . . . . . . . . . . 2.3 Representation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Triangular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Perturbations of Triangular Systems . . . . . . . . . . . . . . . . . . . . . . 2.6 Integrally Small Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Integrally Small Perturbations of Autonomous Systems . . . . . .

27 27 29 31 32 34 35 37

3.

Systems with Slowly Varying Coefficients . . . . . . . . . . . . . . . . . 3.1 The Freezing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Proofs of Theorems 3.1.1 and 3.1.5, and Lemma 3.1.7 . . . . . . . 3.3 Systems with Differentiable Matrices . . . . . . . . . . . . . . . . . . . . . . 3.4 Additional Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 43 46 49

VIII

Table of Contents

3.5 Proofs of Theorems 3.3.1, 3.3.3, and 3.4.2 . . . . . . . . . . . . . . . . . . 3.6 Matrix Lipschitz Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Proof of Theorem 3.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Estimates for z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Lower Solution Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 54 54 56 59 60

Dissipative and Piecewise Constant Systems . . . . . . . . . . . . . . 4.1 The Lozinskii Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linear Systems with Majorants and Minorants . . . . . . . . . . . . . 4.3 Systems with Piecewise Constant Matrices . . . . . . . . . . . . . . . . . 4.4 Perturbations of Systems with Piecewise Constant Matrices . . . . . . . . . . . . . . . . . . . . . . . . 4.5 General Second Order Vector Systems . . . . . . . . . . . . . . . . . . . . . 4.6 Proof of Theorem 4.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Second Order Vector Systems with Differentiable Matrices . . .

61 61 63 64

5.

Nonlinear Systems with Autonomous Linear Parts . . . . . . . . 5.1 Statement of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Theorem 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Stability Conditions in Terms of Determinants . . . . . . . . . . . . . 5.3.1 Statement of the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Global Stability Under Matrix Conditions . . . . . . . . . . . . . . . . . 5.5 Proof of Theorem 5.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Region of Attraction of One-Contour Systems . . . . . . . . . . . . . . 5.7 Proof of Theorem 5.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 74 74 76 77 79 80 82

6.

The Aizerman Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Absolute Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Systems with Matrix Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 90 91

7.

Nonlinear Systems with Time-Variant Linear Parts . . . . . . . 7.1 Systems with General Linear Parts . . . . . . . . . . . . . . . . . . . . . . . 7.2 Systems with the Lipschitz Property . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Statement of the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Proofs of Corollary 7.2.2 and Theorem 7.2.3 . . . . . . . . . 7.3 Systems with Differentiable Linear Parts . . . . . . . . . . . . . . . . . . 7.3.1 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Proof of Theorem 7.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Absolute Stability and Region of Attraction . . . . . . . . .

93 93 95 95 98 99 99 101 104

4.

65 66 68 69

Table of Contents

IX

7.4 Additional Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Proof of Theorem 7.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Absolute Stability and Region of Attraction . . . . . . . . . 7.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106 106 107 108 108

8.

Essentially Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Freezing Method for Nonlinear Systems . . . . . . . . . . . . . . . 8.1.1 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Lyapunov’s Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Proofs of Theorems 8.1.1 and 8.1.4 . . . . . . . . . . . . . . . . . . . . . . . 8.3 Perturbations of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 8.4 Nonlinear Triangular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Perturbations of Triangular Systems . . . . . . . . . . . . . . . . . . . . . . 8.6 Nonlinear Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Nonlinear Systems with Linear Majorants . . . . . . . . . . . . . . . . . 8.8 Second Order Vector Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Proof of Theorem 8.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Scalar Equations with Real Characteristic Roots . . . . . . . . . . .

111 111 111 113 114 115 116 118 119 120 121 124 127

9.

Lur’e Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 132 133

10. Aizerman’s Problem for Nonautonomous Systems . . . . . . . . 10.1 Comparison of the Green Functions . . . . . . . . . . . . . . . . . . . . . . . 10.2 Proof of Theorem 10.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Aizerman’s Type Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Equations with Purely Real Roots . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Equations with Nonreal Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Absolute Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Estimates for Green’s Functions . . . . . . . . . . . . . . . . . . . . 10.6.2 Absolute Stability Conditions . . . . . . . . . . . . . . . . . . . . . . 10.7 Positive Solutions of Nonlinear Equations . . . . . . . . . . . . . . . . . .

135 135 136 137 139 141 141 141 143 144

11. Input-to-State Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Systems with Time-Variant Linear Parts . . . . . . . . . . . . . . . . . . 11.3 Proof of Theorem 11.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 The Input-to-State Version of Aizerman’s Problem . . . . . . . . . . 11.5 Proof of Theorem 11.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 148 149 151

X

Table of Contents

12. Orbital Stability and Forced Oscillations . . . . . . . . . . . . . . . . . . 12.1 Global Orbital Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 One Contour Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Proof of Theorem 12.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Existence and Stability of Forced Oscillations . . . . . . . . . . . . . . 12.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 154 156 157 158

13. Existence of Steady States. Positive and Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . 13.1 Systems of Semilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Fully Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Nontrivial Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Positive Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Systems with Differentiable Entries . . . . . . . . . . . . . . . . . . . . . . .

159 159 160 163 164 165

Appendix A. Bounds for Eigenvalues of Matrices . . . . . . . . . . . . . 167 Appendix B. Positivity of the Green Function . . . . . . . . . . . . . . . . 171 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 List of Main Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

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2

Introduction

Chapters 2–10 are devoted to asymptotic, exponential and absolute stabilities of linear and nonlinear systems. In Chap. 2 we consider perturbations of the linear system x(t) ˙ = A(t)x(t) (x(t) ≡ dx(t)/dt, t ≥ 0) ˙

(2)

with a variable matrix A(t). In addition, the multiplicative representation for solutions is introduced. That representation is used for constructing of majorants and minorants of solutions. Moreover, we establish stability conditions for systems, which are close to triangular ones. In Chap. 3 we investigate linear system (2) in the case when A(t) is a slowly varying matrix. To illustrate a typical result assume that the matrix A(t) satisfies the condition A(t) − A(s) ≤ q0 |t − s| (t, s ≥ 0),

(3)

where q0 is a positive constant. In addition, v := sup g(A(t)) < ∞. t≥0

(4)

Denote by z(q0 , v) the extreme right-hand (positive) root of the algebraic equation z n+1 = q0 P (z), where n−1

P (z) = k=0

(k + 1)v k n−k−1 √ . z k!

By the freezing method and estimate (1), the following result is proved: let conditions (3) and (4) hold. In addition, let the matrix A(t) + z(q0 , v) I be a Hurwitz one for all t ≥ 0. Then system (2) is stable. We also give simple estimates for z(q0 , v). In Chap. 4 we continue to investigate linear multivariable systems. Stability conditions are derived for systems with piecewise constant matrices. Moreover, by the multiplicative representation, the Lozinskii and Wazewski inequalities are proved. In addition the second order vector equation x ¨ + A(t)x˙ + B(t)x = 0 is considered. Here A(t), B(t) are variable matrices. Chapter 5 is devoted to the system x˙ = Ax + F (x, t) (t ≥ 0),

(5)

where A is a constant n × n-matrix and F maps Cn × [0, ∞) into Cn with the property F (h, t) ≤ ν h for all h ∈ Ω(r) and t ≥ 0.

(6)

Introduction

3

Here ν = const ≥ 0, and Ω(r) = {h ∈ Cn : h ≤ r} (r ≤ ∞). By estimate (1) and the first Lyapunov method, conditions for the exponential stability of system (5) are established. Chapter 6 is devoted to the Aizerman conjecture. Let us consider in Rn the equation y˙ = Ay + b f (s, t) (s = cy, t ≥ 0), (7) where A is a constant Hurwitz n × n-matrix, b is a column, c is a row, f maps R1 × [0.∞) into R1 with the property 0 ≤ f (s, t)/s ≤ q (s ∈ R1 , s = 0, t ≥ 0).

(8)

In 1949 M. A. Aizerman formulated the following conjecture: under the condition f (s, t) ≡ f (s), for the absolute stability of the zero solution of system (7) in the class of nonlinearities (8) it is necessary and sufficient that the linear equation y˙ = Ay + q1 bcy be asymptotically stable for any q1 ∈ [0, q]. This conjecture is not, in general, true. Therefore, the following problem arose: to find the class of systems that satisfy Aizerman’s hypothesis. To formulate the relevant result let us introduce the transfer function of the linear part of system (7): W (λ) = c(λI − A)−1 b =

L(λ) . P (λ)

Here P (λ) and L(λ) are polynomials. Besides, let K(t) :=

1 2π

∞ −∞

eiyt W (iy)dy.

That is, K(t) is the corresponding Green (impulse) function. It is proved that under the condition K(t) ≥ 0 (t ≥ 0), the zero solution of system (7) is absolutely stable in the class of nonlinearities (8) if and only if the polynomial P (λ) − qL(λ) is Hurwitzian. Clearly, that result singles out one of the classes of linear parts of systems that satisfy the Aizerman conjecture. It is also proved that the polynomial P (λ) − qL(λ) is Hurwitzian, provided P (0) > qL(0) and the Green function is positive. Moreover, Chapter 6 also contains the generalized Aizerman problem for multivariable systems. Chapter 7 deals with nonlinear systems of the type x˙ = A(t)x + F (x, t) (t ≥ 0),

(9)

where A(t) is a variable n × n-matrix and F maps Cn × [0, ∞) into Cn with the property (6).

4

Introduction

Chapter 8 deals with nonlinear systems of the type x(t) ˙ = B(x(t), t)x(t) (t ≥ 0), where

(10)

B(h, t) = (bjk (h, t))nj,k=1

is an n × n-matrix for every h ∈ Cn and t ≥ 0. In particular, the freezing method is developed to nonlinear systems. In addition, nonlinear dissipative systems, nonlinear triangular systems and their perturbations are investigated. Moreover, the very interesting Levin stability criterion for nonlinear scalar equations with real variable characteristic roots is presented. In Chapter 8 we also consider systems of the type x ¨ + F (x, x, ˙ t)x˙ + G(x, x, ˙ t)x = 0, where F (h, w, t) and G(h, w, t) are matrices continuously dependent on h, w ∈ Rn and t ≥ 0. In Chapter 9 we consider the Lur’e type systems x ¨ + Ax˙ + Bx = Φ(x, t), where A and B are constant matrices, and Φ is a continuous vector-valued function. Stability conditions and bounds for the region of attraction are derived. One contour nonlinear nonautonomous systems with positive Green functions are investigated in Chap. 10. Chapter 11 is devoted to input-to-state stability of nonlinear systems. Here we consider the input-to-state version of Aizerman’s conjecture. Orbital stability and forced oscillations are discussed in Chap. 12. In particular, conditions that provide the existence of forced oscillations are established. Conditions for the existence of positive and nontrivial steady states are investigated in Chap. 13. In Appendix A, some well known bounds for eigenvalues of matrices are collected. In Appendix B, positivity conditions for the Green functions of ordinary differential equations are presented.

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6

1. Preliminaries

where A∗ = (akj )nj,k=1 is the adjoint matrix. Thus, the Frobenius norm does not depend on the choice of an orthogonal normal basis. The relations N (A) > 0 (A = 0); N (λA) = |λ|N (A) (λ ∈ C), N (AB) ≤ N (A)N (B) and N (A + B) ≤ N (A) + N (B) are true for all matrices A and B. Furthermore, Rn denotes the real Euclidean space, I is the unit matrrix and (., .) is the scalar product in Rn or Cn . So x =

(x, x).

1.2 Definitions of Stability Put R+ = [0, ∞) and consider in Cn the differential equation x(t) ˙ = f (t, x(t)) (t > 0; x(t) ˙ ≡ dx/dt),

(2.1)

where f : R+ × Cn → Cn is a continuous vector-valued function. A solution of (2.1) is a differentiable function x : R+ → Cn satisfying that equation for all t > 0. It is further assumed that the function f is of such a nature that equation (2.1) has a unique solution over R+ corresponding to each initial condition x(0) = x0 . For example, if f satisfies the Lipshitz condition (see for instance (Vidyasagar, 1993, Theorem 2.4.25)). A point x1 ∈ Cn is said to be an equilibrium point of the system (2.1) if f (t, x1 ) ≡ 0 for all t ≥ 0. In other words, if the system starts at an equilibrium point, it stays there. The converse is also true. Throughout this book we assume that 0 is an equilibrium point of the system (2.1). This assumption does not result in any loss of generality, because if x1 is an equilibrium point of (2.1), then 0 is an equilibrium point of the system z(t) ˙ = f1 (t, z(t)), where f1 (t, z(t)) = f (t, z(t) + x1 ). Everywhere below in this section, x(t) is a solution of (2.1). Definition 1.2.1 The equilibrium point 0 is said to be stable (in the sense of Lyapunov) if, for every t0 ≥ 0 and > 0, there exists δ(t0 ) > 0, such that the condition x(t0 ) ≤ δ(t0 ) implies x(t) ≤ (t ≥ t0 ).

(2.2)

It is uniformly stable if, for each > 0, there exists δ > 0 independent of t0 , such that the condition x(t0 ) ≤ δ implies inequality (2.2). The equilibrium is unstable, if it is not stable.

1.2 Definitions of Stability

7

Because all norms on Cn are topologically equivalent, it follows that the stability status of an equilibrium does not depend on the particular norm. Definition 1.2.2 The equilibrium 0 is asymptotically stable if it is stable and for each t0 ∈ R+ , there is an η(t0 ) > 0 such that x(t0 ) ≤ η(t0 ) implies x(t) → 0 as t → ∞. The equilibrium 0 is uniformly asymptotically stable if it is stable and there is an η > 0 independent of t0 , such that x(t0 ) ≤ η implies x(t) → 0 as t → ∞. Also, the set Ω(η), defined by Ω(η) = {x ∈ Cn : x ≤ η} is called the region of attraction (the stability domain) for the equilibrium point 0. The equilibrium point 0 is said to be globally (uniformly) asymptotically stable if it is (uniformly) asymptotically stable and the region of attraction Ω(η) = Cn . As shown by Vinograd (1957) (see also (Vidyasagar, 1993, p. 141)), attractivity and stability are really independent properties, i.e., an equilibrium can be attractive without being stable. Definition 1.2.3 The equilibrium point 0 is exponentially stable if for any t0 ≥ 0 there exist constants r, a, b > 0 such that x(t) ≤ a x(t0 ) e−b(t−t0 ) (t ≥ t0 ), if x(t0 ) ≤ r.

(2.3)

It is uniformly exponentially stable if constants r, a, b > 0 in (2.3) are independent of t0 . The equilibrium point 0 is globally exponentially stable if inequality (2.3) holds with r = ∞. Consider the linear equation u(t) ˙ = A(t)u(t) (t ≥ 0)

(2.4)

with a variable n × n-matrix A(t). Since the zero is the unique equilibrium point of a linear equation, we will say that (2.4) is stable (uniformly stable, asymptotically stable, exponentially stable) if the zero solution of (2.4) is stable (uniformly stable, asymptotically stable, exponentially stable). For linear equations, the notions of the global asymptotic (exponential) stability and asymptotic (exponential) stability coincide. In addition, we will say that (2.4) is uniformly asymptotically (exponentially) stable if the zero solution of (2.4) is uniformly asymptotically (exponentially) stable.

8

1. Preliminaries

1.3 Eigenvalues of Matrices Recall that for an n × n-matrix (a linear operator in Cn ) A = (ajk )nj,k=1 , A∗ = (akj )nj,k=1 , is the adjoint matrix. In other words (Ax, y) = (x, A∗ y) (x, y ∈ Cn ). A is a Hermitian matrix if A∗ = A. It is positive definite if it is Hermitian and (Ah, h) ≥ 0 for any nonzero h ∈ Cn ; A is a normal matrix if AA∗ = A∗ A. It is nilpotent if An = 0. Let A be an arbitrary matrix. Then the matrices AI = (A − A∗ )/2i and AR = (A + A∗ )/2 are the imaginary Hermitian component and real Hermitian one of A, respectively. By A−1 the matrix inverse to A is denoted: AA−1 = A−1 A = I. Let A be an arbitrary matrix. Then if for some λ ∈ C, the equation Ah = λh has a nontrivial solution, λ is an eigenvalue of A and h is its eigenvector. An eigenvalue λ(A) has the (algebraic) multiplicity r if dim(∪nk=1 ker(A − λI)k ) = r. Let λk (A) (k = 1, . . . , n) be eigenvalues of A, including with their multiplicities. Then the set σ(A) = {λk (A)}nk=1 is the spectrum of A. All eigenvalues of a Hermitian matrix A are real. If, in addition, A is positive (negative) definite, then all its eigenvalues are positive (negative). Furthermore, rs (A) = max |λk (A)| k=1,...,n

is the spectral radius of A. Denote α(A) = max Reλk (A), β(A) = k=1,...,n

min Reλk (A).

k=1,...,n

A matrix A is said to be a Hurwitz one if all its eigenvalues lie in the open left half-plane, i.e., α(A) < 0. A complex number λ is a regular point of A if it does not belong to the spectrum of A, i.e., if λ = λk (A) for any k = 1, . . . , n. Recall some properties of the trace T race A = T r A of A : n

n

akk =

Tr A = k=1

λk (A). k=1

1.4 Matrix-Valued Functions

9

In addition T r (A + B) = T r A + T r B for all matrices A and B. By det(A) the determinant of A is denoted: n

det(A) =

λk (A). k=1

A polynomial

n

p(λ) = det(λI − A) =

(λ − λk (A)) k=1

is said to be the characteristic polynomial of A. All the eigenvalues of A are the roots of its characteristic polynomial. The algebraic multiplicity of an eigenvalue of A coincides with the multiplicity of the corresponding root of the characteristic polynomial. A polynomial P (λ) is said to be a Hurwitz one if all its roots lie in the open left half-plane. Thus, the characteristic polynomial of a Hurwitz matrix is a Hurwitz polynomial.

1.4 Matrix-Valued Functions Let A be a matrix and let f (λ) be a scalar-valued function which is analytical on a neighborhood D of σ(A). We define the function f (A) of A by the generalized integral formula of Cauchy f (A) = −

1 2πi

Γ

f (λ)Rλ (A)dλ,

where Γ ⊂ D is a closed smooth contour surrounding σ(A), and Rλ (A) = (A − λI)−1 is the resolvent of A. If an analytic function f (λ) is represented in the domain {z ∈ C : |z| ≤ rs (A)} by the Taylor series

∞

f (λ) =

c k λk ,

k=0

then

∞

f (A) =

c k Ak .

k=0

In particular, for any matrix A, eA =

∞ k=0

Ak . k!

1. Preliminaries

10

Example 1.4.1 Let  a1 0  0 a2 A=  . ... 0 ... Then



f (a1 )  0 f (A) =   . 0

A be a diagonal matrix:  ... 0 ... 0  . . .  0 an 0 f (a2 ) ... ...

... ... . 0

Example 1.4.2 If a matrix J is  λ0 1 0 ... 0  0 λ0 1 . . . 0   . . . ... .  . . . ... . J =   . . . ... .   0 0 . . . λ0 1 0 0 . . . 0 λ0 then



     f (J) =     

f (λ0 ) 0 . . . 0 0

f (λ0 ) 1!

f (λ0 ) . . . ... ...

 0 0  .  . f (an ) an n × n-Jordan block:      ,    

... ... ... ... ... f (λ0 ) 0

f (n−1) (λ0 ) (n−1)!

. . .

f (λ0 ) 1! f (λ0 )

      .    

1.5 Norms of Matrix Functions Let A = (ajk ) be an n × n-matrix. The following quantity plays a key role in the sequel: g(A) = (N 2 (A) −

n

|λk (A)|2 )1/2 .

(5.1)

k=1

Recall that I is the unit matrix, N (A) is the Frobenius (Hilbert-Schmidt) norm of A, and λk (A) (k = 1, ..., n) are the eigenvalues taken with their multiplicities. Since n

k=1

|λk (A)|2 ≥ |T race A2 |,

1.5 Norms of Matrix Functions

we get

g 2 (A) ≤ N 2 (A) − |T race A2 |.

11

(5.2)

In (Gil, 2003, Section 2.1), the following relations are proved: g 2 (A) ≤ and

1 2 ∗ N (A − A) 2

(5.3)

g(eiτ A + zI) = g(A)

(5.4) ∗

∗

for all τ ∈ R and z ∈ C. If A is a normal matrix: AA = A A, then g(A) = 0. To formulate the result, for a natural n > 1 introduce the numbers γn,k =

k Cn−1 (k = 1, ..., n − 1) and γn,0 = 1. (n − 1)k

Here k Cn−1 =

(n − 1)! (n − k − 1)!k!

are binomial coefficients. Evidently, for all n > 2, 2 γn,k =

1 (n − 1)(n − 2) . . . (n − k) ≤ (k = 1, 2, ..., n − 1). (n − 1)k k! k!

(5.5)

Let B(Cn ) be the set of all linear operators (matrices) in Cn . Theorem 1.5.1 Let A ∈ B(Cn ) and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then n−1

f (A) ≤

sup |f (k) (λ)|g k (A)

k=0 λ∈co(A)

γn,k . k!

The proof of this theorem can be found in (Gil, 2003, Theorem 2.7.1). This theorem is exact: if A is a normal matrix and sup |f (λ)| = sup |f (λ)|,

λ∈co(A)

λ∈σ(A)

then we have the equality f (A) = supλ∈σ(A) |f (λ)|. The previous theorem and (5.5) give us the following Corollary 1.5.2 Let A ∈ B(Cn ) and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then n−1

f (A) ≤

sup |f (k) (λ)|

k=0 λ∈co(A)

g k (A) . (k!)3/2

12

1. Preliminaries

Theorem 1.5.1 and Corollary 1.5.2 give us the following Corollary 1.5.3 For a linear operator A in Cn , exp(At) ≤ eα(A)t

n−1

g k (A)tk

k=0

γn,k ≤ eα(A)t k!

n−1 k=0

g k (A)tk (k!)3/2

(t ≥ 0)

where α(A) = maxk=1,...,n Re λk (A). In addition, Am ≤

n−1 k=0

γn,k m!g k (A)rsm−k (A) ≤ (m − k)!k!

n−1 k=0

m!g k (A)rsm−k (A) (m = 1, 2, ...) (m − k)!(k!)3/2

where rs (A) is the spectral radius. Recall that 1/(m − k)! = 0 if m < k. Furtermore we will use the following result. Corollary 1.5.4 Let A be an n × n-matrix. Then for any h ∈ Cn , the inequality exp(At)h ≥ e[β(A)t] [

n−1

g k (A)tk (k!)−1 γn,k ]−1 h (t ≥ 0)

k=0

is valid. Therefore, in the accordance with (5.5), exp(At)h ≥ e[β(A)t] [

n−1

g k (A)(k!)−3/2 tk ]−1 h (t ≥ 0).

k=0

Indeed, since max Reσ(−A) = − min Reσ(A) = −β(A), and g(−A) = g(A), Corollary 1.5.3 yields exp(−At)v ≤ v e−β(A)t

n−1 k=0

g k (A)tk

γn,k (t ≥ 0) k!

for any v ∈ Cn . Taking into account that the operator exp(−At) is the inverse one to exp(At) and putting exp(−At)v = h, we arrive at the assertion. We will need also the following results. Theorem 1.5.5 Let A be a linear operator in Cn . Then its resolvent Rλ (A) = (A − λI)−1 satisfies the inequality n−1

Rλ (A) ≤ k=0

g k (A)γn,k for any regular point λ of A, ρk+1 (A, λ)

where ρ(A, λ) = mink=1,...,n |λ − λk (A)|.

1.6 Examples

13

The proof of this theorem can be found in (Gil, 2003, Section 2.1). Theorem 1.5.5 is exact: if A is a normal matrix, then g(A) = 0 and Rλ (A) =

1 for all regular points λ of A. ρ(A, λ)

Let A be an invertible n × n-matrix. Then by Theorem 1.5.5, A−1 ≤

n−1

g k (A)

k=0

γn,k , k+1 ρ0 (A)

where ρ0 (A) = ρ(A, 0) is the smallest modulus of the eigenvalues of A: ρ0 (A) =

inf

k=1,...,n

|λk (A)|.

Moreover, Theorem 1.5.5 and inequalities (5.5) imply Corollary 1.5.6 Let A be a linear operator in Cn . Then n−1

√

Rλ (A) ≤ k=0

g k (A) for any regular point λ of A. k!ρk+1 (A, λ)

The following result is proved in (Gil’, 2003, Section 2.11). Lemma 1.5.7 Let A be n × n-matrix (n > 1). Then (Iλ − A)−1 ≤

(N 2 (A) − 2Re (λ T race (A)) + n|λ|2 )(n−1)/2 = |det (λI − A)| (n − 1)(n−1)/2 N n−1 (A − Iλ) |det (λI − A)| (n − 1)(n−1)/2

for any regular λ of A.

1.6 Examples In this section we present some examples of calculations of g(A). Example 1.6.1 Consider the matrix A=

a11 a21

a12 a22

where ajk (j, k = 1, 2) are real numbers. Then due to (5.3) g(A) ≤ |a12 − a21 |.

1. Preliminaries

14

Now let us consider the case of nonreal eigenvalues: λ2 (A) = λ1 (A). It can be written det(A) = λ1 (A)λ1 (A) = |λ1 (A)|2 and |λ1 (A)|2 + |λ2 (A)|2 = 2|λ1 (A)|2 = 2det(A) = 2[a11 a22 − a21 a12 ]. Thus,

g 2 (A) = N 2 (A) − |λ1 (A)|2 − |λ2 (A)|2 = a211 + a212 + a221 + a222 − 2[a11 a22 − a21 a12 ].

Hence,

g(A) =

(a11 − a22 )2 + (a21 + a12 )2 .

(6.1)

Let n = 2 and a matrix A have real entries again, but now the eigenvalues of A are real. Then |λ1 (A)|2 + |λ2 (A)|2 = T race A2 . Obviously, A2 =

a211 + a12 a21 a21 a11 + a21 a22

a11 a12 + a12 a22 a222 + a21 a12

.

We thus get the relation |λ1 (A)|2 + |λ2 (A)|2 = a211 + 2a12 a21 + a222 . Consequently, g 2 (A) = N 2 (A) − |λ1 (A)|2 − |λ2 (A)|2 = a211 + a212 + a221 + a222 − (a211 + 2a12 a21 + a222 ). Hence, Example 1.6.2  a11  0 A=  . 0

g(A) = |a12 − a21 |.

(6.2)

Let A be an upper-triangular matrix:  a12 . . . a1n a21 . . . a2n  . ... . .  . . . 0 ann

Then

n k−1

|ajk |2 ,

g(A) = k=1 j=1

since the eigenvalues of a triangular matrix are its diagonal elements.

(6.3)

1.7 Integral Inequalities

Example 1.6.3  −a1  1 A=  . 0

15

Consider the matrix ... ... ... ...

−an−1 0 . 1

 −an 0   .  0

with complex numbers ak . Such matrices play a key role in the theory of scalar ordinary differential equations. Take into account that   2 a1 − a2 . . . a1 an−1 − a1 a1 an  −a1 ... −an−1 −an    1 . . . 0 0  A2 =  .   . ... . .  0 ... 0 0 Thus, we obtain, T race A2 = a21 − 2a2 . Therefore g 2 (A) ≤ N 2 (A) − |T race A2 | = n − 1 − |a21 − 2a2 | +

n

|ak |2 .

(6.4)

k=1

Hence, g 2 (A) ≤ n − 1 + 2|a2 | +

n

|ak |2 .

(6.5)

k=2

1.7 Integral Inequalities Let h = (hk )nk=1 and w = (wk )nk=1 be real vectors. We will write h ≤ w if wk ≤ hk (k = 1, ..., n). The vector w is non-negative, if wk ≥ 0 (k = 1, ..., n). A vector-valued function is non-negative if its values are non-negative vectors. In the sequel we will often use the following result. Lemma 1.7.1 Let a non-negative continuous vector-valued function φ : R+ → Rn satisfy the inequality φ(t) ≤ f (t) +

t 0

K(t, s)φ(s)ds (t ≥ 0)

with a non-negative continuous vector-valued function f : R+ → Rn and a 2 continuous matrix-valued kernel K : R+ → Rn with non-negative entries. Then φ(t) ≤ ψ(t), where ψ(t) is a solution of the integral equation

16

1. Preliminaries t

ψ(t) = f (t) +

0

K(t, s)ψ(s)ds.

Similarly, the inequality t

φ(t) ≥ f (t) +

0

K(t, s)φ(s)ds (t ≥ 0)

implies φ(t) ≥ ψ(t) (t ≥ 0). Proof: The lemma is due to an obvious application of the well-known Theorem 1.9.3 (Daleckii and Krein, 1974) to the space C(R+ , Rn ) of real continuous vector-valued functions defined on R+ , since the integral operator (Ku)(t) =

t 0

K(t, s)u(s)ds

is a Volterra operator and, as is well known (Gohberg and Krein, 1970), its spectral radius equals zero. ✷ Corollary 1.7.2 (The Gronwall inequality). Suppose a positive continuous scalar-valued function φ subordinates the inequality φ(t) ≤ c +

t 0

h(s)φ(s)ds (c = const > 0, t ≥ 0),

where h(t) is a continuous positive scalar-valued function. Then φ(t) ≤ c exp [

t 0

h(s)ds] (t ≥ 0).

Corollary 1.7.3 Suppose a positive continuous scalar-valued function φ subordinates the inequality φ(t) ≤ c exp [

t 0

t

p(τ )dτ ] +

0

exp [

t s

p(τ )dτ ]h(s)φ(s)ds (t ≥ 0),

where h(t) and p(t) are continuous scalar-valued functions. Moreover, h(t) is positive. Then φ(t) ≤ c exp [

t 0

(p(s) + h(s))ds] (t ≥ 0).

To establish that result it is sufficient to set φ1 (t) = φ(t) exp [− and to apply Corollary 1.7.2.

t 0

p(s)ds]

1.8 Algebraic Equations

17

1.8 Algebraic Equations Let us consider the algebraic equation z n = P (z),

(8.1)

where P (z) is the polynomial n−1

P (z) =

cj z n−j−1

j=0

with non-negative coefficients cj (j = 0, ..., n − 1). Lemma 1.8.1 The extreme right-hand (non-negative) root z0 of equation (8.1) satisfies the estimate z0 ≤ δ0 , where δ0 =

[P (1)]1/n P (1)

if P (1) ≤ 1 if P (1) > 1

For the proof see (Gil’, 2003, Section 1.6). Setting in (8.1) z = ax with a positive constant a, we obtain xn =

n−1

cj a−j−1 xn−j−1 .

(8.2)

j=0

If then

a=2 n−1 j=0

cj a−j−1 ≤

max

j=0,...,n−1

n−1

√

j+1

cj ,

2−j−1 = 1 − 2−n+1 < 1.

j=0

Let x0 be the extreme right-hand root of equation (8.2), then by the previous lemma x0 ≤ 1. Since z0 = ax0 , we have derived Corollary 1.8.2 The extreme right-hand root z0 of equation (8.1) is nonnegative. Moreover, √ j+1 c . z0 ≤ 2 max j j=0,...,n−1

18

1. Preliminaries

1.9 Upper Bounds for Lyapunov’s Equation Recall the famous Lyapunov theorem Theorem 1.9.1 In order for the eigenvalues of a matrix A to lie in the interior of the left half-plane, it is necessary and sufficient that there exists a positive definite Hermitian matrix W , such that the matrix W A + A∗ W is a negative Hermitian one. Moreover, if the eigenvalues of A lie in the interior of the left half-plane, then for any positive definite Hermitian matrix H there exists a positive definite Hermitian matrix WH such that WH A + A∗ WH = −2H. In addition,

∞

WH = 2

0

(9.1)

∗

eA t HeAt dt.

(9.2)

For the proof see for instance, (Daleckii and Krein, 1974, p. 33) or (Vidyasagar, 1993, p. 198). Moreover, WH =

1 π

∞ −∞

(−iIω − A∗0 )−1 H(iIω − A0 )−1 dω,

(9.3)

cf. (Godunov, 1998, p.151). Equation (9.1) is called the Lyapunov equation. In many applications, it is important to know bounds for the norm of the solution WH of the Lyapunov equation. We are beginning with Lemma 1.9.2 Let A be a Hurwitz n × n-matrix. Then a solution WH of the Lyapunov equation (9.1) subordinates the inequality n−1

WH ≤ H j,k=0

g j+k (A)(k + j)! . k!)3/2

2j+k |α(A)|j+k+1 (j!

Proof: Take into account that g(A) = g(A∗ ), and α(A) = α(A∗ ). Then by virtue of Corollary 1.5.3 and equality (9.2), WH ≤ 2 2 H

∞ 0

∞ 0

eA

∗

t

H n−1

exp[2α(A)t] k=0

eAt dt ≤ g k (A)tk (k!)3/2

2

dt.

The integration gives us WH ≤ 2 H

∞ 0

n−1

exp[2α(A)t]

(g(A)t)k+j dt = (j! k!)3/2 j,k=0

1.9 Upper Bounds for Lyapunov’s Equation n−1

H

2 j,k=0

19

(k + j)!g j+k (A) , (2|α(A)|)j+k+1 (j! k!)3/2

as claimed. ✷ Lemma 1.9.3 Let A be a real Hurwitz n × n-matrix. Then a solution WH of the Lyapunov equation (9.1) subordinates the inequality

Proof: have

∞

1 π

WH ≤ H

−∞

(N 2 (A) + nω 2 )n−1 dω . |det (iωI − A)|2 (n − 1)n−1

Since A is real and Im T race A(t) = 0. Due to Lemma 1.5.7 , we

(N 2 (A) + ny 2 )(n−1)/2 (y ∈ R). |det (iyI − A)| (n − 1)(n−1)/2 Hence, the required result is due to (9.3). ✷ (Iiy − A)−1 ≤

Note that, if the spectrum is real, then |det (iωI − A)|2 ≥ (ω 2 + α2 (A))n (ω ∈ R). Thus, WH ≤ H ψ, where ψ :=

∞

2βn π

0

(N 2 (A) + ny 2 )n−1 dy, (y 2 + α2 (A))n

where βn = Clearly,

2βn π|α(A)|2n−1 Take into account that ψ=

n

∞ 0

1 . (n − 1)n−1 (N 2 (A) + nα2 (A)s2 )n−1 ds. (s2 + 1)n

|λk (A|2 ≤ N 2 (A)

k=1

and

|α(A)| ≤

min |λk (A)|.

k=1,...,n

So nα2 (A) ≤ N 2 (A) and ψ≤

2n βn N 2(n−1) (A) π|α(A)|2n−1

∞ 0

ds . s2 + 1

Hence, ψ ≤ ψ1 (A), where ψ1 (A) =

2n−1 N 2(n−1) (A) . (n − 1)n−1 |α(A)|2n−1

20

1. Preliminaries

Now lemma 1.9.3 implies Lemma 1.9.4 Let A be a real Hurwitz n × n-matrix with the real spectrum. Then a solution WH of the Lyapunov equation (9.1) subordinates the inequality WH ≤ H ψ1 (A). Let us consider the Lyapunov equation with H = I: W A + A∗ W = −2I.

(9.4)

Furthermore, let u(t) be a solution of the equation u(t) ˙ = Au(t),

(9.5)

and let W be a solution of equation (9.4). Multiplying equation (9.5) by W and doing the scalar product, we get (W u(t), ˙ u(t)) = (W Au(t), u(t)). Since

d (W u(t), u(t)) = (W u(t), ˙ u(t)) + (u(t), W u(t)), ˙ dt it can be written d (W u(t), u(t)) = (W Au(t), u(t)) + (u(t), W Au(t)) = dt ((W A + A∗ W )u(t), u(t)). Now (9.5) yields d (W u(t), u(t)) = −2(u(t), u(t)) = −2(W −1 W u(t), u(t)) ≤ dt −2b(W )(W u(t), u(t)), where b(W ) = inf n h∈C

(W −1 h, h) (w, w) = inf n = W w∈C (W w, w) (h, h)

−1

.

Solving this inequality, we get (W u(t), u(t)) = W 1/2 u(t)

2

≤ e−t2

W

−1

W 1/2 u(0) 2 .

Since u(t) = eAt u(0), this inequality means that W 1/2 eAt h ≤ W 1/2 h exp [− We thus have derived.

t ] (h ∈ Cn ; t ≥ 0). W

(9.6)

1.10 Lower Bounds for Lyapunov’s Equation

21

Lemma 1.9.5 Let A be a Hurwitz matrix, and W be a solution of equation (9.4). Then inequality (9.6) is true. This lemma, and Lemmas 1.9.2 and 1.9.4 yield Corollary 1.9.6 Let A be a real Hurwitz n × n-matrix, and W be a solution of equation (9.4). Then with the notation n−1

ψ0 (A) := j,k=0

g j+k (A)(k + j)! , k!)3/2

2j+k |α(A)|j+k+1 (j!

the inequalities −1

ψ0 (A) e−tψ0

W 1/2 eAt ≤

−1

W 1/2 eAt ≤

ψ1 (A) e−tψ1

(A)

(A)

and (t ≥ 0)

are true.

1.10 Lower Bounds for Lyapunov’s Equation Lemma 1.10.1 Let A be a real Hurwitz n × n-matrix. Then a solution W of equation (9.4) satisfies the inequality (W h, h) ≥ Proof:

h 2 (h ∈ Cn ). π A

By the Parseval equality we have (W h, h) = =

1 2π

∞ −∞

(A − iIy)−1 h ≥ Thus, (W h, h) ≥ h π as claimed. ✷

∞ 0

0

(eAs h, eAs h)ds

(A − iIy)−1 h

But

2

∞

1 2π

2

dy (h ∈ Cn ).

h h ≥ . A − iIy A + |y| ∞ −∞

h 2 dy = ( A + |y|)2

dy h 2 dy = , 2 ( A + |y|) π A

1. Preliminaries

22

Lemma 1.10.2 Let A be a Hurwitz n × n-matrix. Then a solution WH of the Lyapunov equation (9.1) satisfies the inequality ∞

˜ (WH h, h) ≥ 2θ(H) where

0

e2β(A)t [

n−1 k=0

g k (A)tk −2 ] dt h (k!)3/2

(h ∈ Cn ),

˜ β(A) = min Reλk (A), θ(H) = min λk (H). k

Proof:

2

k

Take into account that g(A) = g(A∗ ) and β(A) = β(A∗ ).

Then by Corollary 1.5.4, (eA∗t HeAt h, h) = (HeAt h, eAt h) ≥ At ˜ h, eAt h) ≥ θ(H)(e 2β(A)t ˜ [ θ(H)e

n−1

g k (A)tk (k!)−3/2 ]−2 h

2

(t ≥ 0).

k=0

Now equality (9.2) yields (WH h, h) = 2 ˜ 2θ(H)

∞ 0

e2β(A)t [

∞ 0

n−1

(eA∗t HeAt h, h)dt ≥

g k (A)tk (k!)−3/2 ]−2 dt h 2 ,

k=0

as claimed. ✷ If A is a normal matrix, then g(A) = 0 and the previous lemma gives us the inequality ˜ θ(H) h 2 (h ∈ Cn ). (WH h, h) ≥ |β(A)|

1.11 Estimates for Contour Integrals Lemma 1.11.1 Let D be the closed convex hull of points x0 , x1 , ..., xn ∈ C and let a scalar-valued function f be regular on a neighborhood D1 of D. In addition, let Γ ⊂ D1 be a Jordan closed contour surrounding the points x0 , x1 , ..., xn . Then the inequality | is valid.

1 2πi

Γ

f (λ)dλ 1 sup |f (n) (λ)| |≤ (λ − x0 )...(λ − xn ) n! λ∈D

1.12 Absolute Values of Matrix Functions

23

Proof: First, let all the points be distinct: xj = xk for j = k (j, k = 0, ..., n), and let Df (x0 , x1 , ..., xn ) be a divided difference of the scalar-valued function f at points x0 , x1 , ..., xn of the complex plane. If f is regular on a neighborhood of the closed convex hull D of the points x0 , x1 , ..., xn , then the divided difference admits the representation Df (x0 , x1 , ..., xn ) =

1 2πi

Γ

f (λ)dλ (λ − x0 )...(λ − xn )

(11.1)

(see (Gelfond, 1967, formula (54)). But, on the other hand, the following estimate is well known: 1 sup |f (n) (λ)| | Df (x0 , x1 , ..., xn ) | ≤ n! λ∈D (Gelfond, 1967, formula (49)). Combining that inequality with relation (11.1), we arrive at the required result. If xj = xk for some j = k, then the claimed inequality can be obtained by small perturbations and the previous reasonings. ✷ Lemma 1.11.2 Let x0 ≤ x1 ≤ ... ≤ xn be real points and let a function f be regular on a neighborhood D1 of the segment [x0 , xn ]. In addition, let Γ ⊂ D1 be a Jordan closed contour surrounding [x0 , xn ]. Then there is a point η ∈ [x0 , xn ], such that the equality 1 2πi

Γ

f (λ)dλ 1 (n) f (η) = (λ − x0 )...(λ − xn ) n!

is true. Proof: First suppose that all the points are distinct: x0 < x1 < ... < xn . Then the divided difference Df (x0 , x1 , ..., xn ) of f in the points x0 , x1 , ..., xn admits the representation Df (x0 , x1 , ..., xn ) =

1 (n) f (η) n!

with some point η ∈ [x0 , xn ] (Gelfond, 1967, formula (43)), (Ostrowski, 1973, p.5). Combining that equality with representation (11.1), we arrive at the required result. If xj = xk for some j = k, then the claimed inequality can be obtained by small perturbations and the previous reasonings. ✷

1.12 Absolute Values of Matrix Functions 1.12.1 Statement of the Result Let A = (alj )nj,l=1 be a matrix, and let mlj (λ) be the cofactor of the element δlj λ − alj of the matrix λI − A. Here δlj is the Kronecker symbol: δlj = 0 if

24

1. Preliminaries

j = l, and δjj = 1. Assume that after division by the coinciding zeros, the equality mlj (λ) µlj (λ) = (j, l = 1, ..., n) (12.1) dlj (λ) det(Iλ − A) holds, where dlj (λ) and µlj (λ) are polynomials which have no coinciding zeros. That is, µlj (λ)/dlj (λ) are the entries of the matrix (Iλ − A)−1 . Furthermore, let the degree of dlj (λ) be denoted by nlj . Obviously, if mlj (λ) and det(Iλ − A) have no coinciding zeros, then nlj = n and mlj (λ) = µlj (λ). For k = 0, ..., n − 1, put bjl (k) =

1 (n −k−1) (λ)| . sup |µ lj k!(nlj − 1 − k)! λ∈co(A) lj

Since (nlj − j)! = 0 for j > nlj , we have bjl (k) = 0 for k = nlj , ..., n − 1 if nlj < n − 1.

(12.2)

Theorem 1.12.1 Let A be an n × n-matrix, and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then the entries flj (A) of the matrix f (A) satisfy the inequalities n−1

|flj (A)| ≤ k=0

max |f (k) (λ)|blj (k) (l, j = 1, ..., n).

λ∈co(A)

(12.3)

The proof of this theorem is presented in the next subsection. Let |A| denote the matrix of the absolute values of the elements of A and let |h| have the similar interpretation for the vector h. Inequalities between vectors and between matrices are interpreted as inequalities between corresponding components. Then (12.3) can be rewritten in the form n−1

|f (A)| ≤ k=0

where

max |f (k) (λ)|Bk ,

λ∈co(A)

Bk = (blj (k))nl,j=1 (k = 0, 1, ..., n − 1).

Corollary 1.12.2 For any n × n-matrix A we have the inequality |eAt | ≤ exp[α(A)t]

n−1

tk Bk (t ≥ 0).

k=0

I.e. the entries elj (At) of eAt satisfy the inequalities n−1

|elj (At)| ≤ exp[tα(A)] k=0

tk blj (k) (t ≥ 0; l, j = 1, ..., n).

1.12 Absolute Values of Matrix Functions

25

1.12.2 Proof of Theorem 1.12.1 Denote by colj (A) the closed convex hull of the roots of the polynomial dlj (λ) defined by (12.1). Lemma 1.12.3 Let A be an n × n-matrix, and let f be a function regular on a neighborhood of the closed convex hull co(A) of the eigenvalues of A. Then the entries fjl (A) of f (A) satisfy the inequalities |fjl (A)| ≤ Proof:

1 dnlj −1 (f (λ)µlj (λ)) sup | | (l, j = 1, ..., n). dλnlj −1 (nlj − 1)! λ∈co(A)

We begin with the representation f (A) =

1 2πi

Γ

f (λ)(Iλ − A)−1 dλ,

where Γ is a smooth contour containing all the eigenvalues of A. From the rule for inversion of matrices it follows that the entries rjl (λ) of the matrix (Iλ − A)−1 are rjl (λ) =

µlj (λ) mlj (λ) = (l, j = 1, ..., n). det(Iλ − A) dlj (λ)

Hence, fjl (A) =

1 2πi

Γ

1 f (λ)µlj (λ) dλ = 2πi dlj (λ)

f (λ)µlj (λ) dλ, (λ − ω1 )...(λ − ωnlj )

Γ

where ω1 , ..., ωnlj are the roots of dlj (λ) which are simultaneously the eigenvalues of A taken with their multiplicities. Now by virtue of Lemma 1.11.1 we conclude that the required assertion is valid. ✷ Proof of Theorem 1.12.1: It can be written dnlj −1 (f (λ)µlj (λ)) |≤ | dλnlj −1

nlj −1 k=0

(n −k−1)

Cnklj −1 |µlj lj

(λ)||f (k) (λ)|.

k Recall that Cm are the binomial coefficients. Hence,

dnlj −1 (f (λ)µlj (λ)) | ≤ (nlj − 1)! sup | dλnlj −1 λ∈co(A)

nlj −1

blj (k) sup |f (k) (λ)|. k=0

Now Lemma 1.12.3 yields the required result. ✷

λ∈co(A)

26

1. Preliminaries

1.13 Banach Spaces By Lp ≡ Lp ([0, ∞), Cn ) (1 ≤ p < ∞) we denote the Banach space of functions defined on [0, ∞) with values in Cn and the norm f

=[

Lp

∞ 0

f (t) p dt]1/p (f ∈ Lp )

and C ≡ C([0, ∞), Cn ) is the Banach space of continuous functions defined on [0, ∞) with values in Cn and the sup-norm norm f

C

= sup f (t) t≥0

(f ∈ C).

We will need the following Lemma 1.13.1 Let 1/p + 1/q = 1 (p, q > 1). Then for any function h ∈ Lp ([0, ∞), Cn ) with the property h˙ ∈ Lq ([0, ∞), Cn ), we have 2

h(t)

∞

≤ 2[

t

h(s) p ds]1/p [

∞ t

dh(s)/ds q ds]1/q .

Proof: Since h(t) is a continuous function and h value theorem it can be written t+1 t

h(s) p ds = h(θ)

p

Lp

< ∞, due to the mean

→ 0 as t → ∞ (t ≤ θ ≤ t + 1).

Consequently, h(∞) = 0. Obviously, h(t)

2

=−

∞ t

d h(s) ds

2

ds = −2

∞ t

h(s)

Taking into account that |

d h(s) | ≤ dh(s)/ds , ds

older’s inequality. ✷ we get the required result due to H´

d h(s) ds. ds

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28

2. Perturbations of Linear Systems

U (t) = etA and U (t, s) = U (t − s) = e(t−s)A . Furthermore, a solution of the non-homogeneous equation u(t) ˙ = A(t)u(t) + f (t) (t ≥ 0)

(1.5)

with a given piecewise continuous function f : R+ → Cn can be obtained in the form u(t) = U (t, s)u(s) +

t s

U (t, τ )f (τ )dτ (t ≥ s ≥ 0).

(1.6)

It is simple to check that, if U (t, s) is the evolution operator of (1.1), then for any complex number α, Uα (t, s) = eα(t−s) U (t, s) is the evolution operator of the equation u(t) ˙ = (A(t) + αI)u(t) (t ≥ 0). (1.7) The Cauchy operator can be represented in the form U (t) = I+

t 0

∞

A(t1 )dt1 + k=2

t 0

0

tk ...

t2 0

A(tk ) A(tk−1 )...A(t1 )dtk dtk−1 ...dt1 .

Lemma 2.1.1 A necessary and sufficient condition for the stability of equation (1.1) is the uniform boundedness of its Cauchy operator: sup U (t) < ∞. t≥0

For the proof see (Daleckii and Krein, 1974, Lemma 3.3.1). Rephrasing the conditions of the lemma, we will say that a necessary and sufficient condition for the stability of equation (1.1) is the existence of a constant M > 0 such that any solution u(t) of equation (1.1) satisfies the estimate u(t) ≤ M u(0) (t ≥ 0). The least value M0 of this constant is clearly given by the equality M0 = sup U (t) . t≥0

Linear equation (1.1) is uniformly stable if there exists a constant M > 0 such that any solution u(t) of this equation satisfies the following estimate for all t ≥ s ≥ 0: u(t) ≤ M u(s) . Since u(t) = U (t, s)u(s), it can be shown that the uniform stability of the equation is equivalent to the condition sup

t≥s≥0

U (t, s) < ∞.

Note that if A(t) ≡ A is a constant matrix, then the stability condition sup etA < ∞ t≥0

coincides with the uniform stability condition.

(1.8)

29

2.2 Perturbations of Evolution Operators

2.2 Perturbations of Evolution Operators Lemma 2.2.1 Let U (t, s) be the evolution operator of equation (1.1) and U1 (t, s) be the evolution operator of the equation u(t) ˙ = A1 (t)u(t) (t ≥ 0)

(2.1)

with a variable n × n-matrix A1 (t) = A(t). Then t

U (t, s) − U1 (t, s) =

s

U (t, τ )(A(τ ) − A1 (τ ))U1 (τ, s)dτ (t ≥ s ≥ 0).

Proof: Subtract (2.1) from (1.1) and put δ(t) = u(t) − u1 (t), where u(t) is a solution of (1.1), and u1 (t) is a solution of (2.1) with the initial conditions u1 (s) = u(s) = h. Here h is a given vector. We get ˙ = A(t)δ(t) + (A(t) − A1 (t))u1 (t). δ(t) Taking into account that δ(s) = 0 and applying formula (1.6), we arrive at the equality t

δ(t) = But

s

U (t, τ )(A(τ ) − A1 (τ ))u1 (τ )dτ.

u1 (τ ) = U1 (t, s)u(s) and δ(t) = (U (t, s) − U1 (t, s))u(s).

This proves the result, since u(s) = h is an arbitrary vector. ✷ Furthermore, Lemma 2.2.1 implies, U (t, s) − U1 (t, s) ≤

t

U (t, τ )

s

A(τ ) − A1 (τ )

U1 (τ, s) dτ (0 ≤ s < t).

Hence, U1 (t, s) ≤ U (t, s) +

t s

U (t, τ )

A(τ ) − A1 (τ )

(0 ≤ s < t).

U1 (τ, s) dτ (2.2)

Lemma 1.7.1 yields Lemma 2.2.2 Let U (t, s) and U1 (t, s) be the evolution operators of the equations (2.1) and (1.1), respectively. Then U1 (t, s) ≤ ηs (t), where ηs (t) is the solution of the equation ηs (t) = U (t, s) +

t s

U (t, τ )

A(τ ) − A1 (τ ) ηs (τ )dτ (0 ≤ s < t).

2. Perturbations of Linear Systems

30

Corollary 2.2.3 Let the evolution operator U (t, s) of equation (1.1) satisfy the inequality U (t, s) ≤ M e−ν(t−s) (t ≥ s ≥ 0) (2.3) with positive constants M and ν. Then the evolution operator U1 (t, s) of equation (2.1) subordinates the inequality U1 (t, s) ≤ M e−ν(t−s) eM

t s

A(τ )−A1 (τ ) dτ

(s ≤ t).

(2.4)

Moreover, U (t, s) − U1 (t, s) ≤ M e−ν(t−s) (eM

t s

A(τ )−A1 (τ ) dτ

− 1).

(2.5)

Indeed, to prove inequality (2.4) we need, according to Lemma 2.2.2, only to check that the right part of that inequality is a solution of the equation t

η(t) = M e−ν(t−s) + M

s

e−ν(t−τ ) A(τ ) − A1 (τ ) η(τ )dτ (s ≤ t).

The checking of inequality (2.5) is left to the reader (see also (Daleckii and Krein, 1974, Section 3.2)). The latter corollary and relation (1.8) imply Corollary 2.2.4 Let equation (1.1) be uniformly exponentially stable and let ∞

A(τ ) − A1 (τ ) dτ < ∞.

0

Then equation (2.1) is also uniformly exponentially stable. If A1 (t) = 0, then by (2.5), we easily get: Corollary 2.2.5 Under condition (2.3) the evolution operator U (t, s) of equation (1.1) satisfies the inequality U (t, s) − I ≤ M e−ν(t−s) (e

t s

A(τ ) dτ

− 1) (s ≤ t).

Lemma 2.2.6 Let U (t, s) and U1 (t, s) be the evolution operators of equations (1.1) and (2.1), respectively. For some positive T ≤ ∞, let the condition J ≡ sup

0≤t≤T

t 0

U (t, s)

A(s) − A1 (s) ds < 1

be fulfilled. Then the inequality sup

0≤t≤T

holds.

U1 (t, 0) ≤ (1 − J)−1 sup

0≤t≤T

U (t, 0)

2.3 Representation of Solutions

Proof:

31

According to (2.2), sup

0≤t≤T

sup

0≤t≤T

U1 (t, 0) ≤ sup

0≤t≤T

U1 (t, 0) sup

0≤t≤T

t

U (t, τ )

0

U (t, 0) +

A(τ ) − A1 (τ ) dτ ≤

U (t, 0) + sup

0≤τ ≤T

U1 (τ, 0) J.

Now the result is due to the condition J < 1. ✷ Lemma 2.2.7 Let U (t, s) and U1 (t, s) be the evolution operators of equations (1.1) and (2.1), respectively. Let the condition U (t, s) ≤ c exp[

t s

a(τ )]dτ (0 ≤ s ≤ t)

be fulfilled with a continuous scalar-valued function a(t) and a positive constant c. Then the inequality t

U1 (t, s) ≤ c exp[

s

(a(τ ) + c A(τ ) − A1 (τ ) )dτ ]

holds. Proof: Thanks to Lemma 2.2.2, U1 (t, 0) ≤ η(t), where η(t) is the solution of the equation η(t) = c exp[

t s

a(τ )dτ ] + c

t s

exp[

t τ

a(τ1 )dτ1 ] A(τ ) − A1 (τ ) η(τ )dτ.

Put z(t) = η(t)exp[− Then z(t) = c + c

t s

t 0

a(τ )dτ ].

A(τ ) − A1 (τ ) z(τ )dτ.

Now the result is due to the Gronwall lemma. ✷

2.3 Representation of Solutions For a bounded Riemann integrable matrix-valued function F defined on a segment [0,t], define the left multiplicative integral as the limit of the sequence of the products

2. Perturbations of Linear Systems

32

←

(M )

1≤k≤M −1

(I + F (tk

(M )

)δk ) := (M )

(M )

(I + F (tM −1 )δM −1 )(I + F (tM −2 )δM −2 )...(I + F (t1 (M )

(M )

as maxk |tk+1 − tk

| tends to zero. Here (M )

0 = t1 and

)δ1 )

(M )

(M )

< t2

(M )

(M )

δk = tk+1 − tk

< ... < tM

=t

(k = 1, ..., M − 1).

I.e., the arrow over the symbol of the product means that the indexes of the co-factors increase from right to left. The left multiplicative integral we denote by the symbol ←

[0,t]

(I + F (s)ds).

Lemma 2.3.1 Any solution x(t) of equation (1.1) with a piecewise continuous matrix A(t) can be represented by the relation x(t) =

← [0,t]

(I + A(s)ds)x(0).

For the proof see (Dollard and Friedman, 1979), (Gil’, 1998, Section 4.1).

2.4 Triangular Systems Let us consider the triangular system j

u˙ j (t) =

ajk (t)uk (t) (t ≥ 0; j = 1, ..., n)

(4.1)

k=1

with piecewise continuous real functions ajk (t) (1 ≤ k ≤ j ≤ n). Then we have uj (t) = e

t 0

ajj (s)ds

uj (0) +

t 0

e

t τ

ajj (s)ds

j−1

ajk (τ )uk (τ )dτ. k=1

Hence we arrive at the inequality |uj (t)| ≤ e with the notation

t 0

α(s)ds

|uj (0)| +

t 0

e

t τ

α(s)ds

j−1

|ajk (τ )||uk (τ )|dτ k=1

2.4 Triangular Systems

α(t) = max ajj (t).

33

(4.2)

j=1,...,n

Furthermore, putting e−

t 0

α(s)ds

|uj (t)| = wj (t),

(4.3)

|ajk (τ )||wk (τ )|dτ (j = 1, ..., n).

(4.4)

we get t j−1

|wj (t)| ≤ |uj (0)| +

0 k=1

Denote by V (t) the lower-triangular matrix with the entries vjk (t) = |ajk (t)| for j > k, and vjk (t) = 0 for j ≤ k (j, k = 1, ...n), then we can write down inequalities (4.4) in the vector form t

|w(t)| ≤ |u(0)| +

0

V (τ )|w(τ )|dτ.

Here w(t) = (w1 (t), ..., wn (t)). For any continuous vector-valued function h(t) defined on the positive half-line, introduce an operator W , defined by the relation t

(W h)(t) =

0

V (τ )h(τ )dτ.

But V (t) is a lower-triangular n × n-matrix with the zero diagonal. Therefore V (τ1 )V (τ2 )...V (τn ) = 0 (τ1 , ..., τn ≥ 0). Consequently W n = 0, and (I − W )−1 =

∞

Wk =

k=0

n−1

W k.

(4.5)

k=0

It can be written as |w(t)| ≤ (I − W )

−1

n−1

|u(0)| =

W k |u(0)|.

k=0

But V (t) ≤ N (V (t)) = 1≤k<j≤n

a2jk (t).

The latter inequality yields the relation (W k h)(t) ≤

t 0

...

tk−1 0

V (t1 )... V (tk ) dtk ...dt1 ≤

(4.6)

2. Perturbations of Linear Systems

34

t 0

...

tk−1 0

N (V (t1 ))... N (V (tk ))dtk ...dt1 =

1 [ k!

t 0

N (V (t1 ))dt1 ]k .

This inequality and (4.4) imply w(t) ≤ (I − W )

n−1

−1

u(0) ≤ u(0) k=0

1 [ k!

t 0

N (V (t1 ))dt1 ]k .

According to (4.3), we arrive at the inequality u(t) ≤ u(0) e

t 0

α(s)ds

n−1 k=0

1 [ k!

t 0

N (V (t1 ))dt1 ]k .

Now instead of 0 let us take an arbitrary real τ . We thus have derived Theorem 2.4.1 With the notations (4.2) and (4.6) the inequality u(t) ≤ u(τ ) e

t τ

α(s)ds

n−1 k=0

1 [ k!

t τ

N (V (t1 ))dt1 ]k (0 ≤ τ ≤ t < ∞)

is true for any solution u(t) of system (4.1). Corollary 2.4.2 Let the relation limt→∞ {

t 0

n−1

α(s)ds + ln [ k=0

1 ( k!

t 0

N (V (t1 ))dt1 )k ]} < +∞

hold. Then system (4.1) is stable. Corollary 2.4.3 Let the relation limt→∞ t−1 {

t 0

n−1

α(s)ds + ln [ k=0

1 ( k!

t 0

N (V (t1 ))dt1 )k ]} < 0

hold. Then system (4.1) is exponentially stable.

2.5 Perturbations of Triangular Systems Let A(t) = (ajk (t))nj,k=1 be a real piecewise continuous n × n-matrix. Denote the corresponding lower-triangular matrix by T (t). That is, T (t) = (Tjk (t))nj,k=1 , where Tjk (t) = ajk (t) if j ≥ k and Tjk (t) = 0 if j < k. We will investigate system (1.1) as a perturbation of the triangular system

2.6 Integrally Small Perturbations

w(t) ˙ = T (t)w.

35

(5.1)

Due to Theorem 2.4.1, a solution w(t) of equation (5.1) is subject to the inequality w(t) ≤ e

t τ

α(s)ds

n−1 k=0

t

1 [ k!

τ

N (V (t1 ))dt1 ]k w(τ ) .

(5.2)

Here α(t) is defined by (4.2) and N (V (t)) is defined by (4.6). Put q(t) = A(t) − T (t) . Employing Lemma 2.2.6 and inequality (5.2), we get Theorem 2.5.1 Let the condition WA ≡ sup t≥0

t 0

q(τ )e

t τ

α(s)ds

n−1 k=0

1 [ k!

t τ

N (V (t1 ))dt1 ]k dτ < 1

be fulfilled. Then equation (1.1) is stable. Moreover, the Cauchy operator U (t) of equation (1.1) satisfies the inequality U (t) ≤ (1 − WA )−1 sup e

t 0

α(s)ds

t≥0

n−1 k=0

1 [ k!

t 0

N (V (t1 ))dt1 ]k .

2.6 Integrally Small Perturbations Consider the equation

x˙ = A(t)x + B(t)x,

(6.1)

where B(t) is a variable n × n-matrix. Introduce the notations J(t) =

t 0

t

q(J) := sup t≥0

and

B(s)ds,

0

B(s)ds ,

m(t) = A(t)J(t) − J(t)(A(t) + B(t)) .

Theorem 2.6.1 Let U (t, s) be the evolution operator of equation (1.1) and the condition η(J) ≡ q(J) + sup t≥0

t 0

U (t, s) m(s)ds < 1

(6.2)

2. Perturbations of Linear Systems

36

hold. Then for any solution x(t) of (6.1) the estimate x(t) ≤

x(0) max U (t, 0) 1 − η(J) t≥0

(6.3)

is true. Proof:

Rewrite (6.1) in the form x(t) = U (t, 0)x(0) +

Evidently,

t 0

U (t, s)B(s)x(s)ds.

(6.4)

t

d [U (t, s)J(s)x(s)]ds = J(t)x(t), ds 0 since U (t, t) is the unit matrix. Exploiting equality (1.3), we obtain J(t)x(t) =

t 0

U (t, s)[−A(s)J(s)x(s)+

B(s)x(s) + J(s)x(s)]ds. ˙

(6.5)

Substituting the right part of (6.1) instead of x(s) ˙ into (6.5), we arrive at the equality t 0

t

U (t, s)B(s)x(s)ds = J(t)x(t) +

0

U (t, s)[A(s)J(s)−

J(s)(A(s) + B(s))]x(s)ds. Now (6.4) and the latter equality imply x(t) = U (t, 0)x(0) + J(t)x(t) +

t 0

U (t, s)[A(s)J(s)−

J(s)(A(s) + B(s))]x(s)ds. Hence, x(t) ≤ x(0) U (t, 0) + q(J) x(t) +

t 0

U (t, s) m(s) x(s) ds.

This inequality implies y(t) ≤ U (t, 0) + q(J)y(t) +

t 0

U (t, s) m(s)y(s)ds ≤

U (t, 0) + η(J) sup y(t), t≥0

where

x(t) . x(0) Now condition (6.2) ensures the required result. ✷ y(t) =

2.7 Integrally Small Perturbations of Autonomous Systems

37

2.7 Integrally Small Perturbations of Autonomous Systems Assume that A(t) ≡ A is a constant matrix and consider the equation dx/dt = Ax + B(t)x.

(7.1)

Suppose that m0 := sup AJ(t) − J(t)(A + B(t)) < ∞ t≥0

and

n−1

√

q(J) + m0 k=0

g k (A) < 1. k!|α(A)|k+1

(7.2)

Recall that g(A) is defined in Section 1.5. Theorem 2.7.1 Let condition (7.2) hold. Then system (7.1) is stable. Proof: In the case of a constant matrix the evolution operator U (t, s) of the equation (1.1) is equal to exp[A(t − s)]. By Corollary 1.5.3 we easily get t 0

exp[A(t − s)] ds ≤ n−1

√ k=0

∞ 0

exp[At] ds ≤

g k (A) . k!|α(A)|k+1

Now Theorem 2.6.1 implies the stated result ✷ . Let C(t) be a τ -periodic matrix. Take A = τ −1

τ 0

C(t)dt, B(t) = C(t) − A.

Besides, J(τ ) =

τ 0

B(t)dt = 0,

q(J) = sup

J(t) ,

(7.4)

AJ(t) − J(t)(A + B(t)) .

(7.5)

x(t) ˙ = C(t)x

(7.6)

0≤t≤τ

and

m0 = sup

0≤t≤τ

The periodic system

(7.3)

can be rewritten in the shape (7.1). Now Theorem 2.7.1 gives us the stability condition for system (7.6).

38

2. Perturbations of Linear Systems

Example 2.7.2 Consider system (7.6) wih real τ -periodic matrix C(t) = (cjk (t))2j,k=1 . That is n = 2. Define A, B(t), q(J) and m0 by (7.3)-(7.5). That is, τ

ajk = τ −1

0

cjk (t)dt, B(t) = C(t) − A.

According to Example 1.6.1, g(A) ≤ |a12 − a21 |. Thus according to Theorem 2.7.1, the condition q(J) +

|a12 − a21 | m0 )<1 (1 + |α(A)| |α(A)|

provides the stability of the considered periodic system.

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40

3. Systems with Slowly Varying Coefficients

and

∞

θ0 := q0

0

tp1 (t)dt < 1.

(1.6)

Then equation (1.1) is uniformly stable. Moreover, estimate (1.4) is true with θ = θ0 and χ = χ0 , where χ0 := sup p0 (t). t≥0

Recall that the quantity g(A) is defined in Section 1.5. In addition, for any n × n-matrix A Corollary 1.5.3 gives us the inequality: exp[A(s)t] ≤ p˜(t, A(s)) (t, s ≥ 0), where

n−1

p˜(t, A(s)) := exp[α(A(s))t] k=0

(1.7)

g k (A(s))tk (k!)3/2

with α(A(s)) = maxk Re λk (A(s)). Let χ ˜ := sup p˜(t, A(t)) < ∞. t≥0

Now Theorem 3.1.1 implies Corollary 3.1.3 Let the condition ζ˜ := sup t≥0

t 0

p˜(t − t1 , A(t))q(t, t1 )dt1 < 1

hold. Then equation (1.1) is stable. Moreover, the estimate x(t) ≤

χ ˜ x(s) 1 − ζ˜

(t > s ≥ 0)

is true for any solution x(t) of (1.1). Let

v := sup g(A(t)) < ∞

(1.8)

ρ := − sup α(A(t)) > 0.

(1.9)

t≥0

and

t≥0

Denote

n−1

p˜0 (t) := exp[−ρt] k=0

and

tk v k , (k!)3/2

χ ˜0 := sup p˜0 (t). t≥0

3.1 The Freezing Method

41

Simple calculations show that ∞ 0

where

t˜ p0 (t)dt = ζ1 , n−1

ζ1 := k=0

(k + 1)v k √ . k!ρ2+k

Now Corollary 3.1.2 and a small perturbation imply Corollary 3.1.4 Let conditions (1.5), (1.8) and (1.9) hold. In addition, let q0 ζ1 < 1. Then equation (1.1) is exponentially stable. Moreover, the evolution operator U (t, s) of equation (1.1) satisfies the inequality U (t, s) ≤

χ ˜0 (t, s ≥ 0). 1 − q0 ζ1

Furthermore, denote by z(q0 , v) the extreme right-hand root of the algebraic equation z n+1 = q0 P (z), (1.10) where

n−1

P (z) = k=0

(k + 1)v k n−k−1 √ . z k!

Recall that I is the unit matrix. Theorem 3.1.5 Let conditions (1.5) and (1.8) hold. In addition, let the matrix A(t) + ( + z(q0 , v))I be a Hurwitz one for an > 0 and all t ≥ 0. Then equation (1.1) is exponentially stable. Moreover, there is a positive constant a, such that any solution x(t) of equation (1.1) satisfies the inequalities a−1 exp[(β0 − z(q0 , v))t] ≤ where

x(t) ≤ a exp[(−ρ + z(q0 , v))t] (t ≥ 0), x(0)

β0 = inf min Reλk (A(t)). t≥0

k

Setting z = vy in (1.10), we have the equation y n+1 = q0 v −2

n−1 k=0

k + 1 n−k−1 √ y . k!

Hence due to Lemma 1.8.1, with the notation

42

3. Systems with Slowly Varying Coefficients n−1

wn = k=0

k+1 √ , k!

we get z(q0 , v) ≤ δ(q0 , v), where 2

δ(q0 , v) =

v 1− n+1 [q0 wn ]1/(n+1) q0 /v

if q0 wn ≥ v 2 , if q0 wn ≤ v 2 .

Theorem 3.1.5 yields Corollary 3.1.6 Let conditions (1.5) and (1.8) hold. If, in addition, the matrix A(t) + ( + δ(q0 , v))I is a Hurwitz one for an > 0 and all t ≥ 0, then system (1.1) is exponentially stable. Below we also will prove the following Lemma 3.1.7 Let conditions (1.5), (1.8) and (1.9) be fulfilled. In addition, let ρn+1 . (1.11) q0 < P (ρ) Then the matrix A(t) + z(q0 , v)I is a Hurwitz one. Thus under (1.5), (1.8), (1.9) and (1.11) system (1.1) is stable. Example 3.1.8 Let us consider the equation y¨ + a(t)y˙ + b(t)y = 0 (t ≥ 0). Here a(t) and b(t) are bounded on [0, ∞) positive scalar-valued functions with the property |a(t) − a(s)| + |b(t) − b(s)| ≤ q0 |t − s| (t, s ≥ 0). According to Example 1.6.1 g(A(t)) ≤ v ≡ 1 + sup b(t) (t ≥ 0). t≥0

Let us suppose that ρ = − sup α(A(t)) = inf Re [a(t)/2 − t≥0

t≥0

Then we can write ζ1 =

a2 (t)/4 − b(t)] > 0.

1 2v + 3. ρ2 ρ

So if ζ1 q0 < 1, then the considered equation is exponentially stable due to Corollary 3.1.4.

3.2 Proofs of Theorems 3.1.1 and 3.1.5, and Lemma 3.1.7

43

3.2 Proofs of Theorems 3.1.1 and 3.1.5, and Lemma 3.1.7 Proof of Theorem 3.1.1: Equation (1.1) can be rewritten in the form dx/dt − A(τ )x = [A(t) − A(τ )]x with an arbitrary fixed τ ≥ 0. This equation is equivalent to the following one: t

x(t) = exp[A(τ )t]x(0) +

0

exp[A(τ )(t − t1 )][A(t1 ) − A(τ )]x(t1 )dt1 .

According to (1.2), t

x(t) ≤ p(t, A(τ )) x(0) +

0

p(t − t1 , A(τ ))q(τ, t1 ) x(t1 ) dt1 .

With τ = t this relation gives sup

0≤t≤T

x(t) ≤ χ x(0) + sup

0≤t≤T

x(t) θ

for any positive finite T . By the condition θ < 1 we arrive at the inequality sup

0≤t≤T

x(t) ≤ χ x(0) (1 − θ)−1 .

Since the right-hand part of the latter inequality does not depend on T , we get sup x(t) ≤ χ x(0) (1 − θ)−1 . t≥0

This is the required inequality for s = 0. It can be similarly proved for any s ≤ t. ✷ Set ζ0 :=

∞ 0

p˜0 (t)dt.

Simple calculations show that n−1

√

ζ0 = k=0

vk . k!ρ1+k

Lemma 3.2.1 Let the conditions (1.5), (1.8), (1.9) and q0 ζ1 < 1 be fulfilled. Then the evolution operator U (t, s) of (1.1) satisfies the estimate t 0

U (t, w) dw ≤

ζ0 for all t ≥ 0. 1 − q0 ζ1

3. Systems with Slowly Varying Coefficients

44

Proof:

Again rewrite equation (1.1) in the form dx/dt − A(τ )x = [A(t) − A(τ )]x

with a fixed τ ≥ 0. Hence x(t) = exp[A(τ )t]x(w) +

t w

exp[A(τ )(t − t1 )][A(t1 ) − A(τ )]x(t1 )dt1

(0 ≤ w ≤ t < ∞). Using relation (1.5), we get t

x(t) ≤ eA(τ )(t−w) x(w) +

eA(τ )(t−t1 ) q0 |t1 − τ | x(t1 ) dt1 .

w

Hence, with the notation p0 (t) ≡ p˜0 (t), we have t

x(t) ≤ p0 (t − w) x(w) +

w

p0 (t − t1 )q0 |t1 − τ | x(t1 ) dt1 .

When τ = t this relation implies the inequality t

y(t, w) ≤ p0 (t − w) + q0

0

with y(t, w) =

p0 (t − t1 )(t − t1 )y(t1 , w)dt1 x(t) . x(w)

The integration with respect to w yields t 0

y(t, w)dw ≤ ζ0 + q0

because

t 0

Obviously,

t

u

0

0

t

t

0

w

p0 (t − t1 )(t − t1 )y(t1 , w)dt1 dw,

w

∞ 0

p0 (s)ds = ζ0 .

p0 (t − u)(t − u)y(u, w)du dw =

w t

Hence,

0

t

p0 (t − w)dw ≤ t

0

t

p0 (t − u)(t − u)y(u, w)dw du.

p0 (t − u)(t − u)y(u, w)dudw ≤

(2.1)

3.2 Proofs of Theorems 3.1.1 and 3.1.5, and Lemma 3.1.7 t

p0 (t − u)(t − u)du max u≤t

0

u 0

45

y(u, w)dw.

Now, due to (2.1) we can write down t 0

u

y(t, w)dw ≤ ζ0 + q0 max u≤t

0

y(u, w)dw

ζ0 + q0 ζ1 max u≤t

u 0

t 0

p0 (t − u)(t − u)du ≤

y(u, w)dw.

For any fixed positive T , this implies max

0≤t≤T

t 0

y(t, w)dw ≤ ζ0 + q0 ζ1 max

0≤u≤T

u 0

y(u, w)dw.

Thus, due to the inequality q0 ζ1 < 1, it can be written max

0≤t≤T

t 0

y(t, w)dw ≤ (1 − q0 ζ1 )−1 ζ0 .

The right hand part of the latter inequality does not depend on T . Therefore, max t≥0

t 0

y(t, w)dw ≤ (1 − ζ1 )−1 ζ0 .

This result proves the lemma, since x(t) is an arbitrary solution. ✷ Proof of Theorem 3.1.5: Under the hypothesis of the theorem we have the inequality ρ > z(q0 , v). Dividing (1.10) by z n+1 , and taking into account that z(q0 , v) is the extreme right-hand root of (1.10), we can write down n−1

1 > q0 k=0

(k + 1)v k √ ≡ q0 ζ 1 . k!ρk+2

Now, Corollary 3.1.5 provides the stability. Let us prove the upper solution estimate. Substituting the equality x(t) = xc (t)exp(−tc) with a real number c in equation (1.1), we get the equation dxc /dt = (A(t) + Ic)xc . Take c = ρ − z(q0 , v) − with an

(2.2)

> 0. Then we can affirm that the matrix

A(t) + I(z(q0 , v) + c) is a Hurwitz one. As it is shown in Section 1.5,

46

3. Systems with Slowly Varying Coefficients

g(A(t) + Ic) = g(A(t)). Therefore, due to Corollary 3.1.4 equation (2.2) is stable, i.e. there is a constant a such that xc (t) ≤ a xc (0) for all t ≥ 0. This ensures the upper estimate, since x(t) = xc (t) exp(−tc), and is arbitrary. Now let us prove the lower estimate. Take into account that the Cauchy operator U (t) of equation (1.1) has the following property: its inverse operator U −1 (t) is the Cauchy operator of the equation u˙ = −A(t)u. But

sup max Reλk (−A(t)) = − inf min Reλk (A(t)). t≥0

t≥0

k

k

Due to the proven upper estimate U −1 (t)h ≤ a h exp [(−β0 + z(q0 , v))t], since g(−A) = g(A). Putting w = U −1 (t)h, we arrive at the relation U (t)w ≥ a−1 exp [(β0 − z(q0 , v))t] w (w ∈ Cn ). That inequality gives the lower estimate. ✷ Proof of Lemma 3.1.7: Take into account that by (1.10), q0 =

z n+1 (q0 , v) . P (z(q0 , v))

But the function xn+1 P −1 (x) increases as x > 0 increases. Hence by (1.11), ρ > z(q0 , v). This fact yields the stated result. ✷

3.3 Systems with Differentiable Matrices Let A(t) = (ajk (t))nj,k=1 be a real differentiable matrix, which is Hurwitzian for each t ≥ 0 and uniformly bounded on [0, ∞). In this section we establish stability conditions for a linear system in terms of the determinant of the variable matrix. Denote 1 ∞ ψ(t) = (A(t) − iIω)−1 3 dω (t ≥ 0). π −∞

3.3 Systems with Differentiable Matrices

Theorem 3.3.1 Let

˙ A(t) ψ(t) ≤ 1 (t ≥ 0).

47

(3.1)

Then system (1.1) is stable. This theorem and the next one are proved in Section 3.5. Now let ˙ sup A(t) ψ(t) < 1.

(3.2)

t≥t0

With an > 0 substitute the equality x(t) = u (t)e− have the equation

t

into (1.1). Then we

u˙ (t) = (A(t) + I )u (t) (t ≥ 0). Put ψ (t) =

1 π

∞ −∞

(A(t) − (1 − )Iω)−1 3 dω (t ≥ 0).

˙ Due to (3.2) we can take sufficiently small such that A(t) ψ (t) ≤ 1 (t ≥ t0 ). Thanks to Lemma 3.3.1, there is a constant M , such that u (t) ≤ M x(0) Thus

x(t) ≤ M x(0) e−

(t ≥ t0 ). t

(t ≥ t0 ).

This implies Corollary 3.3.2 Let condition (3.2) hold. Then system (1.1) is exponentially stable. Put

˜ := 1 ψ(t) π

∞ −∞

(N 2 (A(t)) + nω 2 )3(n−1)/2 dω . |det (iωI − A(t))|3 (n − 1)3(n−1)/2

Recall that N (A0 ) is the Frobenius norm of a matrix A0 . Theorem 3.3.3 For some t0 ≥ 0, let ˜ ≤ 1 (t ≥ t0 ). ˙ A(t) ψ(t) Then system (1.1) is stable. Moreover, if ˜ < 1. ˙ sup A(t) ψ(t)

t≥t0

Then system (1.1) is exponentially stable.

3. Systems with Slowly Varying Coefficients

48

Note that if all the eigenvalues of the matrix are real, then |det (iωI − A(t))| ≥ (ω 2 + α2 (t))n/2 (ω ∈ R), where

α(t) : = α(A(t)) = max Re λk (A(t)) k=1,...,n

and λk (A(t)) (k = 1, ..., n) are the eigenvalues of an n × n-matrix A(t) with their multiplicities. Thus, ∞

˜ ≤ γn ψ(t) π

0

(N 2 (A(t)) + ny 2 )3(n−1)/2 dy, (y 2 + α2 (t))3n/2

where γn =

1 . (n − 1)3(n−1)/2

Hence, ˜ ≤ ψ(t)

∞

γn π|α(t)|3n−1

0

(N 2 (A(t)) + nα2 (t)s2 )3(n−1)/2 ds. (s2 + 1)3n/2

Take into account that n

|λk (A(t))|2 ≤ N 2 (A(t))

k=1

and

|α(t)| ≤ min |λk (A(t))|. k

So and

nα2 (t) ≤ N 2 (A(t)) 3(n−1)/2 γn N 3(n−1) (A) ˜ ≤2 ψ(t) π|α(t)|3n−1

∞ 0

(s2

1 dy. + 1)3/2

It can be checked by differentiation that t 0

(s2

1 t ds = 2 . 3/2 + 1) (t + 1)1/2

˜ ≤ ψ0 (t), where Thus ψ(t) ψ0 (t) = In the case n = 2,

23(n−1)/2 γn N 3(n−1) (A(t)) . π|α(t)|3n−1

√ 4 2N 3 (A(t)) . ψ0 (t) = π|α(t)|5

Now Theorem 3.3.3 implies

(3.3)

3.4 Additional Stability Conditions

49

Corollary 3.3.4 For a t0 ≥ 0, let all the eigenvalues A(t) be real, and ˙ A(t) ψ0 (t) ≤ 1 (t ≥ t0 ). Then system (1.1) is stable. Example 3.3.5 Let us consider the system x˙ 1 = −3a(t)x1 + b(t)x2 , x˙ 2 = a(t)x2 − b(t)x1

(3.4)

where a(t), b(t) are positive differentiable functions, and 3a2 (t) < b2 (t) < 4a2 (t). Then

λ1,2 (A(t)) = −a(t) ±

4a2 (t) − b2 (t),

N 2 (A(t)) = 10a2 (t) + 2b2 (t) and by (3.3) ψ0 (t) =

16(5a2 (t) + b2 (t))3/2 . π|α(t)|5

Moreover, simple calculations show that ˙ ˙ A(t) ≤ 3|a(t)| ˙ + |b(t)| Thanks to Corollary 3.3.4, system (3.4) is stable, provided (5a2 (t) + b2 (t))3/2 ˙ 16(3|a(t)| ˙ + |b(t)|) ≤ 1 (t ≥ 0). π|α(t)|5

3.4 Additional Stability Conditions In this section as well as in the previous section, A(t) is a real differentiable matrix, which is Hurwitzian for each t ≥ 0, but the stability conditions are formulated in terms of the individual eigenvalues. Denote ∞ η(t) =

0

eA(t)s

2

ds.

Theorem 3.4.1 For a t0 ≥ 0, let ˙ A(t) η 2 (t) ≤ 1 (t ≥ t0 ).

(4.1)

Then system (1.1) is stable. Moreover, if ˙ sup A(t) η 2 (t) < 1,

t≥t0

then system (1.1) is exponentially stable.

(4.2)

3. Systems with Slowly Varying Coefficients

50

This theorem is proved in the next section. Recall that g(A) is introduced in Section 1.5 and α(t) = α(A(t)) is defined in the previous section. Put n−1

η˜(t) : =

g j+k (A(t)) (k + j)! . 2j+k+1 |α(t)|j+k+1 (j! k!)3/2 j,k=0

Due to Lemma 1.9.2 η(t) ≤ η˜(t). Now the previous theorem implies Corollary 3.4.2 For a t0 ≥ 0, let ˙ A(t) η˜2 (t) ≤ 1 (t ≥ t0 ). Then system (1.1) is stable. Moreover, if ˙ sup A(t) η˜2 (t) < 1,

t≥t0

then system (1.1) is exponentially stable. In particular, if n = 2, then η˜(t) =

κ2 (t) 1 (1 + κ(t) + ), 2|α(t)| 2

where κ(t) : =

(4.3)

g(A(t)) . |α(t)|

If n = 3, then

√ √ ( 2 + 1) 2 3 2κ3 1 2κ4 κ + (1 + κ + η˜(t) = + ) (κ = κ(t)). 4 3 2 2|α(t)|

(4.4)

Example 3.4.3 Let A(t) = (ajk (t))2j,k=1 be a real differentiable 2×2 matrix. Then Example 1.6.1 implies g(A(t)) ≤ |a12 (t) − a21 (t)| in the general case; if the eigenvalues of A(t) are nonreal, then g(A(t)) =

(a11 (t) − a22 (t))2 + (a21 (t) + a12 (t))2 .

(4.5)

For instance, let us consider the system x˙ 1 = −3a(t)x1 + x2 , x˙ 2 = a(t)x2 − x1 ,

(4.6)

where a(t) is a positive differentiable function satisfying the inequality a(t) ≤ 1/2 (t ≥ 0). ˙ Then A(t) = 3|a(t)|, ˙ α(A(t)) = −a(t) and due to (4.5) g(A(t)) = 4a(t). Thus (4.3) implies

3.5 Proofs of Theorems 3.3.1, 3.3.3, and 3.4.2

η˜(t) =

51

13 . 2a(t)

Thanks to Corollary 3.4.2, system (4.6) is stable, provided 3 . 132 |a(t)| ˙ ≤ 1 (t ≥ 0). 2 4a (t) Example 3.4.4 Let A(t) = (ajk (t))3j,k=1 be a real differentiable 3×3-matrix. According to property (5.3) from Section 1.5, g(A(t)) ≤ v(t) : = [(a12 − a21 )2 + (a13 − a31 )2 + (a23 − a32 )2 ]1/2 . For simplicity assume that 3

ajj (t) +

|ajk (t)| ≤ −ρ0 (t) (t ≥ 0, j = 1, 2, 3), k=1,k=j

where ρ0 (t) is a positive scalar function. Then due to the well known result (Marcus and Minc, 1964, Section 3.3.5), α(A(t)) ≤ −ρ0 (t). Thus (4.4) implies √ √ 3 2κ31 2κ4 1 ( 2 + 1)κ21 (1 + κ1 + + + 1 ), η˜(t) ≤ η1 (t) : = 2ρ0 (t) 2 4 3 where κ1 = κ1 (t) = is stable, provided

v(t) ρ0 (t) .

Thanks to Corollary 3.4.2, the considered system ˙ ≤ 1 (t ≥ 0). η12 (t) A(t)

3.5 Proofs of Theorems 3.3.1, 3.3.3, and 3.4.2 Recall the Lyapunov theorem (see Section 1.9): if the eigenvalues of A0 lie in the interior of the left half-plane, then for any positive definite Hermitian matrix H there exists a positive definite Hermitian matrix WH , such that WH A0 + A∗0 WH = −2H.

(5.1)

Moreover, WH =

1 π

∞ −∞

(−iIω − A∗0 )−1 H(iIω − A0 )−1 dω.

Now let A(t) depend on t. Put W (t) =

1 π

∞ −∞

(−iIω − A∗ (t))−1 (iIω − A(t))−1 dω.

(5.2)

52

3. Systems with Slowly Varying Coefficients

Then W (t) is a solution of the equation. W (t)A(t) + A∗ (t)W (t) = −2I.

(5.3)

Since A(.) is real, (W (t)A(t)h, h) = −(h, h) (h ∈ Rn ). Furthermore, multiplying equation (1.1) by W (t) and doing the scalar product, we get (W (t)x(t), ˙ x(t)) = (W (t)A(t)x(t), x(t)) = −(x(t), x(t)).

(5.4)

Since d ˙ (t)x(t), x(t)) + (W (t)x(t), x(t)) (W (t)x(t), x(t)) = (W (t)x(t), ˙ x(t)) + (W ˙ = dt ˙ (t)x(t), x(t)), 2(W (t)x(t), ˙ x(t)) + (W it can be written d ˙ (t)x(t), x(t)) + 2(W (t)x(t), (W (t)x(t), x(t)) = (W ˙ x(t)). dt Thus due to (5.4) d ˙ (t)x(t), x(t)) − 2(x(t), x(t)). (W (t)x(t), x(t)) = (W dt and the inequality

˙ (t) ≤ 2 (t ≥ t0 ) W

(5.5)

provides the stability, since (W (t)h, h) ≥ c0 (h, h) (c0 = const > 0, t ≥ 0, h ∈ Rn ). Indeed, the latter relation follows from Lemma 1.10.1 and the uniform boundedness of A(t). We thus have proved Lemma 3.5.1 Let condition (5.5) hold. Then system (1.1) is stable. Proof of Theorem 3.3.1: From (5.3) it follows ˙ (t) = 1 W π

∞ −∞

(−iIω − A∗ (t))−1 [A˙ ∗ (t)(−iIω − A∗ (t))−1 +

˙ (−iIω − A(t))−1 A(t)](iIω − A(t))−1 dω. Hence, 1 ˙ (t) ≤ 2 A(t) ˙ W π

∞ −∞

˙ (iIω − A(t))−1 3 dω = 2 A(t) ψ(t).

3.5 Proofs of Theorems 3.3.1, 3.3.3, and 3.4.2

53

This and (5.5) prove the required result. ✷ Proof of Theorem 3.3.3: Recall that for any constant n × n-matrix A0 the inequality (Iλ − A0 )−1 ≤

(N 2 (A0 ) − 2Re (λ T race (A0 )) + n|λ|2 )(n−1)/2 |det (λI − A0 )|(n − 1)(n−1)/2

is true for any regular λ (see Section 1.5). Since A(t) is real and Im T race A(t) = 0, we have (Iiy − A(t))−1 ≤

(N 2 (A(t)) + ny 2 )(n−1)/2 (y ∈ R). |det (iyI − A(t))| (n − 1)(n−1)/2

˜ Hence, ψ(t) ≤ ψ(t). Now Theorem 3.3.1 implies the required result. ✷ Proof of Theorem 3.4.1: Due to (5.2) we can write W (t) = 2

∞ 0

eA

∗

(t)s A(t)s

e

ds

(5.6)

(see Section 1.9). So W (t) is a solution of equation (5.3). We again use the ˙ (t) , let us differentiate equation stability condition (5.5). To estimate W (5.3). Then ˙ (t)A(t) + A∗ (t)W ˙ (t) = −H1 (t), W where

˙ + A˙ ∗ (t)W (t). H1 (t) := W (t)A(t)

Due to (5.1) and (5.6) ˙ (t) = W

∞ 0

eA

∗

(t)s

H1 (t)eA(t)s ds.

˙ η(t). Hence, Moreover, W (t) ≤ 2η(t) and H1 (t) ≤ 2 A(t) ˙ (t) ≤ H1 (t) W

∞ 0

eA(t)s

2

˙ ds = H1 (t) η(t) ≤ 2 A(t) η 2 (t).

Now Lemma 3.5.1 proves the stability. If (4.2) holds, then repeating the arguments of the proof of Corollary 3.3.2 we get the exponential stability. The proof is complete. ✷

54

3. Systems with Slowly Varying Coefficients

3.6 Matrix Lipschitz Conditions 3.6.1 Stability Criterion Consider equation (1.1) in the form dxj = dt

n

ajk (t)xk (t ≥ 0; j = 1, ..., n)

(6.1)

k=1

under the matrix Lipschitz conditions |ajk (t) − ajk (s)| ≤ qjk |t − s| (t, s ≥ 0; j, k = 1, ..., n), where qjk are constants. We will write these conditions in the form |A(t) − A(s)| ≤ Q|t − s| (t, s ≥ 0), where

(6.2)

A(t) = (ajk (t))nj,k=1 and Q = (qjk )nj,k=1

is a constant matrix. Inequality (6.2) permits us to obtain estimates for each coordinate of a solution. Throughout the present section again it is assumed that matrix A(t) is uniformly bounded on [0, ∞). Furthermore, let mij (λ, t) (λ ∈ C, t ≥ 0) be the cofactor of the element δij λ − aij (t) of the matrix λI − A(t). Recall that δij = 0, j = i and δjj = 1. Assume that after division by the coinciding zeros we obtain the equality mij (λ, t) µij (λ, t) = , det(Iλ − A(t)) dij (λ, t) where dij (λ, t) and µij (λ, t) are polynomials with respect to λ, and the degree of dij (λ, t) is denoted by nij (t). Let co(A(t)) be the convex hull of all eigenvalues of matrix A(t). Put bji (k, t) =

1 ∂ (nij −k−1) µij (λ, t) |, sup | k!(nij − 1 − k)! λ∈co(A(t)) ∂λ(nij −k−1) (nij ≡ nij (t); k = 0, ..., n − 1).

Since (−m)! = 0 for a natural m > 0, the equalities bij (k, t) = 0 for k ≥ nij (t) are valid if nij (t) < n − 1. Set bij (k) = sup bij (k, t) t≥0

3.6 Matrix Lipschitz Conditions

and

55

B k = (bij (k))nj,i=1 .

Finally, denote

α0 ≡ sup max

t≥0 k=1,...,n

Reλk (A(t)).

Theorem 3.6.1 If A(t) satisfies condition (6.2), then any solution x(t) of equation (1.1) subordinates the inequality |x(t)| ≤ exp[(α0 + z0 )t]C(t)|x(0)| (t ≥ 0), where C(t) is a non-negative polynomial matrix, and z0 is the extreme righthand (positive) root of the polynomial det(Iλn+1 −

n−1

λn−k−1 (k + 1)!B k Q).

(6.3)

k=0

Below in this section we will derive a simple estimate for z0 . Recall that I is the unit matrix. Corollary 3.6.2 Under condition (6.2), let the matrix A(t) + (z0 + )I be a Hurwitz one for an > 0 and all t ≥ 0. Then system (6.1) is exponentially stable. Example 3.6.3 Let us consider the equation y + a(t)y + y = 0,

(6.4)

where the scalar-valued function a(t) satisfies the conditions |a(t) − a(s)| ≤ q|t − s| and 0 < a(t) ≤ 2 (t, s ≥ 0). Reducing this equation to the form (6.1) by the substitution x1 = y and x2 = y, we find that 1 α0 = − inf a(t), 2 t≥0 and co(A(t)) = [λ1 (A(t)), λ2 (A(t))] is the segment with

and We have Q= and

q 0 0 0

λ1 (A(t)) = −a(t)/2 + i

1 − a2 (t)/4

λ2 (A(t)) = −a(t)/2 − i

1 − a2 (t)/4.

56

3. Systems with Slowly Varying Coefficients

Bk Q =

qb11 (k) qb21 (k)

0 0

with k = 0, 1. Clearly, m11 (λ, t) = −λ and m12 (λ, t) = −1. Hence b21 (1) = 1, b21 (0) = 0, b11 (1) = max |λ| = 1, λ∈co(A(t))

and b11 (0) = 1. Thus ∆(λ) := det (λ3 I − λB 0 Q − 2B 1 Q) = λ3 − λq − 2q −q

det

0 λ3

= λ3 (λ3 − λq − 2q).

The extreme right-hand root z0 of ∆(λ) coincides with the the extreme righthand root of the polynomial λ3 − λq − 2q. The Cardan formula yields z0 = (q +

q 2 − (q/3)3 )1/3 + (q −

q 2 − (q/3)3 )1/3 (q ≤ 27).

Cubing this expression and making elementary calculations, we find that z03 = q(2 + (q +

q 2 − (q/3)3 )1/3 + (q −

q 2 − (q/3)3 )1/3 ).

It is simple to check that the function (q +

q 2 − (q/3)3 )1/3 + (q −

q 2 − (q/3)3 )1/3

monotonically increases as q < 27 increases. Hence for an arbitrary fixed q ≤ 27 we have z0 ≤

3

2q(1 + q 1/3 ).

Theorem 3.6.1 implies the following stability condition of equation (6.4): −a(t)/2 +

3

2q(1 + q 1/3 ) < 0 (t ≥ 0, q ≤ 27).

3.6.2 Proof of Theorem 3.6.1 Let T1 , T2 , ..., Tm be arbitrary constant n × n-matrices. Recall that a number ω is called a characteristic value of the matrix pencil Φ(λ) = λm I − λm−1 T1 − ... − Tm

(6.5)

if there is a non-zero vector h0 such that Φ(ω)h0 = 0. We will use the following Lemma 3.6.4 Each characteristic value of the matrix pencil defined by (6.5) is an eigenvalue of the block matrix   T1 T2 . . . Tm−1 Tm  I 0 ... 0 0   .  . . .  . ... 0 0 ... I 0

3.6 Matrix Lipschitz Conditions

57

For the proof see (Rodman, 1989, Chapter 1). By virtue of the Frobenius theorem for positive matrices (Gantmaher, 1967), we arrive at the following result. Corollary 3.6.5 Let all the matrices Tk (k = 1, ..., m) be non-negative. Then the extreme right characteristic value of the matrix pencil (6.5) is positive. Proof of Theorem 3.6.1: Fix s ≥ 0 and write (6.1) as dx/dt − A(s)x = [A(t) − A(s)]x. This equation is equivalent to the following one: x(t) = eA(s)t x(0) +

t 0

eA(s)(t−τ ) [A(τ ) − A(s)]x(τ )dτ.

Hence (6.2) implies that |x(t)| ≤ |eA(s)t x(0)| +

t

|eA(s)(t−τ ) |Q|τ − s||x(τ )|dτ.

0

Putting t = s and writing t instead of s, we find that |x(t)| ≤ |eA(t)t x(0)| +

t 0

(t − τ )|eA(t)(t−τ ) |Q|x(τ )|dτ.

According to Corollary 1.12.2, |eA(τ )t | ≤ eα0 t Ψ (t) (τ, t ≥ 0), where

n−1

Ψ (t) =

tk B k .

k=0

Consequently, |x(t)|e−α0 t ≤ Ψ (t)|x(0)| +

t 0

Ψ (t − τ )(t − τ )Qe−α0 τ |x(τ )|dτ.

It follows from Lemma 1.7.1 that |x(t)|e−α0 t ≤ η(t),

(6.6)

where η(t) is the solution of the equation η(t) = Ψ (t)|x(0)| +

t 0

Ψ (t − τ )(t − τ )Qη(τ )dτ.

To solve the foregoing equation we apply the Laplace transformation. Let λ be the dual variable, η(λ), Ψ (λ) and L(λ) be the Laplace transforms of

58

3. Systems with Slowly Varying Coefficients

η(t), Ψ (t) and tΨ (t), respectively. Taking the Laplace transform of the latter equation and using the fact that the transform of a convolution is the product of the functions in the convolution, we obtain η(λ) = Ψ (λ)|x(0)| + L(λ)Qη(λ). By the inverse Laplace transform, η(t) =

1 2πi

Reλ>z0

etλ (I − L(λ)Q)−1 Ψ (λ)|x(0)|dλ.

Obviously,

n−1

Ψ (λ) = k=0

and

n−1

L(λ) = k=0

k! λk+1

(6.7)

Bk

(k + 1)! B k Q. λk+2

These relations yield λn+1 Ψ (λ) =

n−1

k!λn−k B k ,

k=0

and λn+1 L(λ) =

n−1

(k + 1)!λn−k−1 B k Q.

k=0

By (6.7) we conclude that η(t) =

1 2πi

Reλ>z0

etλ T −1 (λ)

n−1

k!λn−k B k dλ |x0 |

(6.8)

k=0

with the notation T (λ) = Iλn+1 −

n−1

(k + 1)!λn−k−1 B k Q.

k=0

Take into account that

T1 (λ) , det T (λ) where T1 (λ) is a polynomial matrix. By virtue of Corollary 3.6.4, z0 is a positive number. Now according to the residue theorem, (6.8) implies that T −1 (λ) =

η(t) ≤ exp(z0 t)[C0 + C1 t + .. + Cm tm−1 ]|x(0)|, where Ck (k = 0, ..., m) are constant matrices and m is the multiplicity of the root z0 . The assertion of the theorem now follows from the latter inequality and inequality (6.6). ✷

3.6 Matrix Lipschitz Conditions

59

3.6.3 Estimates for z0 Lemma 3.6.6 Let z1 be the extreme right-hand (positive) characteristic value of the matrix pencil λn I −

n−1

λn−k−1 Qk ,

k=0

where Qk are non-negative matrices. Then z1 ≤ z2 , where z2 is the extreme right-hand root of the algebraic equation λn =

n−1

λn−k−1 Qk .

(6.9)

k=0

Moreover, if

n−1

Qk ≤ 1,

(6.10)

k=0

then z1n ≤

n−1

Qk .

(6.11)

k=0

Proof:

Clearly,

n−1

1≤

z1−k−1 Qk .

k=0

On the other hand, dividing (6.9) by λn and taking into account that z2 is the root of (6.9), we arrive at the equality n−1

1=

z2−k−1 Qk .

k=0

The comparison of the two last relations gives the inequality z1 ≤ z2 . Now inequality (6.11) is due to Lemma 1.8.1. ✷ Corollary 3.6.7 Let z0 be the (positive) extreme right-hand root of the polynomial (6.3), and let n−1

(k + 1)! B k Q ≤ 1. k=0

Then z0 ≤ c(Q), where n−1

c(Q) =

(k + 1)! B k Q .

n+1

k=0

(6.12)

60

3. Systems with Slowly Varying Coefficients

The latter corollary and Corollary 3.6.2 imply Corollary 3.6.8 Under (6.2) and (6.12), for an > 0 and all t ≥ 0, let A(t) + I(c(Q) + ) be a Hurwitz matrix. Then system (6.1) is exponentially stable.

3.7 Lower Solution Estimates Theorem 3.7.1 If (6.2) holds and β0 = inf

min Reλk (A(t)) > −∞,

t≥0 k=1,...,n

then any solution x(t) of system (6.1) is subject to the estimate |x(t)| ≥ exp[(β0 − z0 )t]C −1 (t)|x(0)|, (t ≥ 0), where C(t) is a non-negative polynomial matrix, and z0 is the extreme righthand root of the polynomial (6.3). Proof: Fox a fixed T > 0, put in (6.1) τ = T − t, x(T − τ ) = y(τ ) and −A(T − τ ) = A1 (τ ). We get dy/dτ = A1 (τ )y(τ ) (0 ≤ τ ≤ T ). The eigenvalues of A1 (τ ) are −λ1 (A(T − τ )), ..., −λn (A(T − τ )). Hence, sup

max Reλk (A1 (τ )) ≤ −β0 .

0≤τ ≤T k=1,...,n

Moreover, the closed convex hull of the numbers −λ1 (A(T −τ )), ..., −λn (A(T − τ )) is the mirror image of co(A(t)) in the imaginary axis. Furhermore, the cofactor of the element −aij (T − τ ) − δij λ of A1 (τ ) − λI equals −mij (−λ, T − τ ), i.e. 1 ∂ (nij −k−1) µij (λ, T − t) sup | | ≤ bji (k) k!(nij − k − 1)! λ∈co(−A(T −τ )) ∂λ(nij −k−1) (nij = nij (t); 0 ≤ τ ≤ T ; i, j = 1, ..., n) (see Subsection 3.6.1). Therefore, Theorem 3.6.1 implies that |y(τ )| ≤ exp[(−β0 + z0 )τ ]C(τ )|y(0)| (0 ≤ τ ≤ T ). Hence, we obtain the desired inequality. ✷

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4. Linear Dissipative and Piecewise Constant Systems

62

where δ = maxk δk . The passage to the limit as m → ∞ and Lemma 2.3.1 give us the required estimate. ✷ Corollary 4.1.2 (Lozinskii). Let the relation I + A(t)δ C n − 1 = φ(t) (t ≥ 0) δ hold. Then for any solution x(t) of (1.1) the inequality (1.3) is true. limδ↓0

Indeed, I + A(t)δ C n = 1 + φ(t)δ + o(δ) (δ ↓ 0). Now, Theorem 4.1.1 implies inequality (1.3). We will say that A is a dissipative matrix if I + Aδ C n ≤ 1 for all sufficiently small δ > 0. System (1.1) is said to be dissipative if A(t) is a dissipative matrix for all t ≥ 0. Corollary 4.1.3 Let (1.1) be a dissipative system. Then it is stable. Theorem 4.1.4 Let there be a piecewise continuous function ψ(t), such that for all sufficiently small δ > 0 and h ∈ Cn , the relation (I + A(t)δ)h h Cn

Cn

≥ 1 + ψ(t)δ (t ≥ 0)

holds. Then any solution x(t) of (1.1) is subject to the inequality x(t)

Cn

≥ x(s)

C n exp[

t s

ψ(s1 )ds1 ] (t, s ≥ 0).

The proof of this theorem is analogous to the proof of Theorem 4.1.1. Let AR = (A + A∗ )/2 be the real Hermitian component of A; β(AR ) and α(AR ) be the smallest and largest eigenvalues of AR , respectively. Lemma 4.1.5 Let . be the Euclidean norm. Then for any constant n × nmatrix A and any h ∈ Cn , 1 + β(AR )δ + o(δ) ≤ Proof:

(I + Aδ)h ≤ 1 + α(AR )δ + o(δ) (δ ↓ 0). h

It is obvious that

(I + Aδ)h

2

= h

2

+ 2(AR h, h)δ + (Ah, Ah)δ 2 ≤ 1 + 2α(AR )δ + o(δ)

when h = 1. Here (., .) is he scalar product. But 1 + 2α(AR )δ = 1 + α(AR )δ + o(δ). This clearly forces the upper estimate. The proof of the lower estimate is left to the reader. ✷ Thus, if we take . C n = . -the Euclidean norm, then A is dissipative, if A + A∗ is negative definite. Theorems 4.1.1 and 4.1.4, and Lemma 4.1.5 imply

4.2 Linear Systems with Majorants and Minorants

63

Corollary 4.1.6 (Wazewski). For any solution x(t) of equation (1.1) and all t ≥ s ≥ 0, the estimates exp[

t s

β(AR (s1 ))ds1 ] ≤

x(t) ≤ exp[ x(s)

t s

α(AR (s1 ))ds1 ]

are true. Let E be a positive definite Hermitian matrix. Introduce the scalar product (., .)E by (h, g)E = (Eh, g) (h, g ∈ Cn ). Put h E = (Eh, h). Theorem 4.1.6 yields Corollary 4.1.7 Let EA∗ (t) + A(t)E ≤ 0 (t ≥ 0). Then equation (1.1) is stable.

4.2 Linear Systems with Majorants and Minorants Let A(t) (t ≥ 0) be a variable matrix with entries ajk (t) (j, k = 1, ..., n). Let there be a real piecewise continuous matrix M (t) = (mjk (t))nj,k=1 with non-negative off diagonal entries such that |I + A(t)δ| ≤ I + M (t)δ (t ≥ 0)

(2.1)

for all sufficiently small positive δ. Inequality (2.1) means that |ajk (t)| ≤ mjk (t) for j = k and |1 + δ akk (t)| ≤ 1 + δ mkk (t) (t ≥ 0; j, k = 1, ..., n). Lemma 4.2.1 Let condition (2.1) hold. Then any solution x(t) of system (1.1) satisfies the inequality |x(t)| ≤ z(t) (t ≥ 0), where z(t) is a solution of the equation z˙ = M (t)z (2.2) with the initial condition z(0) = |x(0)|. Proof:

Due to condition (2.1) it can be written ←

| 1≤k≤m

(m)

(I + A(tk )δk )u0 | ≤

← 1≤k≤m

(m)

(I + M (tk )δk )|u0 |.

But according to Lemma 2.3.1, one can assert that the sequences at the lefthand part and at the right-hand part of this inequality tend to the solutions x(t) and z(t) of equations (2.2) and (2.3), respectively. This finishes the proof. ✷ Corollary 4.2.2 Under condition (2.1) let M (t) ≡ M be a constant Hurwitz matrix. Then (1.1) is an exponentially stable system.

64

4. Linear Dissipative and Piecewise Constant Systems

Now we will obtain a lower solution estimate. Suppose that that there is a real matrix P (t) with non-negative off diagonal entries such that |I − A(t)δ| ≤ I + P (t)δ (t ≥ 0)

(2.3)

for all sufficiently small positive δ. Furthermore, let U (t) be the Cauchy operator of equation (1.1). Take into account the operator U −1 (t) is the Cauchy operator of the equation x˙ = −A(t)x.

(2.4)

By the previous lemma |U −1 (t)| ≤ Z(t), where Z(t) is the Cauchy operator of the equation z˙ = P (t)z. Now we easily obtain |U (t)h| ≥ Z −1 (t)|h| (h ∈ Cn , t ≥ 0). But Z −1 (t) is the Cauchy operator of the equation z˙ = −P (t)z.

(2.5)

We thus have derived Lemma 4.2.3 Under condition (2.3) any solution x(t) of equation (1.1) satisfies the inequality |x(t)| ≥ z− (t) (t ≥ 0), where z− (t) is the solution of equation (2.5) with the initial condition z− (0) = |x(0)|. Let (2.3) hold with a constant matrix P (t) ≡ P . Then any solution x(t) of (1.1) is subject to the inequality |x(t)| ≥ e−P t |x(0)| (t ≥ 0).

4.3 Systems with Piecewise Constant Matrices Let an n × n-matrix A(t) be piecewise constant: A(t) = Ak (tk ≤ t < tk+1 ; k = 0, 1, 2, ...)

(3.1)

with constant matrices Ak (k = 0, 1, 2, ...), and 0 = t0 < t1 < t2 .... Let U (t, s) be the evolution operator of (1.1). Since U (t, s)U (s, τ ) = U (t, τ ) (t, s, τ ≥ 0), we have U (tk+1 , tj ) = U (tk+1 , tk )... U (tj+1 , tj ) = U (tk+1 , tj ) = eδk Ak eδk−1 Ak−1 ... eδj

Aj

(k > j ≥ 0)

with δk = tk+1 − tk . Thanks to Corollary 1.5.3 this relation implies U (tk+1 , tj ) ≤ eδk Ak

eδk−1 Ak−1 ... eδj

Aj

≤

eδk α(Ak ) γ(Ak , δk )eδk−1 α(Ak−1 ) γ(Ak−1 , δk−1 )...eδj α(Aj ) γ(Aj , δj )

4.4 Perturbations of Systems with Piecewise Constant Matrices

with the notation

n−1

γ(A, t) = k=0

tk

g k (A) . (k!)3/2

65

(3.2)

Consequently, k

U (tk+1 , tj ) ≤ exp [

k

δm α(Am )] m=j

γ(Am , δm ).

(3.3)

m=j

We thus have derived Theorem 4.3.1 Let the conditions (3.1) and sup α(Ak ) + δk−1 ln γ(Ak , δk ) < 0

k=1,2,...

hold. Then system (1.1) is asymptotically stable.

4.4 Perturbations of Systems with Piecewise Constant Matrices Let an n × n-matrix A(t) be piecewise continuous. With notation (3.2), let ρ˜0 :=

sup

0≤τ ≤1, k=0,1,...,

α(A(k)) +

ln γ(A(k), τ ) < 0. τ

(4.1)

Theorem 4.4.1 Let the conditions (4.1) and sup

0≤t≤1; k=1,2,...

A(t + k) − A(k) <

1 ρ˜0

hold. Then system (1.1) is stable. Proof: Set A0 (t) = A(k) (k ≤ t < k + 1; k = 0, 1, 2, ...). Let U0 (t, s) be the evolution operator of the equation v˙ = A0 (t)v. Then U0 (t, τ ) = e(t−k)A(k) eA(k−1) . . . eA(j)(j−τ ) (k ≤ t < k+1; j−1 ≤ τ < j; k > j). (4.2) Corollary 1.5.3 gives us the inequality U0 (t, s) ds ≤ e(t−s)α(A(k)) γ(A(k), (t − s)) (k ≤ s ≤ t < k + 1). Then under (4.1) U0 (t, s) ds ≤ e−(t−s)ρ˜0 (k ≤ s ≤ t < k + 1). But according to (4.2), this inequality is true for all t ≥ s ≥ 0. Thus

66

4. Linear Dissipative and Piecewise Constant Systems t 0

U0 (t, s) ds ≤ ∞ 0

t 0

e−sρ˜0 ds =

e−(t−s)ρ˜0 ≤ 1 . ρ0

Now Lemma 2.2.6 implies the required result. ✷

4.5 General Second Order Vector Systems Let us consider the system x ¨ + 2A(t)x˙ + B(t)x = 0 (t > 0),

(5.1)

with real piecewise continuous n × n-matrices A(t) and B(t). As above AR = (A + A∗ )/2, AI = (A − A∗ )/2i denote the real and imaginary components of a matrix A, respectively. Let A0 be a symmetric matrix. Then we will write A0 ≥ 0 (A0 > 0) if it is positive definite (strongly positive definite). For two symmetric matrices A0 , B0 we write A0 ≥ B0 if A0 − B0 ≥ 0, and A0 > B0 if A0 − B0 > 0. Take the initial conditions x(t) = x0 , x(0) ˙ = x1 (x0 , x1 ∈ Rn )

(5.2)

Assume that there is a constant mA > 0, such that AR (t) ≥ mA I and 0 ≤ BR (t) ≤ mA (2AR (t) − mA I) (t ≥ 0),

(5.3)

where I is the unit matrix, and put T (t) := 2mA A(t) − B(t).

(5.4)

Theorem 4.5.1 Under condition (5.3), let p0 := sup TR (t) + TI (t) < 2m2A . t≥0

(5.5)

Then equation (5.1) is exponentially stable. Moreover, any solution x(t) of problem (5.1), (5.2) satisfies the estimate x(t) ≤ c2 eΛt ( x0 + x1 ) (t ≥ 0) where Λ := −mA + the initial vectors.

(5.6)

p0 − m2A < 0 and the constant c2 does not depend on

This theorem is proved in the next section.

67

4.5 General Second Order Vector Systems

Corollary 4.5.2 Let matrices A(t) and B(t) be symmetric, and there be a constant mA > 0, such that A(t) ≥ mA I and m2A I ≤ 2mA A(t) − B(t) < 2m2A I (t ≥ 0). Then equation (5.1) is exponentially stable. Moreover, estimate (5.6) is valid with p0 = sup 2mA A(t) − B(t) < 2m2A . t≥0

Example 4.5.3 Consider the equation x ¨ + 2a(t)x˙ + b(t)x = 0

(5.7)

with real positive piecewise continuous scalar functions a(t) and b(t). Let ma ≡ inft≥0 a(t) > 0. Then due to Corollary 4.5.2, equation (5.7) is exponentially stable, provided m2a ≤ 2ma a(t) − b(t) ≤ p˜0 (t ≥ 0) with p˜ < 2m2a . In particular, take 1 1 a(t) = 2 + sin2 (t), b(t) = 3 + cos (t). 2 2 Then ma = 2, and 1 m2a = 4 ≤ 2ma a(t) − b(t) = 8 + 2sin2 t − 3 − cos t < 2m2a = 8. 2 So equation (5.7) is exponentially stable. Example 4.5.4 Let A(t) = (ajk )nj,k=1 (ajk ≡ ajk (t)) be a real symmetric matrix, having the properties n

|ajk | ≤ 1/8 (t ≥ 0, j = 1, ..., n).

ajj (t) ≥ 9/8,

(5.8)

k=1,k=j

Then

n

inf ajj (t) − j

|ajk | ≥ 1 (t ≥ 0). k=1,k=j

Due to the well-known result from Section III.2.2 of the book (Marcus and Minc, 1964) (see also the Appendix A, Section A1) we conclude that A(t) ≥ I. So mA = 1. In addition, take B(t) = diag [bj (t)]nj , such that 0 ≤ 2ajj − 7/4 < bj ≤ 2ajj − 5/4 (bj = bj (t); j = 1, 2, ...). Then

(5.9)

68

4. Linear Dissipative and Piecewise Constant Systems n

2(ajj −

|ajk |) − bj ≥ k=1,k=j

2ajj − bj − 1/4 ≥ 5/4 − 1/4 = 1. Again use the well-known result from Section III.2.2 of the book (Marcus and Minc, 1964) (see also the Appendix A, Section A1). It gives us the inequality T (t) ≥ I. In addition, n

2(ajj +

|ajk |) − bj k=1,k=j

≤ 2ajj − bj + 1/4 < 7/4 + 1/4 = 2. Thus T (t) < 2I. So under (5.8) and (5.9) system (5.1) is exponentially stable due to Corollary 4.5.2.

4.6 Proof of Theorem 4.5.1 Put in (5.1)

x(t) = e−mt y(t) (m ≡ mA ).

Then we have the equation y¨ − 2(m − A(t))y˙ − C(t)y = 0 with

C(t) = 2A(t)m − Im2 − B(t) = T (t) − m2 I.

Reduce this equation to the system y˙ 1 = 2(m − A(t))y1 + C(t)y2 , y˙ 2 = y1 . Doing the scalar product, we get d (y1 , y1 ) = 4((m − A(t))y1 , y1 ) + 2(C(t)y2 , y1 ), dt d (y2 , y2 ) = 2(y2 , y1 ). dt Since the matrices are real, we can write out ((mI − A(t))y1 , y1 ) = ((mI − AR (t))y1 , y1 ) ≤ 0. So Thus

d(y1 , y1 )/dt ≤ 2 C(t)

y2

y1 .

(6.1)

4.7 Second Order Vector Systems with Differentiable Matrices

d y1 (t) /dt ≤ C(t)

y2 (t) , d y2 (t) /dt = y1 (t) .

69

(6.2)

Furthermore, C ≤ C R + C I = T R − m2 I + T I

(C = C(t), T = T (t)).

Under the hypothesis of the present theorem CR is positive. So T R − m2 I = Thus

sup

v∈Rn ,

v =1

(TR v, v) − m2 = TR − m2 .

C ≤ C R + C I = T R − m2 I + T I = TR + TI − m2A ≤ p0 − m2A (t ≥ 0; C = C(t), T = T (t)).

Put

b0 :=

p 0 − m2 .

Then (6.2) implies d y1 (t) /dt ≤ b20 y2 (t) , d y2 (t) /dt ≤ y1 (t) . Hence,

yk (t) ≤ zk (t) (k = 1, 2; t ≥ 0),

(6.3)

where (z1 (t), z2 (t)) is a solution of the coupled system z˙1 = b20 z2 , z˙2 = z1 (z1 (0) = x1 , z2 (0) = x0 ). Simple calculations show that |zk (t)| ≤ const (|z1 | + |z2 |)eb0 t (k = 1, 2) Hence (6.1) and (6.3) yield the required result. ✷

4.7 Second Order Vector Systems with Differentiable Matrices Consider in Rn , the system u ¨ + B(t)u˙ + C 2 (t)u = 0 (t > 0), where B(t) and C(t) are real n × n-matrices.

(7.1)

4. Linear Dissipative and Piecewise Constant Systems

70

Theorem 4.7.1 Let B(t) be a piecewise continuous matrix, and C(t) be a differentiable positive definite one. If, in addition, ˙ K(t) ≡ −C −1 (t)(C(t) + B(t)C(t))

(7.2)

is a dissipative matrix in the Euclidean norm for all t ≥ 0, then system (7.1) is stable. Moreover, any solution u of (7.1) satisfies the inequality u(t) Proof:

2

+ C −1 (t)u(t) ˙

2

≤ u(0)

2

+ C −1 (0)u(0) ˙

2

(t ≥ 0).

(7.3)

Clearly, system (7.1) is equivalent to the following one x˙ = −Bx − C 2 y, y˙ = x,

where B = B(t), C = C(t). Put x(t) = Cz(t). Then the latter system takes ˙ = −BCz − C 2 y, y˙ = Cz. Hence the form C z˙ + Cz z˙ = K(t)z − Cy, y˙ = Cz. Let A(t) be the matrix of the obtained system. Then A(t) + A∗ (t) =

K(t) + K ∗ (t) 0

0 0

.

Clearly, if K(t) is dissipative, then A(t) is dissipative, and the required result is due to Lemma 4.1.5. ✷ Example 4.7.2 Consider the equation u ¨ + b(t)u˙ + c2 (t)u = 0

(7.4)

with a positive piecewise continuous scalar-valued function b(t) and a positive differentiable one c(t). By the previous theorem, the inequality −1 b(t) + c(t)c ˙ (t) ≥ 0 (t ≥ 0)

ensures the stability of equation (7.4). Moreover, the inequality 2 2 u2 (t) + (c−1 (t)u(t)) ˙ ≤ u2 (0) + (c−1 (0)u(0)) ˙ (t ≥ 0)

holds for any solution u(t) of equation (7.4).

5. Nonlinear Systems with Autonomous Linear Parts

5.1 Statement of the Main Result Denote

Ω(r) = {h ∈ Cn : h ≤ r}

and consider the equation x˙ = Ax + F (x, t) (t ≥ 0, x = x(t)),

(1.1)

where A is a constant Hurwitz n × n-matrix and F maps Ω(r) × [0, ∞) into Cn with the property F (h, t) ≤ ν h for all h ∈ Ω(r) and t ≥ 0.

(1.2)

Here ν = ν(r) ≡ const > 0. Put Γ˜ :=

∞ 0

eAt dt and χ ˜ := max eAt . t≥0

Lemma 5.1.1 Under condition (1.2), let the inequality ν Γ˜ < 1 be valid. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, any vector x0 satisfying the condition χ ˜ x0 < r, 1 − ν Γ˜ belongs to a region of attraction of the zero solution and a solution x(t) of (1.1) with (1.3) x(0) = x0 subordinates the estimate x(t) ≤

χ ˜ x0 (t ≥ 0). 1 − ν Γ˜

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 71–84, 2005. © Springer-Verlag Berlin Heidelberg 2005

72

5. Nonlinear Systems with Autonomous Linear Parts

This result immediately follows from Theorem 7.1.1 proved below. Denote n−1

Γ (A) = j=0

g j (A) √ |α(A)|j+1 j!

and

n−1

χ(A) = max exp[α(A)t] t≥0

j=0

g j (A)tj √ . j!

Recall that g(A) is defined in Section 1.5. Corollary 1.5.3 yields χ ˜ ≤ χ(A); Γ˜ ≤ Γ (A). Now Lemma 5.1.1 implies Corollary 5.1.2 Under condition (1.2), let the inequality νΓ (A) < 1

(1.4)

hold. Then the zero solution to equation (1.1) is exponentially stable. Moreover, any vector x0 satisfying the condition χ(A) x0 < r, 1 − νΓ (A)

(1.5)

belongs to a region of attraction of the zero solution. Besides, a solution x(t) of problem (1.1), (1.3) subordinates the estimate x(t) ≤

χ(A) x0 (t ≥ 0). 1 − νΓ (A)

(1.6)

Introduce the algebraic equation zn = ν

n−1 j=0

g j (A) n−j−1 √ z j!

(1.7)

and denote by z(ν, A) the extreme right-hand (unique positive and simple) zero of this equation. Theorem 5.1.3 Under condition (1.2), let the matrix A+z(ν, A)I be a Hurwitz one. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, inequality (1.4) holds and any vector x0 satisfying condition (1.5) belongs to a region of attraction of the zero solution, and estimate (1.6) is valid. The proof of this theorem is presented in the next section. Put n−1 j g (A) √ . p0 = ν j! j=0

5.2 Proof of Theorem 5.1.3

73

Due to Lemma 1.8.1, z(ν, A) ≤ δ(ν, A), where δ(ν, A) =

1/n

p0 p0

if p0 ≤ 1 . if p0 > 1

Now Theorem 5.1.3 implies Corollary 5.1.4 Under condition (1.2), let the matrix A + δ(ν, A)I be a Hurwitz one. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, inequality (1.4) holds and any vector x0 satisfying condition (1.5) belongs to a region of attraction of the zero solution, and estimate (1.6) is valid. We will say that (1.1) is a quasilinear equation if lim

h→0

F (h, t) =0 h

uniformly in t. Recall the following famous result Theorem 5.1.5 (Lyapunov) Let (1.1) be a quasilinear equation. Then if A is a Hurwitz matrix, the zero solution of (1.1) is asymptotically stable. Conversely, if A has an eigenvalue in the interior of the right half-plane, then the zero solution of (1.1) is unstable. For the proof see (Daleckii and Krein, 1974, Section 7.2).

5.2 Proof of Theorem 5.1.3 We need the following Lemma 5.2.1 Let z(ν, A) be the extreme right-hand zero of the algebraic equation (1.7). Then the condition α(A) + z(ν, A) < 0 implies inequality (1.4). Proof:

The hypothesis of the lemma entails the relation |α(A)| > z(ν, A).

(2.1)

Dividing (1.7) by z n+1 , and taking into account that z(ν, A) is its root, we arrive at the equality n−1 g j (A) √ . 1=ν j+1 z (ν, A) j! j=0

74

5. Nonlinear Systems with Autonomous Linear Parts

But according to (2.1) n−1

νΓ (A) = ν j=0 n−1

ν j=0

g j (A) √ < |α(A)|j+1 j!

g j (A) z j+1 (ν, A)

√

j!

= 1.

This proves the stated result. ✷ The assertion of Theorem 5.1.3 immediately follows from Corollary 5.1.2 and Lemma 5.2.1.

5.3 Stability Conditions in Terms of Determinants 5.3.1 Statement of the Result Theorem 5.3.1 Let matrix A be real and Hurwitzian and the conditions (1.2), and 1 b(A) := sup (3.1) (A − Iiy)−1 < ν |y|≤2 A hold. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, with the notations ψ(A) := [

1 2π

and η(A) :=

∞ −∞

(A − Iiy)−1 2 dy ]1/2

2( A + ν)ψ(A) 1 − νb(A)

any vector x0 ∈ Cn , satisfying the inequality x0 η(A) < r

(3.2)

belongs to the region of attraction of the zero solution to equation (1.1). Besides a solution x of problem (1.1), (1.3) satisfies the inequalities sup x(t) ≤ x0 η(A) t≥0

and x

L2

≡[

∞ 0

x(t) 2 dt]1/2 ≤

x0 ψ(A) . 1 − νb(A)

(3.3)

(3.4)

5.3 Stability Conditions in Terms of Determinants

75

The proof of this theorem is given below in this section. Put φ(z) :=

N n−1 (A − z I) (z ∈ C). (n − 1)(n−1)/2 |det (A − z I)|

Thanks to Lemma 1.5.7, the relation (A − zI)−1 ≤ φ(z) holds, provided the matrix A − z I is invertible. Recall that N () is introduced in Section 1.1. Clearly, the integral ∞

1 ˜ ψ(A) := [ 2π

−∞

φ2 (iy)dy]1/2

˜ ˜ converges and ψ(A) ≤ ψ(A). Below we will give simple estimates for ψ(A). The previous theorem implies Corollary 5.3.2 Let matrix A be Hurwitzian and the conditions (1.2) and ˜b(A) :=

sup

|y|≤2 A

φ(iy) <

1 ν

hold. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, with the notation ˜ 2( A + ν)ψ(A) 1 − ν˜b(A)

η˜(A) :=

any vector x0 ∈ Cn , satisfying the inequality x0 η˜(A) < r belong to the region of attraction of the zero solution to equation (1.1). Besides, any solution x of problem (1.1), (1.3) satisfies the inequalities supt≥0 x(t) ≤ x0 η˜(A) and x

L2

≤

˜ x0 ψ(A) . 1 − ν˜b(A)

It is simple to check that if the spectrum is real, then |det (A − iyI)| ≥ |α − iy|n = (α2 + y 2 )n/2 , where α = α(A) = max Re λk (A). Thus, ψ 2 (A) ≤

γn2 2π

∞ −∞

N 2n−2 (A − iIy) dy, (α2 + y 2 )n

76

5. Nonlinear Systems with Autonomous Linear Parts

where

1 . (n − 1)(n−1)/2

γn = But

N (A − Iiy) ≤

Therefore

∞

γ2 ψ˜2 (A) ≤ n π

0

√

n|y| + N (A).

√ ( ny + N (A))2n−2 dy . (α2 + y 2 )2n

This integral is simple calculated. 5.3.2 Proof of Theorem 5.3.1 Lemma 5.3.3 The relation sup (A − Iiy)−1 =

y∈R1

sup

|y|≤2 A

(A − Iiy)−1 = b(A)

is valid. Proof:

Clearly, sup (A − Iiy)−1 ≥ A−1 ≥

y∈R1

But

(A − Iiy)−1 ≤

1 1 ≤ |y| − A A

1 . A

(|y| ≥ 2 A ).

So the supremum is attained on [−2 A| , 2 A| ]. As claimed. ✷ Furthermore, recall that the space L2 = L2 (R+ , Cn ) is introduced in Section 1.13. Due to the Laplace transform, Parseval equality and previous lemma, we get ∞

t

0

0

eA(t−s) f (s)ds 2 dt = ≤ b2 (A)

∞

1 2π

−∞

1 2π

∞ −∞

(A − Iiy)−1 f˜(iy) 2 dy

f˜(iy) 2 dy = b2 (A) f

2 L2 ,

(3.5)

where f˜ is the Laplace transform to f ∈ L2 . Moreover, ∞ 0

eA(s) 2 ds =

1 2π

∞ −∞

(A − Iiy)−1 2 dy = ψ 2 (A).

(3.6)

First, let condition (1.2) hold with r = ∞. Then for any h ∈ L2 , (1.2) implies F (h, t)

L2

≤ν h

L2 .

5.4 Global Stability Under Matrix Conditions

77

Rewrite (1.1) as t

x(t) = eAt x0 +

0

eA(t−s) F (x(s), s)ds.

(3.7)

Due to (3.6) and (3.5) x

L2

≤ ψ(A) x0 + νb(A) x

x

L2

≤ ψ(A) x0 (1 − νb(A))−1 .

L2 .

Thus (3.1) yields

Moreover, from (1.1) it follows that x˙

L2

≤ A

x

L2

+ν x

L2 .

Let us use the inequality v 2 (t) ≤ 2[

∞ t

v 2 (s)ds]1/2 [

∞ t

2 (v(s)) ˙ ds]1/2 (v, v˙ ∈ L2 [0, ∞))

(see Section 1.13). Hence, we get sup x(t) ≤ t≥0

2( A + ν) x

L2

≤

2( A + ν)ψ(A) x0 (1 − νb(A))−1 = η(A) x0 . So in the case r = ∞ the theorem is proved. Now let r < ∞. Due to (3.1), for a small enough t0 > 0 a solution w of problem (1.1), (1.2) lies in Ω(r) for all t ≤ t0 . Applying our above reasonings, we get the inequality sup

0≤t≤t0

x(t) ≤ η(A) x0 .

But under (3.2) this inequality can be extended from [0, t0 ] to [0, ∞). This proves the theorem. ✷

5.4 Global Stability Under Matrix Conditions Consider system (1.1) in the form dxj = dt

n

ajk xk + Fj (x, t) k=1

78

5. Nonlinear Systems with Autonomous Linear Parts

(x = (xk (t)) ∈ Cn ; t > 0; j = 1, ..., n) under the conditions

n

(4.1)

qjk |hk | (h = (hk ) ∈ Cn ; t ≥ 0; j = 1, ..., n)

|Fj (h, t)| ≤ k=1

where qjk (j, k = 1, ..., n) are nonnegative constants. We will write these conditions in the form |F (x, t)| ≤ Q|h| (h ∈ Cn ; t ≥ 0) where

(4.2)

Q = (qjk )nj,k=1

is a constant matrix. Inequality (4.2) permits us to obtain estimates for each coordinate of a solution and to take into account information about each element of the matrix more completely than under the customary condition (1.2), although applications of condition (4.2) require more calculations than condition (1.2). Furthermore, let mij (λ) be the cofactor of the element δij λ − aij of the matrix λI − A. Recall that δij = 0, j = i and δjj = 1. Assume that after division by the coinciding zeros we obtain the equality mij (λ) µij (λ) = (i, j = 1, ...n), det(Iλ − A) dij (λ) where dij (λ) and µij (λ, t) are polynomials with respect to λ, and the degree of dij (λ) is equal to nij . Let co(A) be the convex hull of all eigenvalues of matrix A. Put 1 (n −k−1) sup |µ ij (λ)| (4.3) bji (k) = k!(nij − 1 − k)! λ∈co(A) ij (k = 0, ..., n − 1). Since (−m)! = 0 for any natural m, the equalities bij (k) = 0 for k = nij , ..., n − 1 are valid if nij < n − 1. Finally, denote Bk = (bij (k))nj,i=1 . Theorem 5.4.1 If F satisfies condition (4.2), then any solution x(t) of system (4.1) subordinates the inequality |x(t)| ≤ exp[(α(A) + z0 )t]C(t)|x(0)| (t ≥ 0), where C(t) is a non-negative polynomial matrix and z0 is the extreme righthand (positive) root of the polynomial det(Iλn −

n−1 k=0

Recall that I is the unit matrix.

λn−k−1 Bk Q).

(4.4)

5.5 Proof of Theorem 5.4.1

79

Corollary 5.4.2 Under condition (4.2), let the matrix A + z0 I be a Hurwitz one. Then system (4.1) is globally exponentially stable.

5.5 Proof of Theorem 5.4.1 Condition (4.2) implies |x(t)| ≤ |eAt x(0)| +

t 0

|eA(t−τ ) |Q|x(τ )|dτ.

According to Corollary 1.12.2, |eAt | ≤ eα0 t Ψ (t) (τ, t ≥ 0), where

n−1

Ψ (t) =

tk B k .

k=0

Consequently, |x(t)|e−α0 t ≤ Ψ (t)|x(0)| +

t 0

Ψ (t − τ )Qe−α0 τ |x(τ )|dτ.

It follows from Lemma 1.7.1 that |x(t)|e−α0 t ≤ η(t),

(5.1)

where η(t) is the solution of the equation η(t) = Ψ (t)|x(0)| +

t 0

Ψ (t − τ )Qη(τ )dτ.

To solve that equation we apply the Laplace transformation. Let λ be the dual variable, η(λ) and Ψ (λ) be the Laplace transforms of η(t) and Ψ (t). Using the fact that the transform of a convolution is the product of the functions in the convolution, we obtain η(λ) = Ψ (λ)|x(0)| + Ψ (λ)Qη(λ). By the inverse Laplace transform, η(t) =

1 2πi

Reλ>z0

etλ (I − Ψ (λ)Q)−1 Ψ (λ)|x(0)|dλ.

Obviously,

n−1

Ψ (λ) = k=0

k! Bk . λk+1

(5.2)

80

5. Nonlinear Systems with Autonomous Linear Parts

By (5.2) we conclude that η(t) =

1 2πi

Reλ>z0

etλ T −1 (λ)

n−1

k!λn−k Bk dλ |x0 |

(5.3)

k=0

with the notation T (λ) = Iλn −

n−1

k!λn−k−1 Bk Q.

k=0

Take into account that T −1 (λ) =

T1 (λ) , det T (λ)

where T1 (λ) is a polynomial matrix. Clearly, z0 is a positive number. Now according to the residue theorem, the assertion of the theorem follows from inequality (5.1). ✷

5.6 Region of Attraction of One-Contour Systems Let us consider the real scalar equation P (D)y = F (y, t) (t ≥ 0, D ≡ d/dt),

(6.1)

where y = y(t) and P (λ) = λn + a1 λn−1 + ... + an is a real Hurwitz polynomial. Besides, F (y, t) is a real continuous scalarvalued function defined on [−r, r] × [0, ∞) with a positive number r ≤ ∞. It is assumed that there is a positive constant q < ∞, such that |F (x, t)| ≤ q|x| (x ∈ [−r, r], t ≥ 0). Let λ1 , ..., λn be the roots of polynomial P (λ). Denote α = max Re λk , 1≤k≤n

Λ = max |λk |, 1≤k≤n

and

(6.2)

5.6 Region of Attraction of One-Contour Systems n−1

ζj = k=0

81

j!Λj−k (n − k − 1)n−k−1 |eα|n−k−1 (j − k)!k!(n − k − 1)! (j = 0, 1, 2, ...).

Besides, for the initial values y (j) (0) = yj (j = 0, ..., n − 1) set

j

n−1

wi (P, 0) ≡

(6.3)

ak ζj−k+i (i = 0, ...., n − 1).

|yn−j−1 | j=0

k=0

Theorem 5.6.1 Let the conditions (6.2) and q < |α|n

(6.4)

be fulfilled. Then the zero solution of (6.1) is asymptotically stable, and all initial values satisfying the inequality M0 ≡

|α|n w0 (P, 0)
(6.5)

belong to the region of attraction of the zero solution of (6.1). Besides, any solution y(t) of problem (6.1), (6.3) subordinates the estimate sup |y(t)| ≤ M0 (j = 0, 1, ..., n − 1). t≥0

The proof of this Let  −a1  1 A=  . 0

(6.6)

theorem is presented in the next section. ... ... ... ...

−an−1 0 . 1

 −an 0  . .  0

be the matrix of the linear part of equation (6.1). Assume that A + q 1/n I is a Hurwitz matrix. Here I is the unit matrix. Then clearly α + q 1/n < 0. That is, (6.4) holds. Now Theorem 5.6.1 implies the following result Corollary 5.6.2 Let condition (6.2) hold and A+q 1/n I be a Hurwitz matrix. Then the zero solution of (6.1) is asymptotically stable. Moreover, inequality (6.4) is valid, and all initial values satisfying (6.5) belong to the region of attraction of the zero solution of (6.1). In addition, estimate (6.6) is true.

82

5. Nonlinear Systems with Autonomous Linear Parts

5.7 Proof of Theorem 5.6.1 K(t) =

1 2πi

C

etλ dλ , P (λ)

where C is a smooth contour encompassing all the roots of P (λ). Lemma 5.7.1 For each natural j = 0, 1, 2, ..., the inequality |K (j) (t)| ≤ eαt ηj (t) (t ≥ 0) is true, where

n−1

ηj (t) = k=0

Proof:

j!Λj−k tn−k−1 . (j − k)!(n − k − 1)!k!

We can write down K (j) (t) =

1 2πi

C

λj etλ dλ, − λk )

n k=1 (λ

According to Lemma 1.11.1, |K (j) (t)| ≤

dn−1 1 sup | n−1 eλt λj |, (n − 1)! λ∈co(P ) dλ

where co(P ) is the closed convex hull of the numbers λ1 , ..., λn . This clearly forces |K (j) (t)| ≤

n−1 k=0

eαt

n−1 k=0

Since

j!tn−k−1 sup |eλt λj−k | ≤ k!(n − k − 1)!(j − k)! λ∈co(P )

j!tn−k−1 sup |λj−k |. k!(n − k − 1)!(j − k)! λ∈co(P ) sup |λ| = Λ,

λ∈co(P )

this is precisely the assertion of the lemma. ✷ Corollary 5.7.2 For each j = 0, 1, 2, ..., the inequality max |K (j) (t)| ≤ ζj t≥0

is true.

5.7 Proof of Theorem 5.6.1

Indeed,

83

max eαt tk = k k |α|−k e−k (k = 1, 2, ...). t≥0

Hence, n−1

max eαt ηj (t) ≤ t≥0

j!Λj−k (n − k − 1)n−k−1 = ζj . |eα|n−k−1 (j − k)!k!(n − k − 1)!

k=0

(7.1)

Thus, the result follows from Lemma 5.7.1. Corollary 5.7.3 For each j = 0, ..., n − 1 the following inequality is valid: ∞ 0

|K (j) (t)|dt ≤

(Λ + |α|)j . |α|n

In fact, due to Lemma 5.7.1 ∞ 0

|K

(j)

n−1

(t)|dt ≤ k=0 j k=0

j!Λj−k k!(n − k − 1)!(j − k)!

∞ 0

eαt tn−k−1 dt =

j!Λj−k = |α|−n (Λ + |α|)j , k!|α|n−k (j − k)!

as claimed. Lemma 5.7.4 For a solution z(t) of the equation P (D)z = 0,

(7.2)

z (j) (0) = yj (j = 0, ..., n − 1)

(7.3)

with the initial conditions

the following estimate is true: |z (m) (t)| ≤ eαt

n−1

i

|yn−i−1 |

Proof: have

as ηi−s+m (t) (t ≥ 0, m = 0, 1, ...). s=0

i=0

By the Laplace transformation (D´ oetch, 1961, formula (14.6)), we n−1

z(t) =

z (j) (0)

j=0

n−j

ak−1 K (n−k−j) (t).

k=1

This implies z (m) (t) =

n−1 j=0

z (j) (0)

n−j k=1

ak−1 K (n−k−j+m) (t) =

5. Nonlinear Systems with Autonomous Linear Parts

84 n−1

z

(n−i−1)

i+1

ak−1 K

(0)

i=0

(i+1−k+m)

n−1

(t) =

z

(n−i−1)

i s=0

i=0

k=1

as K (i−s+m) (t).

(0)

Now the required result follows from Lemma 5.7.1. ✷ Lemma 5.7.4 and (7.1) yield Corollary 5.7.5 For a solution z(t) of problem (7.2), (7.3), the estimate maxt≥0 |z (m) (t)| ≤ wm (P, 0) (m = 0, 1, 2, ...) is true. Proof of Theorem 5.6.1: The equation (6.1) is equivalent to the following one: y(t) = z(t) +

t

0

K(t − s)F (y, s)ds

(7.4)

where z(t) is a solution of the linear problem (7.2), (7.3). Since (6.5) implies that |y(0)| < r, there is a sufficiently small t0 such that |y(t)| ≤ r for 0 ≤ t ≤ t0 . We conclude from (7.4) that |y(t)| ≤ |z(t)| +

t 0

|K(t − s)|q|y(s)|ds, 0 ≤ t ≤ t0 .

Hence, max |y(t)| ≤ maxt≥0 |z(t)| + q max |y(t)| t≤t0

t≤t0

∞ 0

|K(s)|ds.

Due to Corollaries 5.7.5 and 5.7.3 we thus get max |y(t)| ≤ max |z(t)| + q|α|−n ≤ w0 (P, 0) + q|α|−n max |y(t)|. t≤t0

t≥0

t≤t0

But q|α|−n < 1. Therefore, max |y(t)| ≤ θ0 ≡ (1 − q|α|−n )−1 w0 (P, 0) (0 ≤ t ≤ t0 ).

0≤t≤t0

(7.5)

According to (6.5) we can extend this estimate to all t ≥ 0. So we have obtained estimate (6.6). Thus, the stability of the zero solution has been proved. The asymptotic stability can be obtained by a small perturbation of equation (6.1). The detailed verification of this fact is left to the reader. ✷

6. The Aizerman Problem

6.1 Absolute Stability Consider the equation y˙ = Ay + b f (s, t) (s = cy, t ≥ 0),

(1.1)

where A is a real constant Hurwitz n × n-matrix, b is a real column, c is a real row, f maps R1 × [0, ∞) into R1 with the property |f (s, t)| ≤ q|s| for all s ∈ R1 and t ≥ 0.

(1.2)

Definition 6.1.1 We will say that the zero solution of system (1.1) is absolutely exponentially stable in the class of nonlinearities (1.2) if there are constants M, > 0 which do not depend on a concrete form of f (but which depend on q) such that |cy(t)| ≤ M exp(− t) y(0)

(t ≥ 0)

for any solution y(t) of (1.1). Introduce the linear equation y˙ = Ay + q1 bcy

(1.3)

In 1949 M. A. Aizerman conjectured the following hypothesis: under the condition f (s, t) ≡ f (s), for the absolute stability of the zero solution of (1.1) in the class of nonlinearities satisfying the condition 0 ≤ f (s)/s ≤ q for all s ∈ R1 , s = 0 and t ≥ 0 it is necessary and sufficient that (1.3) be asymptotically stable for any q1 ∈ [0, q] (Aizerman, 1949). This hypothesis caused great interest among the specialists. Counterexamples were set up that demonstrated it was not, in general, true (see (Naredra and Taylor, 1973), (Reissig et al., 1974), (Willems, 1971), and references therein). Therefore, the following problem arose: to find the class of systems that satisfy Aizerman’s hypothesis. To formulate the relevant theorem, let us introduce the transfer function W (λ) of the linear part of system (1.1): , M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 85–92, 2005. © Springer-Verlag Berlin Heidelberg 2005

6. The Aizerman Problem

86

W (λ) = c(λI − A)−1 b = Here

L(λ) (λ ∈ C). P (λ)

P (λ) = λn + a1 λn−1 + ... + an (a1 , ..., an > 0)

and

m

L(λ) =

bk λm−k (m < n)

k=0

are polynomials. It is assumed that P (λ) and L(λ) have no coinciding roots. In addition, let ∞ 1 K(t) := exp[iωt]W (iω) dω 2π −∞ be the impulse (Green) function. Theorem 6.1.2 Let the condition K(t) ≥ 0 for all t ≥ 0

(1.4)

be fulfilled. Then for the absolute exponential stability of the zero solution of (1.1) in the class of nonlinearities (1.2), it is necessary and sufficient that the polynomial P (λ) − qL(λ) be Hurwitzian. That theorem is proved in the next section Clearly, Theorem 6.1.2 singles out one of the classes of linear parts of systems that satisfy the conjecture of M.A. Aizerman. In the next section we will prove Lemma 6.1.3 Let condition (1.4) be fulfilled. Then the polynomial P (λ) − qL(λ) is a Hurwitz one, if and only if qW (0) < 1.

(1.5)

Now Theorem 6.1.2 implies Corollary 6.1.4 Let condition (1.4) be fulfilled. Then for the absolute exponential stability of the zero solution of (1.1) in the class of nonlinearities (1.2), it is necessary and sufficient that condition (1.5) holds. Furthermore, immediately from Lemma 1.11.2 it follows Lemma 6.1.5 Let all the roots of polynomial P (λ) belong to a real segment [a, b] (b < 0) and L(k) (z) ≥ 0 (k = 0, ..., deg L(λ); a ≤ z ≤ b). Then K(t) ≥ 0 (t ≥ 0). This result, Theorem 6.1.2 and Lemma 6.1.3 imply

(1.6)

6.2 Proofs

87

Corollary 6.1.6 Let all the roots of P (λ) belong to a real segment [a, b] and condition (1.6) hold. Then for the absolute exponential stability of the zero solution of (1.1) in the class of nonlinearities (1.2) it is necessary and sufficient that the polynomial P (λ) − qL(λ) be Hurwitzian. That is, an = P (0) > qL(0) = qbm . In particular, let

L(λ) ≡ 1

and all the roots of P (λ) be real. Then for the absolute exponential stability of the zero solution of (1.1) in the class of nonlinearities (1.2) it is necessary and sufficient that an > q. In the next section we also prove the following result Lemma 6.1.7 Let a real Hurwitz polynomial P3 (λ) = λ3 + a1 λ2 + a2 λ + a3 have a pair of complex conjugate roots: −γ±iω, and a real root −z0 (z0 , γ, ω > 0). Then the function K3 (t) ≡

1 2π

∞ −∞

exp(ity)P3−1 (iy)dy

is positive, provided γ > z0 . Since the Laplace transform of a convolution is a product of the Laplace transforms, we can extend this result to an arbitrary n > 3. The previous lemma, and Lemma 6.1.3 imply Corollary 6.1.8 Let a real Hurwitz polynomial P3 (λ) = λ3 +a1 λ2 +a2 λ+a3 have a pair of complex conjugate roots: −γ ± iω, and a real root −z0 > −γ (z0 , γ, ω > 0). Then for the absolute exponential stability of the zero solution of (1.1) with W (λ) = P3−1 (λ) in the class of nonlinearities (1.2) it is necessary and sufficient that a3 > q.

6.2 Proofs Proof of Theorem 6.1.2: First, we will prove the sufficiency of the conditions. Note that K(t) = c exp[At]b. Furthermore, equation (1.1) is equivalent to following one: y(t) = exp[At]y(0) +

t 0

exp[A(t − τ )]bf (s(τ ), τ )dτ (s(t) = cy(t)).

Multiplying this equation by c, we have

6. The Aizerman Problem

88

s(t) = z(t) + with

t 0

K(t − τ )f (s(τ ), τ )dτ

z(t) = c exp[At]y(0).

Taking into account condition (1.2), we arrive at the inequality |s(t)| ≤ |z(t)| +

t 0

K(t − τ )q|s(τ )|dτ

From Lemma 1.7.1 it follows that |s(t)| ≤ η(t),

(2.1)

where η is the solution of the equation η(t) = |z(t)| +

t 0

K(t − τ )qη(τ )dτ.

We will solve this equation by the Laplace transformation. Since the transform of the convolution is equal to the product of the transforms, after straightforward calculations, we get the equation η(λ) = g(λ) + P −1 (λ)L(λ)qη(λ), where η(λ) and g(λ) are the Laplace transforms of η(t) and of |z(t)|, respectively, λ is the dual variable. Thus we can write down η(t) =

1 2πi

i∞+c0 −i∞+c0

It is easy to see that

exp[λt](P (λ) − L(λ)q)−1 P (λ)g(λ)dλ (c0 = const).

|z(t)| ≤ exp[α(A)t]G(t),

where G(t) is a polynomial. Recall that α(A) is the real part of the extreme right-hand eigenvalue of A. Thus, thanks the residue theorem η(t) ≤ exp(m1 t)G1 (t) for all t ≥ 0, where G1 (t) is a polynomial, and m1 = max{α(A), β} . Here β is the real part of the extreme right-hand zero of the polynomial det(P (λ) − L(λ)q). This and (2.1) prove the sufficiency of conditions. To prove the necessity of the conditions put f (s, t) = qs. Then (1.1) take the form y˙ = Ay + bqs (t ≥ 0, s = cy). Hence,

λy(λ) − y(0) = Ay + bqcy(λ),

6.2 Proofs

89

where y(λ) is the Laplace transform of y(t). Consequently, cy(λ) = −c(A − Iλ)−1 (y(0) + bqcy(λ)) = h(λ) + W (λ)bqcy(λ) where h(λ) = −c(A − Iλ)−1 y(0). Therefore, cy(λ) = h(λ) + P −1 (λ)L(λ)qcy(λ) and

cy(λ) = (P (λ) − L(λ)q)−1 P (λ)h(λ).

Hence the required result follows. ✷ Proof of Lemma 6.1.3: Since P (λ) is a Hurwitz polynomial, according to the Rauscher theorem we need only to check that |P (iω)| > q|L(iω)| or q|W (iω)| < 1 (ω ∈ R), but due to (1.4) this inequality follows from the equality max |W (is)| = W (0) > 0, s∈R

since |W (is)| = | = ✷

0

∞ 0 ∞

e−its K(t)dt| ≤

∞ 0

|K(t)|dt

K(t)dt = W (0) (s ∈ R).

Proof of Lemma 6.1.7: Taking into account that K3 (t) is a solution of the equation P3 (D)x(t) = 0, ¨ = 1 (D¨ oetsch, 1961, with the initial condition K3 (0) = K˙ 3 (0) = 0, K(0) Section 12), we get K3 (t) = c[e−z0 t − e−γt (cos(ωt) + b sin(ωt))] (c = const > 0) with

b = (γ − z0 )ω −1 > 0.

Consider the equation K3 (t) = 0. Multiplying it by eγt and substituting s = ωt, we have the equation ebs = cos s + b sin s. By virtue of the Taylor series that equation can be rewritten as ebs = 1 + bs + g(s) = cos s + b sin s (g(s) > 0). Since

|cos s| ≤ 1, |sin s| ≤ s (s ≥ 0),

one can assert that this equation has no zeros for s > 0. Therefore, K3 (t) has no zeros. ✷

90

6. The Aizerman Problem

6.3 Systems with Matrix Conditions Now we consider the system y˙ = Ay + Bf (s, t), (t ≥ 0, s = Cy)

(3.1)

where A is a real Hurwitz n × n-matrix, B : Rm → Rn and C : Rn → Rm are linear operators also, and m < n. Besides, F (s, t) maps Rm × [0, ∞) into Rm with the property |F (x, t)| ≤ Q|x| (x ∈ Rm , t ≥ 0).

(3.2)

Here Q is a non-negative m × m-matrix ; |F (x, t)| and |x| are vectors whose coordinates are the moduli of the vectors F (x, t) and x, respectively. Let W (λ) be the transfer matrix of the linear part of the system from the input F to the output y: W (λ) = C(λI − A)−1 B. It is simple to check that W (λ) = P −1 (λ)L(λ), where L(λ) and P (λ) are matrices whose elements are polynomials in λ. So each element Wjk (λ) of W (λ) (j, k = 1, ..., m) is a rational quotient function and the degree of the numerator of Wjk (λ) is less than the degree of the denominator. It is assumed that W (λ) is non-degenerate, i.e. the poles of this matrix are not canceled by the zeros. In particular, (3.1) can take the form P (D)y = L(D)F (y, t) (t ≥ 0; D = d/dt). Let K(t) =

1 2π

i∞ −i∞

exp[iωt]W (iω)dω

(3.3)

be the impulse (Green) matrix of the linear part of the system. We will say that the zero solution of the system (3.1) is absolutely exponentially stable in the class of nonlinearities (3.2) if there are scalar constants N, > 0 independent of the form of F , such that Cy(t)

Rm

≤ N exp(− t) y(0)

Rn

and any solution y(t) of system (3.1). Here . norms in Rn and Rm , respectively.

for all t ≥ 0

Rn

and .

Rm

are arbitrary

Theorem 6.3.1 Let K(t) ≥ 0 for all t ≥ 0. Then for the absolute exponential stability of the zero solution of equation (3.1) in the class of nonlinearities (3.2) it is necessary and sufficient that the polynomial det(P (λ) − L(λ)Q) be a Hurwitz one.

6.4 Examples

91

The proof of this theorem is similar to the proof of Theorem 6.1.2. Clearly, Theorem 6.3.1 generalizes Theorem 6.1.2. Furthermore, each element Wjk (λ) (j, k = 1, ..., m) of the transfer function W (λ) has the form mjk (λ) Wjk (λ) = , (3.4) pjk (λ) where pjk (λ) and mjk (λ) are polynomials in λ. In addition pjj are monic polynomials. That is their coefficients of the largest powers are equal to one. Lemma 1.11.2 and Theorem 6.3.1 imply Corollary 6.3.2 Under (3.4), for all j, k = 1, ..., m, let all the roots of pjk (λ) belong to a real segment [ajk , b] and suppose that dν mjk (λ) ≥ 0 (λ ∈ [ajk , bjk ], t ≥ 0; ν = 0, 1, ..., deg mjk (λ)). dλν Then for the absolute exponential stability of the zero solution of (3.1) in the class of nonlinearities (3.2) it is necessary and sufficient that the polynomial det(P (λ) − L(λ)Q) be a Hurwitz one.

6.4 Examples Example 6.4.1 Consider the equation d2 x dx dφ(x) (a1 , a2 , b1 = const > 0), + a2 x = b1 φ(x) + + a1 2 dt dt dt where φ(s) satisfies the condition |φ(s)| ≤ q|s| (s ∈ R, t ≥ 0). Let the polynomial

(4.1)

(4.2)

P (λ) = λ2 + a1 λ + a2

have real roots λ1 ≤ λ2 < 0. Then under the condition b1 + λ 1 ≥ 0 we have

(4.3)

d L(λ) = 1 (λ1 ≤ λ ≤ λ2 ). dλ By Corollary 6.1.6, the zero solution of equation (4.1) is absolutely exponentially stable in the class of nonlinearities (4.2), provided the conditions (4.3) and a2 > qb1 hold. L(λ) = λ + b1 ≥ 0,

6. The Aizerman Problem

92

Example 6.4.2 Consider the equation d2 x dx d3 x dφ(x) + a3 x = b1 φ(x) + + a + a2 1 3 2 dt dt dt dt (a1 , a2 , a3 , b1 = const > 0),

(4.4)

where φ(x) satisfies condition (4.2). Let the polynomial P (λ) = λ3 + a1 λ2 + a2 λ + a3 have real roots λ1 ≤ λ2 ≤ λ3 < 0. Then under condition (4.3), relations L(λ) = λ + b1 ≥ 0,

d L(λ) = 1 (λ1 ≤ λ ≤ λ3 ) dλ

hold. By Corollary 6.1.6, the zero solution of equation (4.4) is absolutely exponentially stable in the class of nonlinearities (4.2), provided the conditions (4.3) and a3 > qb1 hold. Example 6.4.3 Consider the system p11 (D)x1 + p12 (D)x2 = φ1 (x1 , x2 , t), p22 (D)x2 = φ2 (x1 , x2 , t),

(4.5)

where pjk (λ) are polynomials. In addition p11 (λ) and p22 (λ) are Hurwitzian monic polynomials. The scalar-valued functions φ1 and φ2 satisfy the conditions |φj (z1 , z2 , t)| ≤ qj1 |z1 | + qj2 |z2 | (j = 1, 2) (4.6) for all z1 , z2 ∈ R1 and t ≥ 0. Assume that all the zeros of p11 (λ) and p22 (λ) belong to a real segment [a, b] (a < b < 0). In the considered case L(λ) = I; Q = (qjk )2j,k=1 ; W (λ) = (Wjk (λ))2j,k=1 , where Wjj (λ) =

1 p12 (λ) (λ) (j = 1, 2), W12 (λ) = − , W21 (λ) = 0. pjj p11 (λ)p22 (λ)

In addition, P (λ) = If

p11 (λ) 0

p12 (λ) p22 (λ) (k)

.

p12 (λ) ≤ 0 (k = 0, 1, ..., deg p12 (λ))

(4.7)

for all λ ∈ [a, b], then due to Corollary 6.3.2 for the absolute exponential stability of the zero solution of system (4.5) in the class of nonlinearities (4.6) it is necessary and sufficient that det(P (λ) − Q) be a Hurwitz polynomial.

7. Nonlinear Systems with Time-Variant Linear Parts

7.1 Systems with General Linear Parts Let us consider in Cn the equation x˙ = A(t)x + F (x, t) (t ≥ 0),

(1.1)

where A(t) is a piecewise continuous Hurwitz n × n-matrix, and F maps Ω(r) × [0, ∞) into Cn . Recall that Ω(r) = {h ∈ Cn : h ≤ r} for a positive number r. It is assumed that there exists a non-negative piecewise continuous function ν(t) = ν(t, r) bounded on [0, ∞), such that F (h, t) ≤ ν(t) h for all h ∈ Ω(r) and t ≥ 0.

(1.2)

For any fixed τ ≥ 0, let the condition exp[A(τ )t] ≤ p(t, A(τ )) (t ≥ 0)

(1.3)

hold, where p(t, A(τ )) is a piecewise continuous in t for each τ ≥ 0 function, such that χ : = sup p(t, A(t)) < ∞. t≥0

Put

q(t, s) ≡ A(t) − A(s)

(t, s ≥ 0).

Theorem 7.1.1 Let the conditions (1.2), (1.3) and θ(A(.), F ) ≡ sup t≥0

t 0

p(t − s, A(t))[q(t, s) + ν(s)]ds < 1

(1.4)

be fulfilled. Then the zero solution to equation (1.1) is stable. Moreover, if χ x(0) < r, 1 − θ(A(.), F )

(1.5)

then a solution x(t) of (1.1) satisfies the estimate x(t) ≤

χ x(t) (t ≥ 0) 1 − θ(A(.), F )

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 93–109, 2005. © Springer-Verlag Berlin Heidelberg 2005

(1.6)

7. Nonlinear Systems with Time-Variant Linear Parts

94

Proof:

Rewrite equation (1.1) in the form dx/dt − A(τ )x = [A(t) − A(τ )]x + F (x, t),

regarding an arbitrary τ ≥ 0 as fixed. This equation is equivalent to the following one: x(t) = exp[A(τ )t]x(0)+ t 0

exp[(A(τ )(t − s)][(A(s) − A(τ ))x(s) + F (x(s), s)]ds.

(1.7)

Since the solutions continuously depend on the initial vector, the inequality x(t) < r (0 ≤ t ≤ t0 ) is true for a sufficiently small t0 . Due to (1.2) and (1.7), the latter inequality implies the relation x(t) ≤ p(t, A(τ )) x(0) +

t 0

p(t − s, A(τ ))[q(τ, s) + ν(s)] x(s) ds (t ≤ t0 ).

Taking τ = t, we get x(t) ≤ p(t, A(t)) x(0) +

t 0

p(t − s, A(t))[q(t, s) + ν(s)] x(s) ds (t ≤ t0 ).

This relation according to the definitions of θ(A(.), F ) and χ yields sup x(s) < χ x(0) + sup x(s) θ(A(.), F ).

s≤t0

s≤t0

Consequently, due to (1.4) sup x(s) ≤ χ x(0) (1 − θ(A(.), F ))−1

s≤t0

for a sufficiently small t0 . But condition (1.5) ensures this bound for all t ≥ 0. That bound provides the (Lyapunov) stability. ✷ Put

n−1

p˜(t, A(s)) = exp[α(A(s))t] k=0

g k (A(s))tk (k!)3/2

with α(A(s)) = maxk Re λk (A(s)). Assume that χ ˜ : = sup p˜(t, A(t)) < ∞. t≥0

Due to Corollary 1.5.3 and the previous theorem we get

7.2 Systems with the Lipschitz Property

95

Theorem 7.1.2 Let the conditions (1.2) and ˜ θ(A(.), F ) := sup t≥0

t 0

p˜(t − s, A(t))[q(t, s) + ν(s)]ds < 1

be fulfilled. Then the zero solution to equation (1.1) is stable. Moreover, if χ ˜ x(0) < r, ˜ 1 − θ(A(.), F) then the estimate x(t) ≤

χ ˜ x(0) (t ≥ 0) ˜ 1 − θ(A(.), F)

is true for a solution x(t) of (1.1).

7.2 Systems with the Lipschitz Property 7.2.1 Statement of the Result Assume that the following conditions are satisfied:

and

A(t) − A(s) ≤ q0 |t − s| (t, s ≥ 0),

(2.1)

F (h, t) ≤ ν0 h

(2.2)

(h ∈ Ω(r), t ≥ 0).

Here q0 and ν0 are non-negative constants. In addition, let exp[A(τ )t] ≤ p0 (t) (t, τ ≥ 0), where p0 (t) is a piecewise continuous function, such that χ0 : = sup p0 (t) < ∞. t≥0

Then,

t 0

p0 (t − s)[q0 (t − s) + ν0 ]ds ≤ θ0 (A(.), ν0 ),

where θ0 (A(.), ν0 ) :=

∞ 0

(q0 t + ν0 )p(t)dt.

Now Theorem 7.1.1 and a small perturbation of equation (1.1) imply

(2.3)

7. Nonlinear Systems with Time-Variant Linear Parts

96

Theorem 7.2.1 Let the conditions (2.1-2.3) and θ0 (A(.), ν0 ) < 1 hold. Then the zero solution to equation (1.1) is exponentially stable. In addition, any initial vector x0 satisfying the inequality χ 0 x0 < r, 1 − θ0 (A(.), ν0 ) belongs to the region of attraction of the zero solution. Moreover, the corresponding solution x(t) of equation (1.1) is subject to the estimate x(t) ≤

χ 0 x0 (t ≥ 0). 1 − θ0 (A(.), ν0 )

Furthermore, let g(.) and α be defined as in Sections 1.5 and 1.3, respectively, and (2.4) v := sup g(A(t)) < ∞, and ρ := − sup α(A(t)) > 0. t≥0

t≥0

Put χ ˜0 := sup e−ρt t≥0

n−1 j=0

vj . (j!)3/2

Below we will prove that the previous theorem implies Corollary 7.2.2 Let the conditions (2.1), (2.2), (2.4) and n−1

Γ0 (A(.), ν0 ) := j=0

v j q (j + 1) ν0 √ ( 0 j+2 + j+1 )<1 ρ ρ j!

hold. Then the zero solution to equation (1.1) is exponentially stable. In addition, any initial vector x0 satisfying the inequality χ ˜ 0 x0 < r, 1 − Γ0 (A(.), ν0 )

(2.5)

belongs to the region of attraction of the zero solution. Moreover, the corresponding solution x(t) of equation (1.1) is subject to the estimate x(t) ≤

χ ˜ 0 x0 (t ≥ 0). 1 − Γ0 (A(.), ν0 )

Furthermore, denote bj =

jq0 v j−1

ν0 v j + √ for j = 1, ..., n − 1, j! (j − 1)!

(2.6)

97

7.2 Systems with the Lipschitz Property

and b0 = ν0 , bn = q0

nv n−1 (n − 1)!

.

(2.7)

Let z(A(.), ν0 ) be the extreme right-hand (unique positive and simple) root of the algebraic equation z n+1 =

n

bj z n−j .

(2.8)

j=0

Theorem 7.2.3 Let conditions (2.1), (2.2) and (2.4) hold. If, in addition, the matrix A(t) + ( + z(A(.), ν0 ))I is Hurwitzian for an > 0 and all t ≥ 0, then the zero solution to equation (1.1) is exponentially stable, and inequality (2.4) is valid. Moreover, any vector x0 satisfying (2.5), belongs to the region of attraction of the zero solution and estimate (2.6) is true for the corresponding solution. Example 7.2.4 Consider the scalar equation y¨ + p(t)y˙ + w(t)y = f (t, y, y), ˙

(2.9)

where p(t) and w(t) are non-negative scalar-valued functions with the property |p(t) − p(s)| + |w(t) − w(s)| ≤ q0 |t − s| for all t, s ≥ 0.

(2.10)

It is assumed that f is a real function defined on [0, ∞) × R2 and satisfying the condition f 2 (t, y, z) ≤ ν02 (y 2 + z 2 ) (t ≥ 0; z, y ∈ R1 , y 2 + z 2 ≤ r2 ; r ≤ ∞).

(2.11)

Let A(t) be the matrix of the linear part of equation (2.9), and α(A(t)) ≤ −ρ < 0 for all t ≥ 0. Let us suppose that supt≥0 w(t) < ∞. Then according to Example 1.6.1, g(A(t)) ≤ v ≡ 1 + sup w(t). t≥0

In addition, assume that Γ0 (A(.), ν0 ) =

q0 ν0 ν0 2q0 + + v( 3 + 2 ) < 1. ρ2 ρ ρ ρ

Then by Corollary 7.2.2 the zero solution of (2.9) is exponentially stable. Each vector x(0) = (y(0), y (0)) satisfying (2.5), belongs to the region of attraction of the zero solution. Note that for the considered equation χ ˜0 ≤ max exp[−tρ] (1 + vt). t≥0

98

7. Nonlinear Systems with Time-Variant Linear Parts

7.2.2 Proofs of Corollary 7.2.2 and Theorem 7.2.3 Proof of Corollary 7.2.2: We have ∞ 0

e−ρt

n−1 k=0

where

v k tk+1 dt = θ1 (A), (k!)3/2 n−1

θ1 (A) = k=0

Moreover,

∞ 0

e−ρt

n−1 k=0

where

(k + 1)v k √ . k!ρk+2

v k sk ds = θ0 (A), (k!)3/2 n−1

θ0 (A) = k=0

vk √ . ρk+1 k!

Now the result is due to Theorem 7.2.1. ✷ Lemma 7.2.5 Let z(A(.), ν0 ) be the extreme right-hand root of equation (2.8). Then the inequality α(A(t)) + z(A(.), ν0 ) < 0 for all t ≥ 0 implies the inequality Γ0 (A(.), ν0 ) < 1. Proof:

The hypothesis of the lemma entails the inequality ρ ≡ − sup α(A(t)) > z(A(.), ν0 ).

Dividing (2.8) by z n+1 , and taking into account that z(A(.), ν0 ) is its root, we arrive at the inequality n

bj z

1=

−j−1

n

(A(.), ν0 ) <

j=0

bj ρ−j−1 .

j=0

But according to (2.7), n

Γ0 (A(.), ν0 ) =

bj ρ−j−1 < 1.

j=0

As claimed. ✷ The assertion of Theorem 7.2.3 immediately follows from Corollary 7.2.1 and Lemma 7.2.5.

7.3 Systems with Differentiable Linear Parts

99

7.3 Systems with Differentiable Linear Parts In this section we consider system (1.1) with a real differentiable matrix A(t) = (ajk (t))nj,k=1 , which is Hurwitzian for each t ≥ 0 and uniformly bounded on [0, ∞). We establish stability conditions in terms of the determinant of the variable matrix. 7.3.1 Stability Conditions Denote

1 π

ψ(t) = and ζ(t) =

∞

1 π

−∞

∞ −∞

(A(t) − iIω)−1 3 dω

(A(t) − iIω)−1 2 dω (t ≥ 0).

Theorem 7.3.1 Let the conditions (1.2) and ˙ A(t) ψ(t) + ν(t)ζ(t) ≤ 1 (t ≥ 0).

(3.1)

hold. Then the zero solution to equation (1.1) is stable. Moreover, there is a constant M0 ≥ 1 such that x(t) ≤ M0 x(0)

(t ≥ 0)

(3.2)

for any solution x(t) of (1.1), provided M0 x(0) < r.

(3.3)

The proof of this theorem is presented in the next subsection. Below we suggest estimates for M0 . We will also prove that Theorem 7.3.1 implies the following result. Corollary 7.3.2 Let the conditions (1.2) and ˙ A(t) ψ(t) + ν(t)ζ(t) < 1

sup t≥0

(3.4)

hold. Then the zero solution to equation (1.1) is exponentially stable. Moreover, there is a constant M0 ≥ 1 such that estimate (3.2) is valid, provided condition (3.3) holds. Now put ˜ := 1 ψ(t) π and

∞ −∞

(N 2 (A(t)) + nω 2 )3(n−1)/2 dω |det (iωI − A(t))|3 (n − 1)3(n−1)/2

∞

(N 2 (A(t)) + nω 2 )n−1 dω . n−1 2 −∞ |det (iωI − A(t))| (n − 1) ˜ and ψ(t) ≤ ψ(t). ˜ We will check that ζ(t) ≤ ζ(t) So the previous theorem implies ˜ := 1 ζ(t) π

7. Nonlinear Systems with Time-Variant Linear Parts

100

Corollary 7.3.3 Let the conditions (1.2) and ˜ + ν(t)ζ(t) ˜ ≤ 1 (t ≥ 0) ˙ A(t) ψ(t) hold. Then the zero solution to equation (1.1) is stable. Moreover, there is a constant M0 ≥ 1 such that (3.2) is valid for any solution x(t) of (1.1), provided (3.3) holds. Note that if all the eigenvalues of the matrix are real, then |det (iωI − A(t))| ≥ (ω 2 + α2 (t))n/2 (ω ∈ R), where

α(t) := α(A(t)) = max Re λk (A(t)) k=1,...,n

and

λk (A(t)) (k = 1, ..., n)

are the eigenvalues of matrix A(t) with their multiplicities. Thus, ∞

˜ ≤ 1 ζ(t) π

−∞

(N 2 (A(t)) + ny 2 )n−1 dy . (y 2 + α2 (A))n (n − 1)n−1

Take into account that n

|λk (A(t))|2 ≤ N 2 (A(t))

k=1

and

|α(t)| ≤ min |λk (A(t))|. k

So and

nα2 (t) ≤ N 2 (A(t)) 2 n−1 ˜ ≤ 2(2N (A(t))) ζ(t) π

∞ 0

ds . s2 + 1

˜ ≤ ζ0 (t), where Consequently, ζ(t) ζ0 (t) = Similarly,

2n−1 N 2(n−1) (A(t)) . |α(t)|2n−1 (n − 1)n−1 ˜ ≤ ψ0 (t), ψ(t)

where ψ0 (t) =

2(3n−1)/2 N 3(n−1) (A(t)) . (n − 1)3(n−1)/2 π|α(t)|3n−1

7.3 Systems with Differentiable Linear Parts

(see Section 3.3). In the case n = 2, √ 2N 2 (A(t)) 4 2N 3 (A(t)) ψ0 (t) = (t) = . and ζ 0 |α(t)|3 π|α(t)|5

101

(3.5)

Corollary 7.3.2 implies Corollary 7.3.4 Let all the eigenvalues of A(t) be real and the conditions (1.2), and ˙ sup A(t) ψ0 (t) + ν(t)ζ0 (t) < 1 t≥0

hold. Then the zero solution to equation (1.1) is exponentially stable. Moreover, there is a constant M0 ≥ 1, such that estimate (3.2) is valid, provided condition (3.3) holds. 7.3.2 Proof of Theorem 7.3.1 Lemma 7.3.5 Let condition (1.2) with r = ∞ and (3.1) hold. Then there is a constant M0 , such that (3.2) is valid. Proof: Recall the Lyapunov theorem (see Section 1.9): if the eigenvalues of a constant matrix A0 lie in the interior of the left half-plane, then for any positive definite Hermitian matrix H there exists a positive definite Hermitian matrix WH such that WH A0 + A∗0 WH = −2H. Moreover, WH =

1 π

∞ −∞

(−iIω − A∗0 )−1 H(iIω − A0 )−1 dω.

Now let A(t) depend on t. Put W (t) =

1 π

∞ −∞

(−iIω − A∗ (t))−1 (iIω − A(t))−1 dω.

(3.6)

Then W (t) is a solution of the equation. W (t)A(t) + A∗ (t)W (t) = −2I.

(3.7)

Since A(.) is real, (W (t)A(t)h, h) = −(h, h) (h ∈ Rn ). Furthermore, multiplying equation (1.1) by W (t) and doing the scalar product, we get

7. Nonlinear Systems with Time-Variant Linear Parts

102

(W (t)x(t), ˙ x(t)) = (W (t)A(t)x(t), x(t)) + (W (t)F (x(t), t), x(t)). But d ˙ (t)x(t), x(t)) + (W (t)x(t), x(t)) (W (t)x(t), x(t)) = (W (t)x(t), ˙ x(t)) + (W ˙ = dt 2(W (t)x(t), ˙ x(t)) + (x(t), ˙ x(t)). Thus, it can be written d ˙ (t)x(t), x(t)) − 2(x(t), x(t)) + 2(W (t)F (x(t), t), x(t)). (W (t)x(t), x(t)) = (W dt Hence the conditions (1.2) and ˙ (t) + 2ν(t) W (t) ≤ 2 (t ≥ 0) W

(3.8)

provides the inequality (W (t)x(t), x(t)) ≤ (W (0)x(0), x(0)).

(3.9)

From Lemma 1.10.1 and the uniform boundedness of A(t) it follows that (W (t)h, h) ≥ c0 (h, h) (c0 = const > 0, t ≥ 0, h ∈ Rn ).

(3.10)

In the next subsection we will establish a lower estimate for c0 . Take into acount tat ˙ (t) = d 1 W dt π 1 π

∞ −∞

∞ −∞

(−iIω − A∗ (t))−1 (iIω − A(t))−1 dω =

(−iIω − A∗ (t))−1 [A˙ ∗ (t)(−iIω − A∗ (t))−1 +

˙ (−iIω − A(t))−1 A(t)](iIω − A(t))−1 dω. Hence, 1 ˙ (t) ≤ 2 A(t) ˙ W π

∞ −∞

˙ (iIω − A(t))−1 3 dω = 2 A(t) ψ(t).

This, (3.8) and (3.9) prove the required result. ✷ Proof of Theorem 7.3.1: Under (3.3) there is a t0 > 0, such that x(t) ∈ Ω(r) for t ≤ t0 . Repeating the reasonings of the proof of Lemma 7.3.4, we get x(t) ≤ M0 x(0) (t ≤ t0 ). But M0 x(0) < r. So we can extend this inequality to [0, ∞). ✷

7.3 Systems with Differentiable Linear Parts

103

˜ and ζ(t) ≤ ζ(t) ˜ are true. Lemma 7.3.6 The inequalities ψ(t) ≤ ψ(t) Proof: Thanks to Lemma 1.5.7, for any constant n × n-matrix A0 the inequality (Iλ − A0 )−1 ≤

(N 2 (A0 ) − 2Re (λ T race (A0 )) + n|λ|2 )(n−1)/2 |det (λI − A0 )|(n − 1)(n−1)/2

is true for any regular λ of A0 . Since A(t) is real and Im T race A(t) = 0, we have (Iiy − A(t))−1 ≤

(N 2 (A(t)) + ny 2 )(n−1)/2 (y ∈ R). |det (iyI − A(t))| (n − 1)(n−1)/2

Hence the required result follows. ✷ Proof of Corollary 7.3.2: Let condition (3.4) hold. With an > 0 substitute the equality x(t) = u (t)e− t into (1.1). Then we have the equation u˙ (t) = (A(t) + I )u (t) + F (u (t), t) (t ≥ 0), where

F (u (t), t) = e t F (u (t)e− t , t).

Due to (1.2),

F (h, t) ≤ ν(t) h

Put

and ζ (t) =

1 π

∞

1 π

ψ (t) =

−∞

∞

(A(t) − iI(ω − i))−1 3 dω

(A(t) − i(ω − i)I)−1 2 dω (t ≥ 0).

−∞

Due to (3.4) we can take

(h ∈ Ω(r), t ≥ 0).

sufficiently small, such that

˙ A(t) ψ (t) + ν(t)ζ (t) ≤ 1 (t ≥ 0). Thanks to Theorem 7.3.1, there is a constant M , such that u (t) ≤ M x0 . Thus

x(t) ≤ M x0 e−

This proves Corollary 7.3.2.

t

(t ≥ 0).

104

7. Nonlinear Systems with Time-Variant Linear Parts

7.3.3 Absolute Stability and Region of Attraction Definition 7.3.7 We will say that the zero solution to equation (1.1) is absolutely stable in the class of nonlinearities (h ∈ Cn , t ≥ 0),

F (h, t) ≤ ν(t) h

(3.11)

if there exist a M0 independent of the specific form of the function F (but dependent on ν(t)) such that for any solution x(t) of (1.1) inequality (3.2) is fulfilled. Theorem 7.3.1 implies Corollary 7.3.8 Let condition (3.1) hold. Then the zero solution to equation (1.1) is absolutely stable in the class of nonlinearities (3.11). Furthermore, (A(t) − iIy)−1 h ≥

h ≥ A(t) − iIy

h (h ∈ Rn , y ∈ R). A(t) + |y| Thus, according to (3.6) (W (t)h, h) ≥

1 π

∞ −∞

h 2 dy = ( A(t) + |y|)2

2 h 2 . π A(t) Moreover, according to the definition of W (t), we easily have W (t) ≤ ζ(t) (t ≥ 0). Hence, due to (3.9)

2 x(t) 2 ≤ ζ(0) x(0) 2 . π A(t)

We thus get Lemma 7.3.9 Under (3.6), let condition (1.2) be fulfilled with r ≤ ∞. Then with the notations a0 := sup A(t) and M0 = t≥0

inequality (3.2) is valid, provided (3.3) holds.

a0 ζ(0)π , 2

7.3 Systems with Differentiable Linear Parts

105

˜ Due to Lemma 7.3.6, ζ(0) ≤ ζ(0). In addition, as it is shown in Subsection 7.3.1, n−1 2(n−1) N (A(0)) ˜ ≤ ζ0 (0) = 2 . ζ(0) |α(0)|2n−1 (n − 1)n−1 Now Corollary 7.3.2 implies Corollary 7.3.10 Under conditions (1.2) and (3.4), the zero solution to equation (1.1) is exponentially stable. Moreover, let M0 = M1 , where M1 =

N n−1 (A(0)) |α(0)|n−1/2

2n−2 a0 π(n − 1)−n+1 .

(3.12)

Then any vector x0 satisfying (3.3), belongs to the region of attraction of the zero solution. Example 7.3.11 Let us consider the system x˙ 1 = −3a(t)x1 + b(t)x2 + f2 (x1 , x2 , t), x˙ 2 = a(t)x2 − b(t)x1 + f2 (x1 , x2 , t),

(3.13)

where a(t), b(t) are positive differentiable functions, fk (x1 , x2 , t) (k = 1, 2) are continuous functions defined on Ω(r) × [0, ∞) with Ω(r) = {z ∈ R2 : z ≤ r}. In addition, f12 (h1 , h2 , t) + f22 (h1 , h2 , t) ≤ ν 2 (t)(h21 + h22 ) (h = (h1 , h2 ) ∈ Ω(r), t ≥ 0) and

3a2 (t) < b2 (t) < 4a2 (t).

Then

˙ ˙ A(t) ≤ 3|a(t)| ˙ + |b(t)|

and α(t) = −a(t) +

4a2 (t) − b2 (t), N 2 (A(t)) = 10a2 (t) + 2b2 (t).

According to (3.5), ψ0 (t) = and ζ0 (t) =

16(5a2 (t) + b2 (t))3/2 . π|α(t)|5 2(10a2 (t) + 2b2 (t)) . |α(t)|3

Thanks to Corollary 7.3.4, system (3.13) is stable, provided (3.1) holds. Besides, Corollary 7.3.10 gives us a bound for the region of attraction of the zero solution.

106

7. Nonlinear Systems with Time-Variant Linear Parts

7.4 Additional Stability Conditions In this section as well as in the previous section, we consider system (1.1) with a real differentiable matrix A(t) = (ajk (t))nj,k=1 , which is Hurwitzian for each t ≥ 0 and uniformly bounded on [0, ∞). But stability conditions are formulated in terms of the individual eigenvalues of the variable matrix. 7.4.1 Stability Criteria Denote η(t) :=

∞ 0

eA(t)s

2

ds.

Theorem 7.4.1 Let the conditions (1.2) and ˙ A(t) η 2 (t) + 2ν(t)η(t) ≤ 1 (t ≥ 0)

(4.1)

hold. Then the zero solution to equation (1.1) is stable. Moreover, there is a constant M0 ≥ 1 such that (3.2) is valid for any solution of (1.1), provided (3.3) holds. The proof of this theorem is presented in the next subsection. Below we suggest additional estimates for M0 . Repeating the arguments of the proof of Corollary 7.3.2 we can show that Theorem 7.4.1 implies the following result. Corollary 7.4.2 Let the conditions (1.2) and sup t≥0

˙ A(t) η 2 (t) + 2ν(t)η(t) < 1

(4.2)

hold. Then the zero solution to equation (1.1) is exponentially stable. Moreover, there is a constant M0 ≥ 1 such that estimate (3.2) is valid, provided condition (3.3) holds. Recall that g(A) is introduced in Section 1.5 and α(t) is defined in the previous section. Put n−1

η˜(t) =

g j+k (A(t)) (k + j)! . 2j+k+1 |α(A((t))|j+k+1 (j! k!)3/2 j,k=0

Due to Lemma 1.9.1, η(t) ≤ η˜(t). Now the previous theorem implies Corollary 7.4.3 Let the conditions (1.2) and ˙ A(t) η˜2 (t) + 2ν(t)˜ η (t) ≤ 1 (t ≥ 0) hold. Then the zero solution to equation (1.1) is stable. Moreover, there is a constant M0 ≥ 1 such that (3.2) is valid for any solution of (1.1), provided (3.3) holds.

7.4 Additional Stability Conditions

107

In particular, if n = 2, then η˜(t) =

κ2 (t) 1 (1 + κ(t) + ). 2|α(A(t)| 2

Here and below κ(t) : =

(4.3)

g(A(t)) . |α(A(t))|

If n = 3, then

√ √ 2κ4 1 ( 2 + 1) 2 3 2κ3 + ) (κ = κ(t)). η˜(t) = (1 + κ + κ + 4 3 2|α(A(t)| 2

(4.4)

7.4.2 Proof of Theorem 7.4.1 Lemma 7.4.4 Let conditions (1.2) hold with r = ∞ and (4.1) hold. Then there is a constant M0 , such that (3.2) holds. Proof:

Recall that

W (t) := 2

∞ 0

eA(t)s eA

∗

(t)s

ds =

1 π

∞ −∞

(−iIω − A∗0 (t))−1 (iIω − A0 (t))−1 dω

is a solution of equation (3.7) (see Section 1.9). So 2η(t) = ζ(t), where ζ(t) is defined in the previous section. Condition to (3.8) provides inequality (3.9). Moreover, as its is shown in Subsection 7.3.3, (W (t)h, h) ≥

2 h 2 . π A(t)

Thus (3.9) implies 2(x(t), x(t)) ≤ A(t) π(W (0)x(0), x(0)).

(4.5)

˙ (t) ≤ A(t) ˙ As it was shown in the proof of Theorem 3.4.1, W η 2 (t) and W (t) ≤ 2η(t) (t ≥ 0). Now inequalities (3.8) and (4.5) prove the lemma. ✷ Lemma 7.4.5 Under (4.4), let condition (1.2) be fulfilled with r ≤ ∞. Then there is a constant M0 , such that (3.2) is valid, provided (3.3) holds. Proof: Under (3.3), x(0) < r and there is a t0 > 0, such that x(t) ∈ Ω(r) for t ≤ t0 . Repeating the arguments of the proof of Lemma 7.4.4, we get x(t) ≤ M0 x(0)

(t ≤ t0 ).

But M0 x(0) < r. So we can extend this inequality to [0, ∞). ✷ The assertion of Theorem 7.4.1 follows from Lemmas 7.4.4 and 7.4.5.

108

7. Nonlinear Systems with Time-Variant Linear Parts

7.4.3 Absolute Stability and Region of Attraction Theorem 7.4.1 implies Corollary 7.4.6 Let condition (4.1) hold. Then the zero solution to equation (1.1) is absolutely stable in the class of nonlinearities (3.11). Due to (4.5)

x(t) 2 ≤ η(0) x(0) 2 . π A(t)

Now Lemma 7.4.4 implies Lemma 7.4.7 Under (2.1), let condition (1.2) be fulfilled with r ≤ ∞. Then with a0 := sup A(t) and M0 = a0 η(0)π t≥0

inequality (3.2) is valid, provided (3.3) holds. Since η(0) ≤ η˜(0), Corollary 7.4.2 implies Corollary 7.4.8 Under conditions (1.2) and (4.3) the zero solution to equa˜ 0 , where tion (1.1) is exponentially stable. Moreover, let M0 = M ˜0 = M

η˜(0)πa0 .

Then the vectors x(0) satisfying (3.3), belongs to the region of attraction. 7.4.4 Examples Example 7.4.9 Let A(t) = (ajk (t))2j,k=1 be a real differentiable 2×2 matrix. In the general case, according to Example 1.6.1, g(A(t)) ≤ |a12 (t) − a21 (t)|. If the eigenvalues of A(t) are nonreal, then as it is shown in that example, g(A(t)) =

(a11 (t) − a22 (t))2 + (a21 (t) + a12 (t))2 .

(4.6)

For instance, let us consider the system x˙ 1 = −3a(t)x1 + x2 + f1 (x1 , x2 , t), x˙ 2 = a(t)x2 − x1 + f2 (x1 , x2 , t), (4.7) where a(t) is a positive differentiable function, fk (x1 , x2 , t) are continuous functions defined on Ω(r) × [0, ∞) with Ω(r) = {z ∈ R2 : z ≤ r}. In addition, f12 (h1 , h2 , t) + f22 (h1 , h2 , t) ≤ ν 2 (t)(h21 + h22 ) (h = (h1 , h2 ) ∈ Ωr , t ≥ 0) and

a(t) ≤ 1/2 (t ≥ 0).

7.4 Additional Stability Conditions

109

˙ Then A(t) = 3|a(t)|, ˙ α(A(t)) = −a(t) and due to (4.1) g(A(t)) = 4a(t). Thus (4.6) implies 13 . η˜(t) = 2a(t) Thanks to Corollary 7.4.2 system (4.7) is stable, provided 13ν(t) 3 . 132 |a(t)| ˙ + ≤ 1 (t ≥ 0). 4a2 (t) a(t) Besides, Corollary 7.4.8, gives us a bound for the region of attraction. Example 7.4.10 Let A(t) = (ajk (t))3j,k=1 be a real differentiable 3 × 3matrix. Consider system (1.1) with condition (1.2) and n = 3. According to inequality (5.3) from Section 1.5, g(A(t)) ≤ v(t) : = [(a12 − a21 )2 + (a13 − a31 )2 + (a23 − a32 )2 ]1/2 . For simplicity assume that 3

ajj (t) +

|ajk (t)| ≤ −ρ0 (t) (t ≥ 0, j = 1, 2, 3) k=1,k=j

where ρ0 is a positive scalar function. Then due to the well known result from the book (Marcus and Minc, 1964, Section 3.3.5), α(A(t)) ≤ −ρ0 (t). Thus (4.4) implies √ √ 3 2κ31 2κ4 1 ( 2 + 1)κ21 + + 1 ), η˜(t) ≤ η1 (t) : = (1 + κ1 + ρ0 2 4 3 where κ1 (t) = ρv(t) . Thanks to Corollary 7.4.3 system (1.1) under consider0 (t) ation is stable, provided ˙ η12 A(t) + 2η1 (t)ν(t) ≤ 1 (t ≥ 0). In addition, Corollary 7.4.8, gives us a bound for the region of attraction.

8. Essentially Nonlinear Systems

8.1 The Freezing Method for Nonlinear Systems 8.1.1 Stability Conditions As above, . is the Euclidean norm and Ω (r) is the ball in Cn with a radius r ≤ ∞ and with the center at zero. Let B(h, t) = (bjk (h, t))nj,k=1 be an n × n-matrix for every h ∈ Ω(r) and t ≥ 0. Everywhere below it is assumed that B(h, t) continuously depends on h ∈ Ω(r) and t ≥ 0. Let us consider in Cn the equation x(t) ˙ = B(x(t), t)x(t) (t ≥ 0).

(1.1)

It is assumed that there are constants µ(r), ν(r), such that B(h1 , t) − B(h, t) ≤ ν(r) h1 − h and B(h1 , t) − B(h, s) ≤ µ(r)|t − s| (h, h1 ∈ Ω(r); t, s ≥ 0).

(1.2)

Put M (r) := Assume that

sup

h∈Ω(r),t≥0

p(r, t) :=

B(h, t)h and q0 (r) = ν(r)M (r) + µ(r).

exp [B(h, s)]t] < ∞

sup

s≥0,h∈Ω(r)

(1.3)

is a continuous function of t, such that χ(r) := sup p(r, t) < ∞. t≥0

(1.4)

Theorem 8.1.1 For a positive r < ∞, let the conditions (1.2) - (1.4) and θ(r) := q0 (r)

∞ 0

tp(r, t)dt < 1

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 111–127, 2005. © Springer-Verlag Berlin Heidelberg 2005

(1.5)

112

8. Essentially Nonlinear Systems

hold. Then the zero solution to equation (1.1) is stable. Moreover, any solution x(t) of (1.1) is subject to the estimate x(t) ≤ x(0) provided x(0)

χ(r) for all t ≥ 0, 1 − θ(r)

(1.6)

χ(r) < r. 1 − θ(r)

(1.7)

The proofs of this theorem and the next one are presented in Section 8.2. Recall that the quantities g(A) and α(A) are introduced in Sections 1.5 and 1.3, respectively. Furthermore, let us suppose that ρ(r) := − and

v(r) :=

sup

α(B(h, t)) > 0

(1.8)

g(B(h, t)) < ∞.

(1.9)

h∈Ω(r),t≥0

sup

h∈Ω(r),t≥0

Put χ(r) ˜ := sup e−ρ(r)t t≥0

n−1 k=0

v k (r)tk . (k!)3/2

Due to Corollary 1.5.3 χ(r) ≤ χ(r) ˜ and ∞ 0

n−1

tp(r, t)dt ≤ j=0

(j + 1)v j (r) √ j+2 . j!ρ (r)

Now Theorem 8.1.1 and a small perturbation imply Corollary 8.1.2 For a positive r < ∞, let the conditions (1.2), (1.8), (1.9) and n−1 (j + 1)v j (r) √ j+2 Γ (r) := q0 (r) <1 (1.10) j!ρ (r) j=0 hold. Then the zero solution to equation (1.1) is exponentially stable. Moreover, with θ(r) = Γ (r) and χ(r) = χ(r), ˜ any solution x(t) of (1.1) is subject to estimate (1.6), provided (1.7) holds. Example 8.1.3 Let us consider the system x˙ = −cos (ax)x − 2y − sin (ax)y, y˙ = −cos (ay)y + 2x + sin (ay)x (0 < a = const < 1). Rewrite this system in the form (1.1) with B(h, t) = B(h) = bjk (h)), where h = (x, y),

(bjk )2j,k=1

(1.11) (bjk =

113

8.1 The Freezing Method for Nonlinear Systems

b11 = −cos (ax), b12 = −2 − sin (ax), b21 = 2 + sin (ay), b22 = −cos (ay). We have λ1,2 (B(h)) = T ± (T 2 − D)1/2 , where T = T (h) = (b11 + b22 )/2 = −(cos (ay) + cos (ax))/2 and D = D(h) = b11 b22 − b12 b21 = cos (ay)cos (ax) + (2 + sin (ax))(2 + sin (ay)). Take r = 1. Then cos (ax) ≥ cos a ≥ cos π/3 = 1/2 (|x| < 1) and T 2 − D ≤ (cos (ay) + cos (ax))2 /4 − cos (ay)cos (ax) − 1 = (cos (ay) − cos (ax))2 /4 − 1 ≤ 0. So Re λ1,2 (B(h)) = −(cos (ay) + cos (ax))/2 ≤ −ρ = −1/2 (|x|, |y| ≤ 1). Due to Example 1.6.1, g(B(h)) ≤ |b12 − b21 | = |4 + sin (ay) + sin (ay)| ≤ v = 6 (|x|, |y| ≤ 1). Moreover, B(h) − B(h1 )

2

n

≤

(bjk (h) − bjk (h1 )2 ≤ ν 2 (r) = 16a2 (|x|, |y| ≤ 1).

j,k=1

In addition, M (r) = M (1) ≤ sup N (B(h)) ≤ (2 + (2 + sin (a))2 )1/2 ≤ h∈Ω(1)

Thus,

√

11.

√ Γ (r) ≤ ν(r)M (r)(1/ρ−2 + 2vρ−3 ) ≤ 4a 11(4 + 96).

For a sufficiently small a we have stability condition (1.10). 8.1.2 Lyapunov’s Exponents Denote by z0 (r) the extreme right-hand root of the algebraic equation z n+1 = q0 (r)

n−1 j=0

v j (r)(j + 1) n−j−1 √ z . j!

(1.12)

8. Essentially Nonlinear Systems

114

Theorem 8.1.4 For some positive r ≤ ∞, let conditions (1.2) and (1.9) hold. If, in addition, the matrix B(h, t) + I(z0 (r) + ) is Hurwitzian for an > 0, all h ∈ Ω(r) and t ≥ 0, then the zero solution of (1.1) is exponentially stable. Besides, the relations (1.8) and (1.10) hold. Moreover, with θ(r) = Γ (r) and χ(r) = χ(r) ˜ any initial vector satisfying condition (1.7), belongs to the region of attraction of the zero solution and the corresponding solution of (1.1) is subject to the estimates (1.6) and x(t) ≤ const exp [(−ρ(r)t + z0 (r))t] (t ≥ 0). Let v(r) = 0. Put

n−1

wn = k=0

k+1 √ . k!

Setting z = v(r)y in (1.12) and applying Lemma 1.11.1, we can assert that z0 (r) ≤ δ0 (r), where δ0 (r) =

v 1−2/(n+1) (r)[q0 (r)wn ]1/(n+1) q0 (r)wn v(r)

if q0 (r)wn ≤ v 2 (r), if q0 (r)wn > v 2 (r)

Thus in the previous theorem we can replace z0 (r) by δ0 (r).

8.2 Proofs of Theorems 8.1.1 and 8.1.4 Proof of Theorem 8.1.1: Let us introduce the linear equation dyh /dt = B(h(t), t)yh ,

(2.1)

where h(t) : [0, ∞) → Ω(r) is a differentiable function. If h(t) = x(t) is a solution of (1.1), then (2.1) and (1.1) coincide. Further, the continuous dependence of solutions on initial data implies that under the condition x(0) < r, there is t0 , such that x(t) ∈ Ω(r) for 0 ≤ t ≤ t0

(2.2)

for a solution x(t) of (1.1). If we put A(t) = B(h(t), t) with some function h(t), then (2.1) takes the form x˙ = A(t)x. Due to (1.2) we have B(h(t), t) − B(h(s), s) ≤ B(h(t), t) − B(h(s), t) + B(h(s), t) − B(h(s), s) ≤ ν(r) h(t) − h(s) + µ(r)|t − s| (t, s ≤ t0 ). Hence, B(x(t), t) − B(x(s), s) ≤ ν(r) x(t) − x(s) + µ(r)|t − s| ≤

8.3 Perturbations of Nonlinear Systems

115

(ν(r) sup x(t) ˙ + µ(r))|t − s| (s, t ≤ t0 ). t≤t0

But according to (1.1) x(t) ˙ ≤ M (r) (t ≤ t0 ). Thus, B(x(t), t) − B(x(s), s) ≤ q0 (r)|t − s| (s, t ≤ t0 ). Applying Corollary 3.1.2 with A(t) = B(x(t), t) to equation (1.1) we have the bound x(t) ≤ χ(r)(1 − θ(r))−1 x(0) for t ≤ t0 . But condition (1.7) allows us to extend this bound to all t ≥ 0. So estimate (1.6) is proved. It yields the Lyapunov stability. Proof of Theorem 8.1.4: Repeating the reasoning of the proof of Theorem 3.1.5, one can show that the Hurwitzness of the matrix B(h, t)+z(r)I implies inequality (1.5). Now Corollary 8.1.2 yields the required result. ✷

8.3 Perturbations of Nonlinear Systems Consider in Cn the equation x(t) ˙ = B(x(t), t)x(t) + C(x(t), t)x(t) (t ≥ 0),

(3.1)

where B(z, t) and C(z, t) are n × n-matrices continuous in z ∈ Cn and t ≥ 0. Let Uh (t, s) (t, s ≥ 0) be the evolution operator of the linear equation y(t) ˙ = B(h(t), t)y(t) (t ≥ 0)

(3.2)

with a differentiable function h : [0, ∞) → Ω(r). Let us assume that there are positive continuous functions φ(r, t, s) and m(t, r) independent of h, such that Uh (t, s) ≤ φ(r, t, s) and C(w, t) ≤ m(t, r) (w ∈ Ω(r); t, s ≥ 0).

(3.3)

Additionally, suppose that (r) := sup φ(r, t, 0) < ∞ and t≥0

η(r) ≡ sup t≥0

t 0

φ(r, t, s)m(s, r)ds < 1.

(3.4)

Theorem 8.3.1 Let conditions (3.3) and (3.4) hold. Then the zero solution of equation (3.1) is stable. If, in addition, an initial vector x(0) satisfies the inequality (r) x(0) < r(1 − η(r)), (3.5) then the corresponding solution x(t) of (3.1) is subject to the estimate x(t) ≤

x(0) (r) (t ≥ 0). 1 − η(r)

116

8. Essentially Nonlinear Systems

Proof: By the property of the evolution operator, equation (3.2) is equivalent to the following one: y(t) = Uh (t, 0)y(0) +

t 0

Uh (t, s)C(h(s), s)y(s)ds (t ≥ 0).

Taking h(t) = x(t)-the solution of (3.1), we get x(t) = Ux (t, 0)x(0) +

t 0

Ux (t, s)C(x(s), s)x(s)ds (t ≥ 0).

Clearly, (3.5) implies x(0) < r. Then for a sufficiently small t0 > 0 we have x(t) ≤ r (t ≤ t0 ). So x(t) ≤ Ux (t, 0) (r) x(0) + Hence,

sup

0≤t≤t0

t

x(0) + t 0

x(t) ≤

0

φ(r, t, s)m(s, r) x(s) ds (t ≤ t0 ). (r) x(0) + sup

0≤t≤t0

and therefore, sup

0≤t≤t0

Ux (t, s) m(s, r) x(s) ds ≤

x(t) η(r),

(r) x(0) ≤ r. 1 − η(r)

x(t) ≤

According to (3.5) this inequality can be extended to [0, ∞). This proves the result. ✷

8.4 Nonlinear Triangular Systems Let us consider the triangular system j

u˙ j (t) =

ajk (u, t)uk (t) k=1

(u = (uk (t))nk=1 , t ≥ 0; j = 1, ..., n) with continuous real functions ajk (., .) : Rn × [0, ∞) → R (1 ≤ k ≤ j ≤ n). For a positive r ≤ ∞ put

(4.1)

8.4 Nonlinear Triangular Systems

α(t, r) := max

max ajj (w, t)

w∈Ω(r) j=1,...,n

g(t, r) := max

w∈Ω(r)

and ψr (t, τ ) := e

t τ

α(s,r)ds

1≤k<j≤n n−1 k=0

a2jk (w, t)

t

1 [ k!

τ

117

(4.2) (4.3)

g(t1 , r)dt1 ]k

(0 ≤ τ ≤ t < ∞), Theorem 8.4.1 Let the condition χ(r) ˜ := sup ψr (t, τ ) < ∞ t≥τ ≥0

(4.4)

be fulfilled. Then the zero solution of equation (4.1) is stable. If an initial vector u0 satisfies the inequality χ(r) ˜ u0 < r,

(4.5)

then a solution u(t) of (4.1) with the initial condition u(τ ) = u0 is subject to the estimate u(t) ≤ ψr (t, τ ) u(τ ) (t ≥ τ ≥ 0). (4.6) Proof:

Consider the triangular linear system j

v˙ j (t) =

ajk (h(t), t)vk (t), (t ≥ 0; j = 1, ..., n)

(4.7)

k=1

with a continuous function h : [0, ∞) → Ω(r). By Theorem 2.4.1, the inequality v(t) ≤ ψr (t, τ ) v(τ )

(t ≥ τ ≥ 0)

(4.8)

is true for any solution v(t) of system (4.7). Clearly, χ(r) ˜ ≥ 1. So condition (4.5) implies the inequality u(τ ) < r. Then for a sufficiently small t0 − τ , u(t) ≤ r (0 ≤ τ ≤ t ≤ t0 ). But equation (4.7) coincides with (4.1) if h(t) = u(t), where u(t) is the solution of (4.1). So by (4.8), u(t) ≤ χ(r) ˜ u0 ≤ r (τ ≤ t ≤ t0 ). According to (4.6) this inequality can be extended to [0, ∞). This proves the result. ✷

118

8. Essentially Nonlinear Systems

Corollary 8.4.2 Let the condition g(t, r) ≤ g0 < ∞ (g0 = const, t ≥ 0)

(4.9)

be fulfilled. Then the zero solution to system (4.1) is stable, provided ajj (w, t) ≤ α0 < 0 (α0 = const; w ∈ Ω(r); t ≥ 0; j = 1, ..., n)

8.5 Perturbations of Triangular Systems Consider system (1.1). It is assumed that B(w, t) = (bjk (w, t))nj,k is continuous in w ∈ Rn and piecewise continuous in t. Also consider the corresponding lower-triangular matrix A(w, t) = (ajk (w, t))nj,k=1 with the entries ajk (w, t) = bjk (w, t) if j ≥ k; and ajk (w, t) = 0 if j < k (w ∈ Cn ). (5.1) Assume that conditions (4.2-4.4) hold. According to (4.8), the evolution operator of the system (4.7) with a continuous function h defined on [0, ∞) with values in Ω(r), satisfies the inequality Uh (t, s) ≤ ψr (t, s) (t, s ≥ 0). Put

m(t, r) = sup

h∈Ω(r)

C(h, t) ,

where C(h, t) = A(h, t) − B(h, t) is the upper triangular matrix with the entries cjk (h, t) = bjk (h, t) (j < k) and cjk (h, t) = 0 (j ≥ k). In addition, suppose that η˜(r) ≡ sup t≥0

t 0

ψr (t, s)m(s, r)ds < 1.

(5.2)

Employing Lemma 8.3.1 and Theorem 8.4.1, we arrive at the following result. Theorem 8.5.1 Let condition (5.2) hold. Then the zero solution of equation (1.1) is stable. In addition, if an initial vector u0 satisfies the inequality χ(r) u0 < r(1 − η˜(r)),

(5.3)

then the corresponding solution u(t) of system (1.1) is subject to the inequality u(t) ≤

χ(r) u0 ≤ r (t ≥ 0). 1 − η˜(r)

Note that the following relation holds: n

m(t, r) ≤ max

j=1,...,n

sup |bjk (h, t)|2 .

k=j+1 h∈Ω(r)

8.6 Nonlinear Dissipative Systems

119

8.6 Nonlinear Dissipative Systems Let .

Cn

be an arbitrary norm in Cn .

Theorem 8.6.1 For any h ∈ Cn with h C n < r and all sufficiently small δ > 0, let I + δB(h, t) C n ≤ 1 + a(t)δ (t ≥ 0) (6.1) where a(t) is a Riemann-integrable function independent of h and having the property t

0

a(s)ds ≤ 0 (t ≥ 0).

(6.2)

Then any solution x(t) of (1.1) satisfies the estimate x(t) provided x(s)

Cn

Cn

≤ x(s)

Cn

exp[

t s

a(τ )dτ ] (t ≥ s ≥ 0),

(6.3)

< r.

Proof: The continuous dependence of solutions on initial data implies that under the condition x(0) < r there is t0 such that relation (2.2) holds. Again consider equation (2.1) with an arbitrary function h(t) : [0, ∞) → Ω(r). Using Theorem 4.1.1 with A(t) = B(h(t), t), we have the required estimate (6.3) for t ≤ t0 . But condition (6.2) permits us to extend it to the whole positive half-line. ✷ Let (., .) be the scalar product in Cn . As above, Ω(r) = {h ∈ Cn : h ≤ r}, where . is the Euclidean norm. Theorem 8.6.2 For any t ≥ 0 and h ∈ Ω(r), let Re (B(h, t)h, h) ≤ Λ(t)(h, h),

(6.4)

where Λ(t) is the Riemann-integrable function independent of h. Then under the condition t

0

Λ(s)ds ≤ 0 (t ≥ 0)

any solution x(t) of equation (3.1) satisfies the estimate x(t) ≤ x(s) exp[ provided x(s) < r.

t s

Λ(τ ) dτ ] (t > s ≥ 0)

120

Proof:

8. Essentially Nonlinear Systems

Doing the scalar product in (1.1) by x and considering that d (x(t), x(t)) = 2 Re(x(t), ˙ x(t)), dt

we get

d (x(t), x(t)) = 2 Re(B(x(t), t)x(t), x(t)). dt For a sufficiently small t0 we have relation (2.2) and, therefore,

(6.5)

d (x(t), x(t)) ≤ 2Λ(t)(x(t), x(t)), 0 ≤ t ≤ t0 . dt Solving this inequality, we arrive at the inequality (x(t), x(t)) ≤ exp[2

t 0

Λ(s)ds](x(0), x(0)), 0 ≤ t ≤ t0 .

Hence the required result easily follows. ✷

8.7 Nonlinear Systems with Linear Majorants Consider in Cn equation (1.1), where B(h, t) is an n × n-matrix for each h ∈ Cn and all t ≥ 0, again. For all sufficiently small positive δ and h ∈ Ω(r) (r ≤ ∞), let there be a variable matrix M (t) independent of h, such that the relation |v + δB(h, t)v| ≤ (I + M (t)δ)|v| (v ∈ Cn , t ≥ 0)

(7.1)

is valid. Then we will say that system (1.1) has in set Ω(r) the linear majorant M (t). Inequality (7.1) means that |bjk (h, t)| ≤ mjk (t) for j = k, and

|1 + δbkk (h, t)| ≤ 1 + δmkk (t) (h ∈ Ω(r), t ≥ 0),

and M (t) = (mjk (t))nj,k=1 . Let us introduce the equation z(t) ˙ = M (t)z(t) (t ≥ 0)

(7.2)

and assume that this equation is stable. This implies that the Cauchy operator V (t) of equation (7.2) is bounded: l = sup V (t) < ∞. t≥0

(7.3)

121

8.8 Second Order Vector Systems

Lemma 8.7.1 Let system (1.1) have a linear majorant M (t) in the ball Ω(r). Then, under condition (7.3), any solution x(t) of (1.1) is subject to the inequality |x(t)| ≤ V (t)|x(0)| (t ≥ 0), (7.4) provided that

x(0) < rl−1 .

(7.5)

Proof: Clearly, l ≥ 1. So x(0) < r, and condition (2.2) holds. Set A(t) = B(x(t), t), where x(t) is the solution of equation (1.1). Now the inequality (7.4) for 0 ≤ t ≤ t0 is due to Lemma 4.2.1. Since x(0) is in the interior of Ω(rl−1 ), the solution remains in Ω(r). This proves the result. ✷ Corollary 8.7.2 Let system (1.1) have a linear constant majorant M in the set Ω(r). In addition, let M be a Hurwitz matrix, such that l ≡ sup eM t < ∞. t≥0

Then any solution x(t) of (1.1) with an initial vector x(0) satisfying (7.5) is subject to the inequality |x(t)| ≤ exp [M t]|x(0)| (t ≥ 0), and, therefore, the zero solution of equation (1.1) is exponentially stable. Note that Corollary 1.5.3 yields the inequality l ≤ max eα(M )t t≥0

n−1 k=0

g k (M )tk . (k!)3/2

8.8 Second Order Vector Systems Again let Rn (n > 1) be a real Euclidean space with the scalar product (., .), the unit matrix I and the norm . = (., .). For a positive r ≤ ∞ put ωr := v, h ∈ Rn : v

2

+ h

2

≤ r2

and consider the equation x ¨ + 2F (x, x, ˙ t)x˙ + G(x, x, ˙ t)x = 0

(8.1)

with real n × n-matrices F (h, w, t) and G(h, w, t) continuously dependent on (h, w) ∈ ωr and t ≥ 0. Take the initial conditions x(t) = x0 , x(0) ˙ = x1 (x0 , x1 ∈ Rn ).

(8.2)

122

8. Essentially Nonlinear Systems

A solution of the problem (8.1), (8.2) is a twice continuously differentiable function x : [0, ∞) → Rn , satisfying (8.1) and (8.2) for all t ≥ 0. Under consideration, problem (8.1), (8.2) has solutions, due to the solution estimates established below, since the entries continuously depend on the arguments. Recall that AR = (A + A∗ )/2 and AI = (A − A∗ )/2i are the real and imaginary components of a matrix A, respectively and A∗ is the matrix adjoint to A. Let A0 be a Hermitian matrix. Then we will write A0 ≥ 0 (A0 > 0) if it is positive definite (strongly positive definite). For two Hermitian matrices A0 , B0 we write A0 ≥ B0 if A0 − B0 ≥ 0, and A0 > B0 if A0 − B0 > 0. So FR (h, w, t) ≡ (F (h, w, t) + F ∗ (h, w, t))/2, GR (h, w, t) ≡ (G(h, w, t) + G∗ (h, w, t))/2. It is assumed that there is a constant mF (r) > 0, such that mF (r)I ≤ FR (h, w, t) and 0 ≤ GR (h, w, t) ≤ mF (r)(2FR (h, w, t) − mF (r)I) Put

( (h, w) ∈ ωr ; t ≥ 0).

(8.3)

Ψ (h, w, t) := 2mF (r)F (h, w, t) − G(h, w, t).

(8.4)

According to the above definitions we write ΨR (h, w, t) = (Ψ + Ψ ∗ )/2, ΨI (h, w, t) = (Ψ − Ψ ∗ )/2i with Ψ = Ψ (h, w, t). Theorem 8.8.1 Under condition (8.3), let ζ0 (r) :=

sup

(h,w)∈ωr , t≥0

ΨR (h, w, t) + ΨI (h, w, t) < 2m2F (r).

(8.5)

Then the zero solution to equation (8.1) is exponentially stable. Moreover, the initial vectors satisfying the condition x0

2

+ x1

2

< r2 max {ζ0 (r) − m2F (r),

1 }, ζ0 (r) − m2F (r)

(8.6)

belong to the region of attraction of the zero solution and the corresponding solution x(t) of problem (8.1), (8.2) satisfies the inequality (ζ0 (r) − m2F (r)) x(t)

2

e2Λ0 (r)t ((ζ0 (r) − m2F (r)) x0 with Λ0 (r) := −mF (r) +

ζ0 (r) −

m2F (r)

+ x(t) ˙

2

≤

+ x1 2 ) (t ≥ 0)

< 0.

This theorem is proved in the next section.

2

(8.7)

123

8.8 Second Order Vector Systems

Corollary 8.8.2 For some positive r and all (h, w) ∈ ωr , and t ≥ 0, let matrices F (h, w, t) and G(h, w, t) be symmetric, and there be a constant mF (r) > 0, such that mF (r)I ≤ F (h, w, t) and m2F (r)I ≤ 2mF (r)F (h, w, t) − G(h, w, t) < 2m2F (r)I ( (h, w) ∈ ωr ; t ≥ 0). Then the zero solution to equation (8.1) is exponentially stable. Moreover, the initial vectors satisfying condition (8.6) with ζ0 (r) =

sup

(h,w)∈ωr ,t≥0

2mF (r)F (h, w, t) − G(h, w, t)

belong to the region of attraction of the zero solution and the corresponding solution x(t) of problem (8.1), (8.2) satisfies the inequality (8.7). Example 8.8.3 Consider the equation x ¨ + 2f (x, x, ˙ t)x˙ + g(x, x, ˙ t)x = 0

(8.8)

with positive continuous scalar functions f and g defined on ω2,r × [0, ∞) with ω2,r := {w, s ∈ R1 : w2 + s2 ≤ r2 }. Let

mf (r) :=

inf

(s,w)∈ω2,r ,t≥0

f (s, w, t) > 0.

Then due to Corollary 8.8.2 the zero solution to equation (8.8) is exponentially stable, provided m2f (r) ≤ 2mf (r) f (s, w, t) − g(s, w, t) ≤ p˜(r) < 2m2f (r) (˜ p(r) = const; (s, w) ∈ ω2,r , t ≥ 0). In particular, take r = 1, f (s, w, t) = 2 +

w3 w2 sin2 (s + t) and g(s, w, t) = 3 + cos (s2 + t2 ). 2 2

Then mf (1) = 2 and m2f (1) = 4 ≤ 2mf (1) f (s, w, t) − g(s, w, t) = 8 + 2w2 sin2 (s + t) − 3− w3 cos (t2 + s2 ) < 2m2f (1) = 8 (t ≥ 0; |w|, |s| < 1). 2 So the zero solution to equation (8.8) is exponentially stable. Example 8.8.4 Let F (h, w, t) = (fjk )nj,k=1 (fjk ≡ fjk (h, w, t)) be a real symmetric matrix, having the properties

8. Essentially Nonlinear Systems

124

n

fjj (h, w, t) ≥ 9/8,

|fjk (h, w, t)| ≤ 1/8 (j = 1, ..., n).

(8.9)

k=1,k=j

Here and below in this section t ≥ 0 and (h, w) ∈ ωr for some positive r ≤ ∞. Then n

inf fjj (h, w, t) − j

|fjk (h, w, t)| ≥ 1 (t ≥ 0). k=1,k=j

Due to the well-known result (Marcus and Minc, 1964, Section III.9.2) (see also the Appendix A, Section A1), F (h, w, t) ≥ I. So we take mF (r) = 1. In addition, let G(h, w, t) = diag [gj (h, w, t)]nj=1 , such that 0 ≤ 2fjj − 7/4 < gj ≤ 2fjj − 5/4 (gj = gj (h, w, t); j = 1, 2, ...). Then

(8.10)

n

2(fjj −

|fjk |) − gj ≥ k=1,k=j

2fjj − gj − 1/4 ≥ 5/4 − 1/4 = 1. Again use the well-known result from (Marcus and Minc, 1964, Section III.9.2) (see also the Appendix A, Section A1). It gives us the inequality Ψ (h, w, t) ≥ I. In addition, n

2(fjj +

|fjk |) − gj k=1,k=j

≤ 2fjj − gj + 1/4 < 7/4 + 1/4 = 2. Hence we have Ψ (h, w, t) < 2I. Therefore, under (8.9) and (8.10) the zero solution to equation (8.1) is exponentially stable due to Corollary 8.8.2.

8.9 Proof of Theorem 8.8.1 Let us consider in Rn the linear problem x ¨ + 2A(t)x˙ + B(t)x = 0 (t > 0),

(9.1)

with real piecewise continuous matrices A(t) and B(t). Assume that there is a constant mA , such that mA I ≤ AR (t) and 0 ≤ BR (t) ≤ mA (2AR (t) − mA I) (t ≥ 0). Put

T (t) := 2mA A(t) − B(t).

(9.2)

8.9 Proof of Theorem 8.8.1

125

Lemma 8.9.1 Under condition (9.2) any solution x(t) of problem (9.1), (8.2) satisfies the estimate (p0 − m2A ) x(t)

2

+ x(t) ˙

2

≤ e2Λt ((p0 − m2A ) x0

2

+ x1 2 )

where p0 := sup TR (t) + TI (t) and Λ := −mA + t≥0

Proof:

p0 − m2A .

Put in (9.1) x(t) = e−mt y(t) (m ≡ mA ).

(9.3)

Then we have the equation y¨ − 2(m − A(t))y˙ − C(t)y = 0 with

C(t) = 2A(t)m − Im2 − B(t) = T (t) − m2 I.

Reduce this equation to the system y˙ 1 = 2(m − A(t))y1 + C(t)y2 , y˙ 2 = y8 . Doing the scalar product, we get d (y1 , y1 ) = 4((m − A(t))y1 , y1 ) + 2(C(t)y2 , y1 ), dt d (y2 , y2 ) = 2(y2 , y1 ). dt Since the matrices are real, we can write out ((mI − A(t))y1 , y1 ) = ((mI − AR (t))y1 , y1 ) ≤ 0. So Thus

d(y1 , y1 )/dt ≤ 2 C(t) d y1 (t) /dt ≤ C(t)

y2

y1 .

y2 (t) , d y2 (t) /dt = y1 (t) .

Furthermore, C ≤ C R + C I = T R − m2 I + T I

(C = C(t), T = T (t)).

Under the hypothesis of the present lemma CR is positive. So T R − m2 I =

sup

v∈Rn , v =1

(TR v, v) − m2 = TR − m2 .

(9.4)

126

8. Essentially Nonlinear Systems

Thus

C ≤ C R + C I = T R − m2 I + T I = TR + TI − m2A ≤ p0 − m2A (t ≥ 0; C = C(t), T = T (t)).

Put

b0 :=

p 0 − m2 .

Then (9.4) implies d y1 (t) /dt ≤ b20 y2 (t) , d y2 (t) /dt = y1 (t) . Hence,

yk (t) ≤ zk (t) (k = 1, 2; t ≥ 0),

(9.5)

where (z1 (t), z2 (t)) is a solution of the coupled system z˙1 = b20 z2 , z˙2 = z1 (z1 (0) = x1 , z2 (0) = x0 ). Simple calculations show that z1 (t) =

1 [(z1 (0) + z2 (0)b0 )eb0 t + (z1 (0) − z2 (0)b0 )e−b0 t ], 2

1 [(z1 (0)/b0 + z2 (0))eb0 t − (z1 (0)/b0 − z2 (0))e−b0 t ]. 2 By the Schwarz inequality, z2 (t) =

z12 (t) ≤ e2b0 t [(z1 (0) + z2 (0)b0 )2 + (z1 (0) − z2 (0)b0 )2 ] ≤ e2b0 t (z12 (0) + z22 (0)b20 ) and

z22 (t)b20 ≤ e2b0 t (z12 (0) + z22 (0)b20 ).

Hence (9.3) and (9.5) yield the required result. ✷ Proof of Theorem 8.8.1: Consider equation (8.1) as (9.1) with A(t) = F (x(t), x(t), ˙ t), B(t) = G(x(t), x(t), ˙ t). If the initial vectors satisfy condition (8.6), then for some t0 > 0, (x(t), x(t)) ˙ ∈ ωr (t ≤ t0 ). Due to Lemma 8.9.1, we have estimate (8.7) for t ≤ t0 . It implies x(t)

2

+ x(t) ˙

2

≤

1 }( x0 ζ0 (r) − m2F (r) (t ≤ t0 ).

e2Λ0 (r)t max {ζ0 (r) − m2F (r),

2

+ x1 2 )

So according to (8.6) the solution remains in ωr and we can thus extend (8.7) to all t ≥ 0, as claimed. ✷

127

8.10 Scalar Equations with Real Characteristic Roots

8.10 Scalar Equations with Real Characteristic Roots Let us consider the scalar equation y (n) + p1 (y, ..., y (n−1) , t)y (n−1) + ... + pn (y, ..., y (n−1) , t)y = 0 (t ≥ 0) (10.1) whose coefficients pj (h1 , ..., hn , t) = pj (h, t) (h = (hk ) ∈ Rn , j = 1, ..., n) are real scalar-valued functions continuous in h = (hk ) ∈ Rn and piecewise continuous in t ≥ 0. Introduce the algebraic equation λn + p1 (h, t)λn−1 + ... + pn (h, t) = 0 (h ∈ Rn ; t ≥ 0).

(10.2)

Theorem 8.10.1 Suppose that, for all h ∈ Rn and a sufficiently large t0 > 0, the roots λ1 (h, t), ..., λn (h, t) of the polynomial (10.2) are real and satisfy the inequalities ν0 ≤ λ1 (h, t) ≤ ν1 ≤ λ2 (h, t) ≤ ... ≤ νn−1 ≤ λn (h, t) ≤ νn < 0 (h ∈ Rn ; t ≥ t0 ), where νi are some constants, such that ν0 < ν1 < ... < νn < 0. Then equation the zero solution to equation (10.1) is exponentially stable. For the proof see (Levin, 1969).

9. Lur’e Type Systems

9.1 Stability Conditions In the present chapter we investigate the asymptotic stability of the system w ¨ + B w(t) ˙ + Cw(t) = F (w, t) (t > 0),

(1.1)

where B and C are real constant n × n-matrices (n > 1); F (h, t) continuously maps Cn × [0, ∞) into Cn . Take the initial conditions ˙ = u1 (u0 , u1 ∈ Cn ). w(0) = u0 , w(0)

(1.2)

A solution of problem (1.1), (1.2) is a function w : [0, ∞) → Cn , having the first and second locally bounded derivatives and satisfying (1.2) and (1.1) for all t ≥ 0. Existence of solutions is assumed. Note that the approach suggested in the present chapter allows us to consider the systems m

Ak w(k) (t) = F (w, t) (m ≥ 2, t ≥ 0),

k=0

where Ak are constant matrices. Assume that there is a constant q = q(r) ≥ 0, such that F (h, t) ≤ q h

(h ∈ Ω(r), t ≥ 0).

(1.3)

Recall that . is the Euclidean norm and Ω(r) = {z ∈ C : z ≤ r} for a positive r ≤ ∞; N (.) is the Frobenius norm (see Section 1.1). Consider the matrix pencil W (z) = Iz 2 + Bz + C (z ∈ C).

(1.4)

It is assumed that pencil W (z) is stable. That is, the determinant det (W (z)) of W (z) is a Hurwitz polynomial. Clearly, the integrals ψk := [

1 2π

∞ −∞

y 2k W −1 (iy) dy]1/2 (k = 0, 1)

converge. , M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 129–134, 2005. © Springer-Verlag Berlin Heidelberg 2005

130

9. Lur’e Type Systems

Theorem 9.1.1 Let pencil W (.) be stable, and the conditions (1.3) and b(W ) := sup W −1 (iy) < y∈R

1 q

(1.5)

hold. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, with the notations ηB (w ˜0 ) = ψ1 u0 + ψ0 u1 + Bu0 and ˜0 ) = ψ1 u1 + ψ0 Cu0 ηC ( w

(w ˜0 = (u0 , u1 )),

all vectors w ˜0 = (u0 , u1 ) ∈ C2n , satisfying the inequality ˜0 )ηC (w ˜0 ))1/2 + ζ(q, w ˜0 ) := (2ηB (w

˜0 ) qψ0 ηB (w
(1.6)

belong to the region of attraction of the zero solution to equation (1.1). Besides, any solution w of problem (1.1), (1.2) satisfies the inequalities w(t) ≤ ζ(q, w ˜0 ) (t ≥ 0) and [

∞ 0

w(t) 2 dt]1/2 ≤

(1.7)

˜0 ) ηB ( w . 1 − qb(W )

(1.8)

The proof of this theorem is divided into a series of lemmas, which are presented in the next two sections. Note that the arguments of the proof of Lemma 5.3.3 allow us to replace b(W ) by sup

|y|≤R0

W −1 (iy) where R0 := B /2 +

Put φ(W (z)) :=

B 2 /4 + 2 C .

N n−1 (W (z)) . (n − 1)(n−1)/2 |det (W (z))|

Due to Lemma 1.5.7 W −1 (z) ≤ φ(W (z)). Thus ψk ≤ ψ˜k , where 1 ψ˜k := [ 2π

∞ −∞

y 2k φ2 (W (iy))dy]1/2 (k = 0, 1).

Below we will give simple estimates for ψ˜k (k = 0, 1). Thus the previous theorem implies Corollary 9.1.2 Let pencil W (.) be stable, and the conditions (1.3) and ˜b(W ) := sup φ(W (iy)) < 1 q y∈R hold. Then the zero solution to equation (1.1) is asymptotically stable. Moreover, with the notations

9.1 Stability Conditions

131

η˜B (w ˜0 ) = ψ˜1 u0 + ψ˜0 u1 + Bu0 and

η˜C (w ˜0 ) = ψ˜1 u1 + ψ˜0 Cu0

all vectors w ˜0 = (u0 , u1 ) ∈ C2n , satisfying the inequality ˜0 ) q ψ˜0 ηB (w ˜ w ζ(q, ˜0 ) := (2˜ ηB ( w ˜0 )˜ ηC ( w ˜0 ))1/2 +
T =

−C 0

is the matrix of the linear part of (1.1). Thus if all the eigenvalues of T are real, then |det (W (iy))| = |det (T − iyI)| ≥ |α − iy|2n = (α2 + y 2 )n , where α = α(T ) = max Re λk (T ), That is, α is the real part of the extreme right root of det (W (z)). Thus, θ2 ψ˜k2 ≤ n 2π

where θn Therefore,

∞

N 2n−2 (W (iy)) dy (α2 + y 2 )2n −∞ √ = (n − 1)−(n−1)/2 . But N (W (iy)) ≤ ny 2 +|y|N (B) +N (C).

θ2 ψ˜k2 ≤ n π

∞ 0

y 2k

√ y 2k ( ny 2 + yN (B) + N (C))2n−2 dy (k = 0, 1). (α2 + y 2 )2n

(1.9)

These integrals are simple calculated. Example 9.1.3 Consider system (1.1) with the diagonal matrices B = diag [b1 , ..., bn ], C = diag [c1 , ..., cn ] having entries bk > 0, ck > 0. Then N 2 (B) =

n

b2k ; N 2 (C) =

k=1

and

n

c2k

k=1 n

det W (z) =

(z 2 + zbk + ck ).

k=1

Hence, α(T ) = max Re {−bk /2 + k

b2k /4 − ck }.

Now we can directly apply Corollary 9.1.2 taken into account. Note that in the general case, one can take bounds for the roots of det (W (z)) from the papers (Gil’, 2003b) and (Gil’, 2003c)

9. Lur’e Type Systems

132

9.2 Preliminaries Recall that L2 ≡ L2 ([0, ∞), Cn ) is the Hilbert space of functions defined on [0, ∞) with values in Cn and the norm f

L2

=[

∞ 0

f (t) 2 dt]1/2

and C([0, ∞), Cn ) is the Banach space of continuous functions defined on [0, ∞) with values in Cn and the sup-norm norm . C . Put G(t) :=

∞

1 2π

−∞

eiyt W −1 (iy)dy (t ≥ 0)

and (G ∗ f )(t) =

t 0

G(t − s)f (s)ds.

Lemma 9.2.1 Let W (.) be stable. Then G∗f

L2

≤ b(W ) f

L2

(f ∈ L2 ).

Proof: Due to the Laplace transform and the Parseval equality with the previous lemma taken into account, G∗f

2 L2

b2 (W )

1 2π

=

1 2π ∞

−∞

∞ −∞

W −1 (iy)f˜(iy) 2 dy ≤

f˜(iy) 2 dy = b2 (W ) f

2 L2 ,

where f˜ is the Laplace transform to f . As claimed. ✷ Consider the equation u ¨ + B u(t) ˙ + Cu(t) = 0.

(2.1)

Lemma 9.2.2 Let W (.) be stable. Then a solution u of problem (2.1), (1.2) satisfies the inequalities u L2 ≤ ηB (w ˜0 ) and u˙ L2 ≤ ηC (w ˜0 ). Moreover, u Proof:

2 C

≤ 2ηB (w ˜0 )ηC (w ˜0 ).

Applying the Laplace transformation to equation (2.1), we have u(z) = u1 + zu0 + Bu0 , (z 2 + Bz + C)˜

where u ˜(z) is the Laplace transform to u and z is the dual variable. Hence, u(t) =

1 2π

∞ −∞

eiyt W −1 (iy)(u1 + Bu0 + iyu0 )dy =

9.3 Proof of Theorem 9.1.1

˙ G(t)u 0 + G(t)(u1 + Bu0 ).

133

(2.2)

Therefore, according to (2.1), ˙ ¨ u(t) ˙ = G(t)u 0 + G(t)(u1 + Bu0 ) = ˙ ˙ (−B G(t) − CG(t))u0 + G(t)(u 1 + Bu0 ). Take into account that ∞

1 ˙ B G(t) + CG(t) := 2π 1 2π

∞ −∞

eiyt (Biy + C)W −1 (iy)dy =

−∞

˙ + G(t)C. eiyt W −1 (iy)(Biy + C)dy = G(t)B

Hence,

˙ u(t) ˙ = G(t)u 1 − G(t)Cu0 .

(2.3) (2.4)

˜0 ). Morever, (2.4) implies the Now (2.2) yields the inequality u L2 ≤ ηB (w inequality u˙ L2 ≤ ηC (w ˜0 ). Furthermore, as it is shown in Section 1.13, for any vector-valued function h ∈ L2 ([0, ∞), Cn ) with the property h˙ ∈ L2 ([0, ∞), Cn ), we have h(t)

2

∞

≤ 2[

t

h(s) 2 ds

∞ t

2 ˙ h(s) ds]1/2 .

This inequality implies the required inequality for u

C.

✷

9.3 Proof of Theorem 9.1.1 Lemma 9.3.1 Let condition (1.3) hold with r = ∞. Then under (1.5), inequalities (1.7) and (1.8) are true. Proof:

For any h ∈ L2 , (1.3) implies F (h, t)

L2

≤q h

L2 .

(3.1)

Rewrite (1.1) as w(t) = u(t) +

t 0

G(t − s)F (w(s), s)ds,

(3.2)

where u is a solution of problem (2.1), (1.2). Due to (3.1) and Lemma 9.2.1, w

L2

≤ u

L2

+ qb(W ) w

L2 .

134

9. Lur’e Type Systems

Thus due to Lemma 9.2.2 and condition (1.5), w

≤

L2

ηB ( w ˜0 ) u L2 ≤ . 1 − qb(W ) 1 − qb(W )

So (1.8) holds. Now (3.1), (3.2), (2.5) and the Schwarz inequality yields w u

C

C

≤ u

+ qψ0 w

C

L2

+q G ≤ u

L2

C

+

w

L2

≤

˜0 ) qψ0 ηB (w . 1 − qb(W )

Hence, due to Lemma 9.2.3, we get w

C

˜0 ))1/2 + ≤ (2ηB (w ˜0 )ηC (w

qψ0 ηB (w ˜0 ) = ζ(q, w ˜0 ). 1 − qb(W ) As claimed. ✷ Proof of Theorem 9.1.1: Let r < ∞. Due to (1.6), for a small enough t0 > 0 a solution w of problem (1.1), (1.2) lies in Ω(r) for all t ≤ t0 . Applying the reasonings of the proof of the previous lemma, we get the inequality sup

0≤t≤t0

w(t) ≤ ζ(q, w ˜0 ).

But under (1.6) this inequality can be extended from [0, t0 ] to [0, ∞). This proves the theorem. ✷

10. Aizerman’s Problem for Nonautonomous Systems

10.1 Comparison of the Green Functions Let bk (t) (t ≥ 0; k = 1, ..., n) be real non-negative scalar-valued functions having continuous derivatives up to n−k-th order. Consider the gradient-type equation n−1 k d (bn−k (t)x(t)) (t ≥ 0), (1.1) P (D)x(t) = dtk k=0

where D≡

d ; P (λ) = λn + c1 λn−1 + ... + cn (ck ≡ const > 0) dt

(1.2)

is a Hurwitzian polynomial. A solution of (1.1) is a function x(.) defined on [0, ∞), having derivatives up to the n-th order, that are bounded for all finite t ≥ 0. In addition, x(.) satisfies (1.1) and the corresponding initial conditions. A scalar valued function W (t, τ ) defined for t ≥ τ ≥ 0 is the Green function to equation (1.1) if it satisfies that equation for t > τ and the initial conditions lim t↓τ

∂ n−1 W (t, τ ) ∂ k W (t, τ ) = 0 (k = 0, ..., n − 2); lim = 1. t↓τ ∂tk ∂tn−1

Put K(t) = and G(t, s) =

1 2π

∞ −∞

1 2π

∞ −∞

eiωt P −1 (iω)dω

eiωt P −1 (iω)Q(ωi, s)dω (s, t ≥ 0),

where

n

Q(z, s) =

bk (s)z n−k (z ∈ C).

k=1

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 135–144, 2005. © Springer-Verlag Berlin Heidelberg 2005

(1.3)

136

10. Aizerman’s Problem for Nonautonomous Systems

Theorem 10.1.1 Let the conditions

and

K(t) ≥ 0

(1.4)

G(t, s) ≥ 0 (t ≥ s ≥ 0)

(1.5)

hold. Then the Green function W (., .) to equation (1.1) is non-negative. Moreover, W (t, τ ) ≥ K(t − τ ) (t ≥ τ ≥ 0). (1.6) The proof of this theorem is presented in the next section.

10.2 Proof of Theorem 10.1.1 Clearly, K(t) is the Green function of the equation P (D)u(t) = 0 (t ≥ 0).

(2.1)

That is, K is a solution of (2.1) with the initial conditions K (k) (0) = 0 (k = 0, ..., n − 2); K (n−1) (0) = 1.

(2.2)

Set W (t, 0) = x(t). Then x(t) satisfies (1.1) with the same initial conditions x(k) (0) = 0 (k = 0, ..., n − 2); x(n−1) (0) = 1.

(2.3)

Due to the variation of constants formula, one can rewrite equation (1.1) in the form n

x(t) = K(t) + k=1

t

0

K(t − s)(bk (s)x(s))(n−k) ds.

(2.4)

Take into account initial conditions (2.2) and (2.3). Then, integrating by parts, we arrive at the relation t 0

K(t − s)(bn−k (s)x(s))(k) ds =

t 0

K (k) (t − s)bn−k (s)x(s)ds.

Moreover, K (k) (t) =

1 2π

∞ −∞

(ωi)k eiωt P −1 (iω)dω (k = 0, ..., n − 1).

(2.5)

Consequently, G(t, s) = K (n−1) (t)b1 (s) + K (n−2) (t)b2 (s) + ... + K(t)bn (s). Thus, from (2.4) it follows

(2.6)

10.3 Aizerman’s Type Problem

x(t) = K(t) +

t 0

G(t − s, s)x(s)ds.

137

(2.7)

Hence, the conditions (1.4) and (1.5) imply x(t) ≥ K(t) ≥ 0 (t ≥ 0). So for τ = 0 the theorem is proved. But due to the variation of constants formula, n

W (t, τ ) = K(t − τ ) + k=1

t τ

K(t − s)(bk (s)W (s, τ ))(n−k) ds.

(2.8)

Repeating our above reasonings for arbitrary τ > 0, according (2.4) we arrive at the required result. ✷

10.3 Aizerman’s Type Problem Consider the equation P (D)x(t) = (b1 (t)x(t))(n−1) + ... + bn (t)x(t) + Q(x(t), t) (t ≥ 0),

(3.1)

where Q is a continuous function defined on R × [0, ∞) and satisfying the condition (3.2) |Q(s, t)| ≤ q|s| (s ∈ R; t ≥ 0) with a constant q ≥ 0. Let x0 be the norm of the initial vector. Definition 10.3.1 We will say that equation (3.1) is absolutely stable in the class of nonlinearities (3.2), if there exists a constant M which does not depend on a concrete form of Q(., .) (but which depends on q), such that |x(t)| ≤ M x0 for all t ≥ 0 and any solution x(t) of (3.1). Problem 10.3.1: To find a class of systems of the type (3.1) that have the following property: for the absolute stability of (3.1) in the class of nonlinearities (3.2), it is sufficient that the linear equation P (D)x(t) = (b1 (t)x(t))(n−1) + ... + bn (t)x(t) + q1 x(t)

(3.3)

be asymptotically stable for some q1 ∈ [0, q]. Lemma 10.3.2 Let the Green function W (., .) to equation (1.1) be nonnegative. Then equation (3.1) is absolutely stable in the class of nonlinearities (3.2) provided (3.3) is asymptotically stable with q1 = 0 and q1 = q.

10. Aizerman’s Problem for Nonautonomous Systems

138

Proof:

Equation (3.1) is equivalent to the equation x(t) = x ˜(t) +

t 0

W (t, s)Q(x(s), s)ds

where x ˜(t) is a solution of (1.1). Clearly, (3.3) takes the form (1.1) when q1 = 0. Thus, t

|x(t)| ≤ |˜ x(t)| + q

0

|W (t, s)||x(s)|ds.

(3.4)

Since (1.1) is stable and the Green function is positive, we have |˜ x(t)| ≤ c2 ≡ const x0 and |x(t)| ≤ c2 + q

t 0

W (t, s)|x(s)|ds.

Due to Lemma 1.7.1, |x(t)| ≤ w(t) where w is the solution of the equation w(t) = c2 + q

t 0

W (t, s)w(s)ds.

But according to the variation of constants formula, this equation is equivalent to (3.3) with q1 = q. Moreover, w(0) = c2 , w(k) (0) = 0 (k = 1, ..., n − 1). Since (3.3) with q1 = q is stable, we have |x(t)| ≤ w(t) ≤ c3 c2 = M x0

(c3 ≡ const; t ≥ 0).

As claimed. ✷ The latter lemma and Theorem 10.1.1 imply Theorem 10.3.3 Let conditions (1.4) and (1.5) hold. Then equation (3.1) is absolutely stable in the class of nonlinearities (3.2), provided (3.3) is asymptotically stable with q1 = 0 and q1 = q. Clearly, the latter theorem separates a class of systems satisfying Problem 10.3.1. Below in Section 10.6 we suggest the explicit absolute stability conditions.

10.4 Equations with Purely Real Roots

139

10.4 Equations with Purely Real Roots Let polynomial P (λ) defined by (1.2) have the real roots β = λ1 ≤ λ2 ≤ .... ≤ λn = α < 0.

(4.1)

Then Lemma 1.11.2 gives us the inequality K(t) ≥

tn−1 eβt (t ≥ 0). (n − 1)!

(4.2)

Moreover, thanks to Lemma 1.11.2, the condition ∂ n−1 ezt Q(z, s) ≥ 0 (λ1 ≤ z ≤ λn ; t, s ≥ 0) ∂z n−1 implies (1.5). But eθt ≥ 0 for any real θ. So, if ∂ j Q(z, s) = ∂z j

n−j k=1

(n − k)!bk (s)z n−k−j ≥0 (n − k − j)!

(j = 0, ..., n − 1; λ1 ≤ z ≤ λn ; s ≥ 0),

(4.3)

then relation (1.5) is valid. Now Theorem 10.1.1 implies Theorem 10.4.1 Let polynomial P (λ) have the purely real roots (4.1). In addition, let condition (4.3) hold. Then the Green function W (., .) to equation (1.1) is non-negative. Moreover, W (t, τ ) ≥ K(t − τ ) ≥

(t − τ )n−1 eβ(t−τ ) (t ≥ τ ≥ 0). (n − 1)!

Let n = 2 and the polynomial P (z) = z 2 + c1 z + c2 have real roots λ1 ≤ λ2 . In this case Q(s, z) ≡ b1 (s)z + b2 (s) (λ1 ≤ z ≤ λ2 ). In addition, and

˙ G(t, s) = K(t)b 1 (s) + K(t)b2 (s) K(t) = (λ2 − λ1 )−1 (eλ2 t − eλ1 t ) if λ1 < λ2 and K(t) = teλ1 t if λ1 = λ2 .

(4.4)

So according to (4.3), the conditions Qz (s, z) ≡ b2 (s) ≥ 0 and b1 (s)z + b2 (s) ≥ 0 (λ1 ≤ z ≤ λ2 ) imply (1.5). Rewrite the latter relations as b2 (s) ≥ 0 and b1 (s)λ1 + b2 (s) ≥ 0 (s ≥ 0).

(4.5)

140

10. Aizerman’s Problem for Nonautonomous Systems

Now Theorem 10.4.1 yields Corollary 10.4.2 Let the polynomial P (z) = z 2 + c1 z + c2 have real roots λ1 ≤ λ2 and condition (4.5) hold. Then the Green function W (., .) of the equation x ¨ + c1 x˙ + c2 x = d(b1 (t)x)/dt + b2 (t)x (x = x(t)) (4.6) is non-negative. Moreover, W (t, τ ) ≥ K(t − τ ) ≥ eλ1 (t−τ ) (t − τ ) (t ≥ τ ≥ 0), where K is defined by (4.4). Note that the equation x ¨ + a1 (t)x˙ + a2 (t)x = 0 with bounded functions a1 (t), a2 (t) can be written as (4.6), if we take ck = maxt≥0 ak (t) (k = 1, 2), b1 (t) = a1 (t) − c1 , b2 (t) = a2 (t) − c2 − a1 (t), provided a1 (t) is differentiable. Consider now the third order equation d3 x d2 x dx + c1 2 + c2 + c3 x = (b2 (t)x) + b3 (t)x (t ≥ 0). 3 dt dt dt

(4.7)

So b1 (t) ≡ 0 and Q(z, s) = b2 (s)z + b3 (s). Let the polynomial P (z) = z 3 + c1 z 2 + c2 z + c3

(4.8)

λ1 ≤ λ2 ≤ λ3 .

(4.9)

b2 (s) ≥ 0 and b3 (s) + b2 (s)λ1 ≥ 0 (s ≥ 0)

(4.10)

have the real roots Then the inequalities

provide condition (4.3). Now Theorem 10.4.1 yields Corollary 10.4.3 Let conditions (4.9) and (4.10) hold. Then the Green function W (., .) to equation (4.7) is non-negative. Moreover, W (t, τ ) ≥

(t − τ )2 λ1 (t−τ ) e (t ≥ τ ≥ 0). 2

10.5 Equations with Nonreal Roots

141

10.5 Equations with Nonreal Roots Recall the following result (see Section 6.1): let the polynomial defined by (4.8) have a pair of complex conjugate roots: −γ ± iω, and a real root −z0 (z0 , γ, ω > 0). Then the function K(t) ≡

1 2πi

i∞ −i∞

is non-negative provided that

λ3

exp(tλ)dλ + c 1 λ2 + c 2 λ + c 3

γ > z0 .

(5.1)

(5.2)

Now Theorem 10.1.1 implies Corollary 10.5.1 Let the polynomial defined by (4.8) have a pair of complex conjugate roots: −γ ± iω, and a real root −z0 (z0 , γ, ω > 0). In addition, let condition (5.2) hold. Then the Green function W (t, τ ) to the equation d3 x(t) d2 x(t) dx(t) + c3 x(t) − b3 (t)x(t) = 0 (b3 (t) ≥ 0, t ≥ 0) + c + c2 1 2 2 dt dt dt is non-negative. Moreover, inequality (1.6) holds, where K is defined by (5.1). Indeed, in this case n = 3 and b1 (t) ≡ b2 (t) ≡ 0. We need only the condition K(t) ≥ 0. Now Theorem 10.1.1 imply the required result. Since the Laplace transform of a convolution is a product of the Laplace transforms, we can use the results of this section in the case n > 3.

10.6 Absolute Stability Conditions 10.6.1 Estimates for Green’s Functions In the present section the positivity of K and G(., .) is not assumed. To establish stability conditions, wee need some estimates for Green’s function of equation (1.1). Let λ1 , ..., λn be the roots of P (λ) taken with their multiplicities, α = maxk Re λk and Λ = maxk |λk |. Due to Lemma 5.7.1, |K (j) (t)| ≤ eαt ηj (t) (t ≥ 0), where

n−1

ηj (t) = k=0

j!Λj−k tn−k−1 . (j − k)!(n − k − 1)!k!

(6.1)

10. Aizerman’s Problem for Nonautonomous Systems

142

In particular,

tn−1 eαt (t ≥ 0). (n − 1)!

|K(t)| ≤

Moreover, from Corollary 5.7.3 it follows, ∞ 0

|K (j) (t)| dt ≤

(Λ + |α|)j . |α|n

(6.2)

Lemma 10.6.1 Let mk := sup |bk (t)| < ∞ (k = 1, ..., n) t≥0

and γ0 :=

1 |α|n

n−1

mj (Λ + |α|)n−j−1 < 1.

(6.3)

(6.4)

j=0

Then the Green function to equation (1.1) satisfies the estimate max

0≤τ ≤t≤∞

|W (t, τ )| ≤

ln (α) , 1 − γ0

where ln (α) = [(n − 1)!]−1 max eαt tn−1 = t≥0

Proof:

(n − 1)n−1 . (n − 1)!en−1 |α|n−1

By (2.6) and (6.3), we have

|G(t, s)| ≤ |K (n−1) (t)|m1 + |K (n−2) (t)|m2 + .. + |K(t)|mn (t, s ≥ 0). With (6.1) taken into account, we get |G(t, s)| ≤ eαt (ηn−1 (t)m1 + ... + η0 (t)mn ).

(6.5)

According to (2.8), |W (t, τ )| ≤ |K(t − τ )| +

So,

t τ

[|K (n−1) (t − s)|m1 + ...

+|K(t − s)|mn ]|W (s, τ )|ds.

(6.6)

max |W (t, τ )| ≤ max |K(t − τ )| + γ max |W (t, τ )|,

(6.7)

t≥τ ≥0

t≥τ ≥0

where γ≡

∞ 0

t≥τ ≥0

[ |K (n−1) (s)|m1 + ... + |K(s)|mn ]ds.

But due to (2.6) γ ≤ γ0 . In addition, (6.1) implies

10.6 Absolute Stability Conditions

143

max |K(t)| ≤ ln (α). t≥0

Thus,

max |W (t, τ )| ≤ ln (α) + γ0 max |W (t, τ )|.

t≥τ ≥0

t≥τ ≥0

Then, thanks to (6.4), we get the required result. ✷ Lemma 10.6.2 Under conditions (6.3) and (6.4) the Green function to equation (1.1) satisfies the inequality max t≥0

Proof:

t 0

|W (t, τ )|dτ ≤

1 . (1 − γ0 ) |α|n

According to (6.6), t 0

|W (t, τ )|dτ ≤

t 0

t

|K(t − τ )|dτ +

t

0

τ

[|K (n−1) (t − s)|m1 +

... + |K(t − s)|mn ]|W (s, τ )|ds dτ.

(6.8)

But t 0

t τ

|K (k) (t − s)| |W (s, τ )| ds dτ = ≤

t 0

s

0

0 s

|K (k) (t − s)|ds maxs≥0 ∞

0

t

|K (k) (s)|ds maxs≥0

0 s 0

|K (k) (t − s)||W (s, τ )|dτ ds |W (s, τ )|dτ ≤

|W (s, τ )|dτ.

Taking into account (6.2), from (6.8) we get max t≥0

t 0

|W (t, τ )|dτ ≤ |α|−n + γ0 max t≥0

t 0

|W (t, τ )|dτ.

Now (6.4) yields the required result. ✷

10.6.2 Absolute Stability Conditions Theorem 10.6.3 Let the conditions (6.3), (6.4) and q <1 (1 − γ0 ) |α|n

(6.9)

hold. Then equation (3.1) is absolutely stable in the class of nonlinearities (3.2).

10. Aizerman’s Problem for Nonautonomous Systems

144

Proof:

Inequality (3.4) and Lemma 10.6.2 imply x(t)| + q sup |x(s)| sup |x(t)| ≤ sup |˜ t≥0

s≥0

t≥0

1 . (1 − γ0 ) |α|n

Now the result is due to condition (6.9). ✷

10.7 Positive Solutions of Nonlinear Equations Consider the equation P (D)x(t) = (b1 (t)x(t))(n−1) + ... + bn (t)x(t)+ F (x(t), x(t), ˙ ..., x(n−1) (t), t) (t ≥ 0),

(7.1) n

where F is a scalar valued continuous function defined on R × [0, ∞). Denote n R+ ≡ {y ≡ column (y1 , ..., yn ) ∈ Rn : y1 ≥ 0}. n That is, R+ is the subset of vectors whose first coordinate is non-negative.

Lemma 10.7.1 Let the Green function W (t, τ ) to (1.1) be non-negative and n F (y, t) ≡ F (y1 , y2 , ..., yn , t) ≥ 0 (y ∈ R+ , t ≥ 0).

(7.2)

Then equation (7.1) possesses non-negative solutions. Proof: Due to the variation of constants formula, equation (7.1) with the initial conditions (2.3) can be written in the form x(t) = W (t, 0) + where

t 0

W (t, s)F (x(s), s)ds,

x(t) = column (x(t), x(t), ˙ ..., x(n−1) (t)).

Since, W (t, 0) ≥ 0, for a sufficiently small t0 we have x(t) ≥ 0 (t ≤ t0 ). So x(t) ≥ W (t, 0) ≥ 0 (t ≤ t0 ). Extending this relation to all t ≥ 0, we arrive at the required result. ✷ The latter lemma and Theorem 10.1.1 yields Theorem 10.7.2 Let conditions (1.4), (1.5) and (7.2) hold. Then equation (7.1) possesses non-negative solutions.

11. Input-to-State Stability

11.1 Definitions and Preliminaries Let us consider a system described by the following equation in a Euclidean space Cn : x(t) ˙ = Φ(x(t), u(t), t) (t ≥ 0), (1.1) where x : R+ → Cn is the state, u : R+ → Cm is the input and Φ maps Cn × Cm × R+ into Cn with the property Φ(0, 0, t) ≡ 0. Here R+ = [0, ∞). In the present chapter it is convenient for us to denote by . C n the Euclidean norm in Cn . In addition, w Lp ,n is the norm in Lp (R+ , Cn ) (1 ≤ p ≤ ∞). That is, w and

Lp ,n

=[

w

∞ 0

L∞ ,n

w(t)

p 1/p C n dt]

= ess sup w(t) t≥0

(1 ≤ p < ∞; w ∈ Lp (R+ , Cn ))

Cn

(w ∈ L∞ (R+ , Cn ))

(see also Section 1.13). Let X and U be Banach spaces of vector-valued functions defined on R+ with values in Cn and Cm , respectively. For instance, X = Lp (R+ , Cn ) (1 ≤ p ≤ ∞) and U = Lp1 (R+ , Cm ) (1 ≤ p1 ≤ ∞). Let . X and . U be the norms in X and U , respectively. Definition 11.1.1 System (1.1) is said to be input-to-state (U X)-stable, if for any > 0, there is a δ > 0, such that the conditions u U ≤ δ and x(0) = 0

(1.2)

imply x X ≤ . System (1.1) is said to be globally input-to-state (U X)-stable if the conditions (1.2) and u ∈ U imply x ∈ X. Let us consider the system x(t) ˙ = B(x(t), t)x(t) + F (x(t), u(t), t),

(1.3)

where x : [0, ∞) → Cn is the state, u : [0, ∞) → Cm is the input, again. Besides, B(h, t) for each h ∈ Cn is a variable n × n-matrix continuously

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 145–152, 2005. © Springer-Verlag Berlin Heidelberg 2005

11. Input-to-State Stability

146

dependent on its arguments, and F continuously maps Cn × Cm × [0, ∞) into Cn . It is assumed that there are constants νS , νI ≥ 0 such that F (h, z, t)

Cn

≤ νS h

+ νI z

Cn

Cm

(h ∈ Cn ; z ∈ Cm , t ≥ 0)

(1.4)

Denote by Wv (t, s) the evolution operator of the equation x(t) ˙ = B(v(t), t)x(t)

(1.5)

for an arbitrary differentiable function v : R+ → Cn . Theorem 11.1.2 Let the conditions (1.4) and t

κ0 := sup v,t

0

Wv (t, s) ds <

1 νS

hold, where the supremum is taken over all t ∈ R+ and differentiable functions v : R+ → C n . Then with X = L∞ (R+ , Cn ) and U = L∞ (R+ , Cm ) system (1.3) is globally input-to-state (U X)-stable. Proof:

Rewrite equation (1.3) in the form x(t) =

t

Wx (t, s)F (x(s), u(s), s)]ds.

0

Hence x

L∞ ,n

≤ νS x

L∞ ,n

where c(u) = νI sup t≥0

t 0

sup t≥0

t 0

Wx (t, s)

Wx (t, s)

Cn

u(s)

C n ds

+ c(u),

C m ds.

This proves the stated result. The details are left to the reader. ✷

11.2 Systems with Time-Variant Linear Parts Let us consider the system x(t) ˙ = A(t)x(t) + F (x(t), u(t), t),

(2.1)

where x : [0, ∞) → Cn is the state, u : [0, ∞) → Cm is the input, again. Besides, A(t) is a variable n × n-matrix and F continuously maps Cn × Cm × [0, ∞) into Cn . It is assumed that

11.2 Systems with Time-Variant Linear Parts

147

A(t) − A(s) ≤ q |t − s| (t, s ≥ 0; q = const > 0).

(2.2)

v = sup g(A(t)) < ∞, and ρ ≡ − sup α(A(t)) > 0.

(2.3)

In addition, t≥0

and

t≥0

n−1 j=0

v j q(j + 1) νS √ [ j+2 + j+1 ] < 1. ρ ρ j!

(2.4)

Recall that g(A) and α(A) are defined in Sections 1.5 and 1.3, respectively. Theorem 11.2.1 Let conditions (1.4) and (2.2)-(2.4) hold. Then with X = L∞ (R+ , Cn ) and U = L∞ (R+ , Cm ) system (2.1) is globally input-to-state (U X)-stable. The proof of this theorem is presented in the next section. Example 11.2.2 Consider the one contour system with a scalar input y¨ + a(t)y˙ + b(t)y = f (t, y, y) ˙ + u(t),

(2.5)

where a(t), b(t) are positive scalar-valued bounded functions with the property |a(t) − a(s)| + |b(t) − b(s)| ≤ q0 |t − s| (2.6) for all t, s ≥ 0. Besides, it is assumed that the scalar-valued function f : [0, ∞) × R2 → R satisfies the condition |f (t, y, z)|2 ≤ νS2 (|y|2 + |z|2 )

(2.7)

for all t ≥ 0; y, z ∈ R. Due to Example 1.6.1, we easily get v ≤ 1 + sup b(t). t≥0

Assume that for some ρ > 0, α(A(t)) = Re{− and

a(t) + 2

a2 (t) − b(t)} ≤ −ρ (t ≥ 0) 4

q0 2vq0 νS νS + 3 + 2 < 1. + ρ2 ρ ρ ρ

Then due to Theorem 11.2.1, system (2.5) under conditions (2.7) and (2.6) is globally input-to-state (U X) - stable with X = U = L∞ (R+ ).

148

11. Input-to-State Stability

11.3 Proof of Theorem 11.2.1 By Corollary 1.5.3 we have exp[A(τ )t]

≤ Γ (t, A(.)) (t, τ ≥ 0),

Cn

where Γ (t, A(.)) = e−ρt

n−1 k=0

(3.1)

v k tk . (k!)3/2

Rewrite equation (2.1) in the form x˙ − A(τ )x = [A(t) − A(τ )]x + F (x, u, t),

(3.2)

regarding arbitrary τ ≥ 0 as fixed. With x(0) = 0, this equation is equivalent to the following one: x(t) =

t 0

exp[(A(τ )(t − s)][(A(s)−

A(τ ))x(s) + F (x(s), u(s), s)]ds. Due to (3.1) and (3.2) the latter equality implies the relation x(t)

Cn

t

≤

0

{[q|τ − s| + νS x(s) Hence

x(t) t 0

Cn

Γ (t − s, A(.))× Cn

+ νI u(s)

≤ sup x(s) s≤t

C m }ds.

Cn ×

Γ (t − s, A(.)){q(t − s)ds + νS }ds + b(u)

where b(u) = νI sup t≥0

t 0

Γ (t − s, A(.)) u(s)

C m ds.

Simple calculations show that t 0

Γ (t − s, A(.))(t − s)ds ≤ θ1 (A)

with

n−1

θ1 (A) = k=0

(k + 1)v k √ . k!ρk+2

11.4 The Input-to-State Version of Aizerman’s Problem

149

Moreover, t 0

n−1

Γ (t − s, A(.))ds ≤ θ0 (A) ≡ k=0

Consequently,

vk √

ρk+1 k!

(t ≥ 0).

b(u) ≤ νI θ0 ess sup u(t) t≥0

By (3.2) it can be written sup x(s) s≥0

Cn

≤ sup x(s) s≥0

Due to (2.4) sup x(s) s≤t

Cn

≤

C n (qθ1

+ νθ0 ) + b(u).

b(u) . 1 − qθ1 − νθ0

This proves the stated result. ✷

11.4 The Input-to-State Version of Aizerman’s Problem Consider the system x(t) ˙ = Ax(t) + b f (s(t), u(t), t) (s(t) = cx(t), t ≥ 0),

(4.1)

where x(t) : R+ → Rn is the state, u(t) : R+ → Rm is the input, A is a constant real Hurwitz matrix, b is a real column, c is a real row, f maps R1 × Rm × R+ into R1 with the property: there are constants qS , qI ≥ 0 such that |f (s, h, t)| ≤ qS |s| + qI h

Cm

for all s ∈ R1 , h ∈ Rm and t ≥ 0.

(4.2)

In the present section we consider the following version of the Aizerman problem: to separate a class of systems such that the asymptotic stability of the linear system x(t) ˙ = Ax(t) + b q1 cx (4.3) with some q1 ∈ [0, qS ] provides the global input-to-state stability. Let Q(λ) W (λ) = c(λI − A)−1 b = P (λ) be the transfer function. Here P (λ), Q(λ) are polynomials. Besides P is monic and Hurwitzian. Recall that K(t) := is the impulse function.

1 2π

∞ −∞

exp[iωt]W (iω) dω

150

11. Input-to-State Stability

Theorem 11.4.1 Let the conditions (4.2) and K(t) ≥ 0 for all t ≥ 0.

(4.4)

hold, and let the polynomial P (λ) − qS Q(λ) be Hurwitzian. Then with X = L∞ (R+ , R) and U = L∞ (R+ , Cm ), system (4.1) is globally input-to-state (U X)-stable. The proof of this theoren is presented in the next section. It is simple to check that the Hurwitzness of the polynomials P (λ) − qS Q(λ) is equivalent to the asymptotic stability of linear system (4.3) with q1 = qS . Due to Lemma 6.1.3, under (4.4), the polynomial P (z) − qS Q(z) is Hurwitzian, if P (0) > qS Q(0) > 0. Now the previous theorem implies Corollary 11.4.2 Let the conditions (4.2), (4.4) and P (0) > qS Q(0) > 0 hold. Then with X = L∞ (R+ , R) and U = L∞ (R+ , Cm ), system (4.1) is globally input-to-state (U X)-stable. Example 11.4.3 Consider the following one contour system with a scalar input: d2 x dx dφ(x, u) + a2 x = b1 φ(x, u) + , + a1 2 dt dt dt where φ(s, u) : R2 → R1 satisfies the condition |φ(s, z)| ≤ qS |s| + qI |z| (s, z ∈ R).

(4.5)

(4.6)

Let the polynomial P (λ) = λ2 + a1 λ + a2 have real roots λ1 ≤ λ2 < 0. Then under (4.6) according to the results of Section 6.1, relation (4.4) holds. That is, if P (0) = a2 > qS Q(0) = qS b1 , then by Corollary 11.4.2, system (4.5) is globally input-to-state (U X)-stable, with X = U = L∞ (R+ , R), provided (4.6) holds. Example 11.4.4 Consider the following one contour system with a scalar input: d3 x d2 x dx + a3 x = φ(x, u), + a1 2 + a2 (4.7) 3 dt dt dt where φ(x, u) is the same as in the previous example. Let the polynomial P (λ) = λ3 + a1 λ2 + a2 λ + a3 have negative real roots. Then according to the results of Section 6.1, relation (4.4) holds. Thus, if a3 > qS , then system (4.7) is globally input-to-state (U X)-stable, with X = U = L∞ (R+ , R), provided (4.6) holds.

11.5 Proof of Theorem 11.4.1

151

11.5 Proof of Theorem 11.4.1 Equation (4.1) is equivalent to following one: x(t) = exp[At]x(0) +

t 0

exp[A(t − τ )]bf (s(τ ), u(τ ), τ )dτ.

Multiplying this equation by c, and taking into account that x(0) = 0, we have t

s(t) =

K(t − τ )f (s(τ ), u(τ ), τ )dτ,

0

since c exp[At]b = K(t). Due to (4.2), we arrive at the inequality |s(t)| ≤

t 0

K(t − τ )(qS |s(τ )| + qI u(τ )

Or |s(t)| ≤

t 0

C m )dτ.

K(t − τ )qS |s(τ )|dτ + h(t)

with the notation h(t) = qI

t 0

K(t − τ ) u(τ )

C m dτ.

Since A is a Hurwitz matrix, there are constants α0 , c1 > 0 such that 0 ≤ c exp[At]B = K(t) ≤ c1 e−α0 t (t ≥ 0). So h(t) ≤ qI c1 u where

t L∞ ,m

0

e−α0 (t−τ ) dτ ≤ b(u) (t ≥ 0),

b(u) = qI c1 α0−1 u

Consequently, |s(t)| ≤

t 0

(5.1)

L∞ ,m .

K(t − τ )qS |s(τ )|dτ + b(u).

From Lemma 1.7.1 it follows that |s(t)| ≤ η(t),

(5.2)

where η is the solution of the equation η(t) = b(u) +

t 0

K(t − τ )qS η(τ )dτ.

We will solve this equation by the Laplace transformation. Since the transform of the convolution is equal to the product of the transforms, after

152

11. Input-to-State Stability

straightforward calculations and taking into account that P −1 (λ)Q(λ) is the Laplace transform of K(t), we get the equation η(λ) =

b(u) + P −1 (λ)Q(λ)qS η(λ), λ

where η(λ) is the Laplace transform of η(t). Thus, η(t) =

1 2πi

i∞ −i∞

exp[λt](P (λ) − qS Q(λ))−1 P (λ)λ−1 dλ b(u).

Since P (λ) − qS Q(λ) is a Hurwitz polynomial, the function under the integral has no the poles in the right open half-plane, and has one simple pole on the imaginary axis. So, thanks to the residue theorem η(t) ≤ const b(u) (t ≥ 0). Consequently,

η(t) ≤ const u

This and (5.2) prove the result. ✷

L∞ ,m

(t ≥ 0).

12. Orbital Stability and Forced Oscillations

12.1 Global Orbital Stability Let us consider a system described by the equation x(t) ˙ = A(t)x(t) + F (x(t), t),

(1.1)

where A(t) is a variable n × n-matrix, and F maps Cn × [0, ∞) into Cn . Definition 12.1.1 We will say that system (1.1) is globally exponentially orbitally stable, if there exist constants N, > 0, such that x1 (t) − x2 (t) ≤ N exp(− t) x2 (0) − x1 (0)

(t ≥ 0)

for arbitrary solutions x1 (t), x2 (t) of (1.1). Here . means the Euclidean norm in Cn , again. It is assumed that there are constants q and ν such that A(t) − A(s) ≤ q |t − s| (t, s ≥ 0), and

(1.2)

F (h1 , t) − F (h2 , t) ≤ ν h1 − h2 (h1 , h2 ∈ Cn ; t ≥ 0).

(1.3)

Furthermore, it is supposed that v = sup g(A(t)) < ∞ and ρ ≡ − sup α(A(t)) > 0, t≥0

and

t≥0

n−1 j=0

v j q(j + 1) ν √ [ j+2 + j+1 ] < 1. ρ ρ j!

(1.4)

(1.5)

Theorem 12.1.2 Let conditions (1.2)-(1.5) hold. Then system (1.1) is globally exponentially orbitally stable

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 153–158, 2005. © Springer-Verlag Berlin Heidelberg 2005

12. Orbital Stability and Forced Oscillations

154

Proof:

Let x1 and x2 be arbitrary solutions of (1.1): x˙k = A(t)xk + F (xk , t) (k = 1, 2; t ≥ 0).

(1.6)

Hence, taking the difference of this equations with k = 1 and k = 2, we get x˙ = A(t)x + Φ(x, t) where

(1.7)

x = x2 − x1 , Φ(x, t) = F (x1 + x, t) − F (x1 , t).

According to (1.3) Φ(x, t) ≤ q x (x ∈ Cn , t ≥ 0). Now applying to this equation Corollary 7.2.2, we arrive at the required result. ✷

12.2 One Contour Systems Consider the equation y˙ = Ay + b f (s, t) (s = cy, t ≥ 0),

(2.1)

where A is a real constant Hurwitz n × n-matrix, b is a real column, c is a real row, f maps R1 × [0, ∞) into R1 . Let us assume that f (s, t) is continuous in t and has the Lipschitz property in the first argument: |f (s1 , t) − f (s2 , t)| ≤ ν|s1 − s2 | for all s1 , s2 ∈ R1 and t ≥ 0.

(2.2)

Certainly, under this condition, equation (2.1) has solutions for all t ≥ 0. Let W (λ) be the transfer function of the linear part of system (2.1): W (λ) = c(λI − A)−1 b = P −1 (λ)L(λ) (λ ∈ C), where P (λ) and L(λ) are polynomials, again. In addition, P (λ) = λn + a1 λn−1 + ... + an is a Hurwitz polynomial and n ≡ deg P (λ) > m ≡ deg L(λ). Besides, P (λ) and L(λ) have no common roots. Let ∞ 1 K(t) ≡ exp[iωt]W (iω) dω (t ≥ 0) 2π −∞ be the impulse function of the linear part of (2.1).

12.2 One Contour Systems

155

Theorem 12.2.1 Let the conditions (2.2) and K(t) ≥ 0 for all t ≥ 0

(2.3)

be fulfilled. Then for the global orbital exponential stability of (2.1) it is sufficient that the polynomial P (λ) − νL(λ) be Hurwitzian. This theorem is proved in the next section. Under (2.3), thanks to Lemma 6.1.3, the polynomial P (λ) − qL(λ) is a Hurwitz one, provided P (0) > νL(0). (2.4) This relation and Theorem 12.2.1 imply Corollary 12.2.2 Let conditions (2.2)-(2.4) be fulfilled. Then equation (2.1) is globally exponentially orbitally stable. Combining this corollary with Lemma 6.1.5, we get Corollary 12.2.3 Let all roots of P (λ) belong to a real segment [a, b] (b ≤ 0). In addition, let L(k) (λ) ≥ 0 (a ≤ λ ≤ b; k = 0, ..., degL(λ)). Then equation (2.1) under (2.2) is is globally exponentially orbitally stable, provided inequality (2.4) is fulfilled. In particular, let L(λ) ≡ 1, and let all the roots of P (λ) be real. Then equation (2.1) is globally orbitally stable if P (0) = an > ν. With a real Hurwitz polynomial P3 (λ) = λ3 + a1 λ2 + a2 λ + a3 let us consider the system P3 (D)x = f (x, t) (D = d/dt).

(2.5)

Thanks to Lemma 6.1.7, we get Corollary 12.2.4 Let P3 (λ) have a pair of complex conjugate roots: −γ ±iω, and a real root −r (r, γ, ω > 0). In addition, let γ > r. Then equation (2.5) is globally orbitally exponentially stable, provided the relations (2.2) and a3 > ν hold.

156

12. Orbital Stability and Forced Oscillations

12.3 Proof of Theorem 12.2.1 Let y1 and y2 be arbitrary solutions of (2.1): y˙k = Ayk + b f (sk , t) (sk = cyk ; k = 1, 2; t ≥ 0).

(3.1)

Hence, taking the difference of this equations with k = 1 and k = 2, we get x˙ = Ax + b Φ(cx, t)

(3.2)

where x = y2 − y1 , Φ(cx, t) = f (cs2 , t) − f (cs1 , t) = f (cy1 + cx, t) − f (cy1 , t). According to (2.2)

|Φ(s, t)| ≤ ν|s| (s ∈ R, t ≥ 0).

(3.3)

Equation (3.2) is equivalent to following one: x(t) = exp[At]x(0) +

t 0

exp[A(t − τ )]bΦ(cx(τ ), τ )dτ.

Multiplying this equation by c and taking into account that c exp[At]b = K(t), we have

where

t

K(t − τ )Φ(s(τ ), τ )dτ,

(3.4)

s(t) = cx(t) and h(t) = cexp[At]x(0).

(3.5)

s(t) = h(t) +

0

Due to (3.3), we arrive at the inequality |s(t)| ≤ |h(t)| +

t 0

K(t − τ )ν|s(τ )|dτ.

Since A is a Hurwitz matrix, there are constants α0 , c1 > 0 such that |h(t)| ≤ c1 e−α0 t (t ≥ 0). In addition,

c1 ≤ a1 x(0)

Rn

(a1 = const)

(3.6)

Consequently, |s(t)| ≤ c1 e−α0 t +

t 0

K(t − τ )ν|s(τ )|dτ.

From the Lemma 1.7.1 it follows that |s(t)| ≤ η(t),

(3.7)

12.4 Existence and Stability of Forced Oscillations

157

where η is the solution of the equation η(t) = c1 e−α0 t +

t 0

K(t − τ )νη(τ )dτ.

We will solve this equation by the Laplace transformation. Since the transform of the convolution is equal to the product of the transforms, after straightforward calculations and taking into account that P −1 (λ)L(λ) is the Laplace transform of K(t), we get the equation η(λ) =

c1 + P −1 (λ)L(λ)νη(λ), λ + α0

where η(λ) is the Laplace transform of η(t). Thus, η(t) = c1

1 2πi

i∞ −i∞

exp[λt](P (λ)−

νL(λ))−1 P (λ)(λ + α0 )−1 dλ. Since P (λ) − νL(λ) is a Hurwitz polynomial, all the poles of the function under the integral lie in the open left half-plane. So, thanks to the residue theorem there is an > 0, such that η(t) ≤ c1 c2 e− t (c2 = const, t ≥ 0). Now the relations (3.6) and (3.7) yield the required result. ✷

12.4 Existence and Stability of Forced Oscillations Theorem 12.4.1 Under conditions (2.2)-(2.4), let f (s, t) be T -periodic in t: f (s, t) = f (s, t + T ) for all s ∈ R1 and t ≥ 0.

(4.1)

Then system (2.1) is globally orbitally exponentially stable and has a T periodic solution. Proof:

Since f is continuous, from (2.2) it follows that |f (s, t)| ≤ ν|s| + l (s ∈ R; l = const > 0).

(4.2)

Then due to the well-known Theorem 26.1 (M. Krasnosel’skii et al, 1989), under the condition (4.3) ν max |W (ikT )| < 1 k=1,2,...

equation (2.1) has at least one periodic solution. As it is proved in Section 6.1, condition (2.3) implies

158

12. Orbital Stability and Forced Oscillations

max |W (iω)| = W (0). ω∈R

So conditions (2.3) and (2.4) imply (4.3). Thus the existence of the periodic solution is proved. The global exponential stability follow from Theorem 12.2.1. ✷ Clearly, the periodic solution is nontrivial if supt |f (0, t)| > 0.

12.5 Examples Example 12.5.1 Consider the second order equation dx df (x, t) d2 x + a2 x = b1 f (x, t) + + a1 dt2 dt dt (x = x(t), a1 , a2 , b1 = const > 0).

(5.1)

2

Let the polynomial P (λ) = λ + a1 λ + a2 have real roots λ1 ≤ λ2 < 0. Then under the condition b1 + λ 1 ≥ 0 (5.2) we have L(λ) = λ + b1 ≥ 0, Thus, if the conditions (5.2) and

d L(λ) = 1 (λ1 ≤ λ ≤ λ2 ). dλ

P (0) = a2 > νL(0) = νb1

(5.3)

hold, then by Corollary 12.2.3, the equation (5.1) under (2.2) is globally exponentially orbitally stable. If, in addition, f is periodic in t, then due to Theorem 12.4.1, equation (5.1) has a periodic solution. Example 12.5.2 Consider the equation d2 x dx d3 x df (x, t) + a3 x = b1 f (x, t) + + a + a2 1 3 2 dt dt dt dt (a1 , a2 , a3 , b1 = const > 0).

(5.4)

Let the polynomial P (λ) = λ3 + a1 λ2 + a2 λ + a3 have real roots λ1 ≤ λ2 ≤ λ3 < 0. Then under condition (5.2), relations L(λ) = λ + b1 ≥ 0, hold. That is, if

d L(λ) = 1 (λ1 ≤ λ ≤ λ3 ) dλ

P (0) − νL(0) = a3 − νb1 > 0

(5.5)

then by Corollary 12.2.3, equation (5.4) under (2.1) is globally exponentially orbitally stable. If, in addition, f is periodic in t, then due to Theorem 12.4.1 under (2.2), (5.2) and (5.5), equation (5.4) has a periodic solution.

13. Existence of Steady States. Positive and Nontrivial Steady States

13.1 Systems of Semilinear Equations Let us consider in Cn the nonlinear equation Ax = F (x),

(1.1)

where A is an invertible matrix, F continuously maps Ω(r) into Cn for a positive r ≤ ∞. Recall that . is the Euclidean norm and Ω(r) = {x ∈ Cn : x ≤ r}. Assume that there are positive constants q and l, such that F (h) ≤ q h + l (h ∈ Ω(r)).

(1.2)

Lemma 13.1.1 Under condition (1.2), let A−1 (qr + l) ≤ r.

(1.3)

Then equation (1.1) has at least one solution x ∈ Ω(r), satisfying the inequality A−1 l . (1.4) x ≤ 1 − q A−1 Proof:

Set

Ψ (y) = A−1 F (y) (y ∈ Cn ).

Hence, Ψ (y) ≤ A−1 (q y + l) ≤ A−1 (qr + l) ≤ r (y ∈ Ω(r)).

(1.5)

So due to the Brouwer Fixed Point Theorem, equation (1.1) has a solution. Moreover, due to (1.3), A−1 q < 1. Now, using (1.5), we easily get (1.4). ✷ Put

n−1

R(A) =

g k (A) √

dk+1 (A) k=0 0

k!

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 159–166, 2005. © Springer-Verlag Berlin Heidelberg 2005

13. Existence of Steady States. Positive and Nontrivial Steady States

160

where g(A) is defined in Section 1.5, d0 (A) is the lower spectral radius. That is, d0 (A) is the minimum of the absolute values of the eigenvalues λ1 (A), ..., λn (A) of A: d0 (A) := Due to Lemma 1.5.6,

min |λk (A)|.

k=1,...,n

A−1 ≤ R(A).

Now the previous lemma implies Theorem 13.1.2 Under condition (1.2), let R(A)(qr + l) ≤ r. Then equation (1.1) has at least one solution x ∈ Ω(r), satisfying the inequality R(A) l . x ≤ 1 − qR(A)

13.2 Fully Nonlinear Systems Consider the coupled system n

ajk (x)xk = fj k=1

(j = 1, ..., n; x = (xj )nj=1 ∈ Cn ), where

(2.1)

ajk : Ω(r) → C (j, k = 1, ..., n)

are continuous functions and f = (fj ) ∈ Cn is given. We can write out system (2.1) in the form A(x)x = f (2.2) with the matrix

A(z) = (ajk (z))nj,k=1 (z ∈ Ω(r)).

Theorem 13.2.1 Let inf d0 (A(z)) ≡ inf min |λk (A(z))| > 0

z∈Ω(r)

z∈Ω(r)

and

n−1

θr := sup

z∈Ω(r) k=0

√

k

g k (A(z)) r . ≤ f k!dk+1 (A(z)) 0

(2.3)

Then system (2.1) has at least one solution x ∈ Ω(r), satisfying the estimate x ≤ θr f .

(2.4)

13.2 Fully Nonlinear Systems

Proof:

161

Tanks to Lemma 1.5.6, A−1 (z) ≤ θr (z ∈ Ω(r)).

Rewrite (2.2) as Due to (2.3)

x = Ψ (x) ≡ A−1 (x)f.

(2.5)

Ψ (z) ≤ θr f ≤ r (z ∈ Ω(r)).

So Ψ maps Ω(r) into itself. Now the required result is due to the Brouwer Fixed Point theorem. ✷ Corollary 13.2.2 Let matrix A(z) be normal: A∗ (z)A(z) = A(z)A∗ (z) (z ∈ Ω(r)). If, in addition,

f ≤ r inf d0 (A(z)), z∈Ω(r)

(2.6)

then system (2.1) has at least one solution x satisfying the estimate f . inf z∈Ω(r) d0 (A(z))

x ≤

Indeed, if A(z) is normal, then g(A(z)) ≡ 0 and θr =

1 . inf z∈Ω(r) d0 (A(z))

Corollary 13.2.3 Let matrix A(z) be upper triangular: n

ajk (x)xk = fj (j = 1, ..., n).

(2.7)

k=j

In addition with the notations |ajk (z)|2 ,

τ (A(z)) := 1≤j
and

a ˜(z) :=

let

n−1

sup

z∈Ω(r) k=0

min |ajj (A(z)|,

j=1,...,n

τ k (A(z)) r √ ≤ . f k!˜ ak+1 (z)

Then system (2.7) has at least one solution x ∈ Ω(r).

(2.8)

13. Existence of Steady States. Positive and Nontrivial Steady States

162

Indeed, this result is due to Theorem 13.2.1, since the eigenvalues of a triangular matrix are its diagonal entries, and g(A(z)) = τ (A(z)) and d0 (A(z)) = a ˜(z). The similar result is true for lower triangular systems. Note that according to the relations g(A0 ) ≤

1/2N (A∗0 − A0 )

for any constant matrix A0 (see Section 1.5), in the general case, g(A(z)) can be replaced by the simple calculated quantity 1/2N (A∗ (z) − A(z)) = [

v(A(z)) =

n

|ajk (z) − akj (z)|2 ]1/2 .

(2.9)

k=1

Example 13.2.4 Let us consider the system aj1 (x)x1 + aj2 (x)x2 = fj (x = (x1 , x2 ) ∈ C2 ), where fj are given numbers, and continuous scalar-valued functions ajk (j, k = 1, 2) are defined on Due to (2.9)

Ω(r) ≡ {z ∈ C2 : z ≤ r}. g 2 (A(z)) ≤ |a21 (z) − a12 (z)|2 .

In addition, λ1,2 (A(z)) are the roots of the polynomial y 2 − t(z)y + b(z), where and Then

t(z) = T race (A(z)) = a11 (z) + a22 (z) b(z) = det (A(z)) = a11 (z)a22 (z) − a12 (z)a21 (z). d0 (A(z)) := min |λk (A(z))|. k=1,2

Moreover, θr ≤ θ˜r := sup

˜ z∈Ω(r)

If

1 |a21 (z) − a12 (z)| + . d0 (A(z)) d20 (A(z)) f θ˜r ≤ r,

then due to Theorem 13.2.1, system (2.10) has a solution x ∈ Ω(r).

(2.10)

13.3 Nontrivial Steady States

163

13.3 Nontrivial Steady States Consider the system Fj (x) = 0

(j = 1, ..., n; x = (xj )nj=1 ∈ Cn ),

(3.1)

where functions Fj : Ω(r) → C admit the representation n

Fj (x) = φj (x) (

ajk (x)xk − fj ) (j = 1, ..., n).

(3.2)

k=1

Here φj : Ω(r) → C are functions with the property φj (0) = 0, ajk : Ω(r) → C are continuous functions, and fj (j, k = 1, ..., n) are given numbers. In the sequel it is assumed that at least one of the numbers fj is non-zero. Then from (3.2) it follows that A(x)

x ≥ f > 0.

for any solution x ∈ Ω(r) of (2.2) (if it exists). So x is non-trivial. Obviously, (3.1) has the trivial (zero) solution. Moreover, if (2.1) has a solution x ∈ Ω(r), then x is simultaneously a solution of (3.1). Now Theorem 13.2.1 implies Theorem 13.3.1 Let condition (2.3) hold. Then system (3.1) under (3.2) has at least two solutions: the trivial solution and a nontrivial one satisfying estimate (2.4). In addition, the previous theorem and Corollary 13.2.2 yield Corollary 13.3.2 Let matrix A(z) be normal for any z ∈ Ω(r) and condition (2.6) hold. Then system (3.1) under (3.2) has at least two solutions: the trivial solution and a nontrivial one belonging to Ω(r). Theorem 13.3.1 and Corollary 13.2.3 imply Corollary 13.3.3 Let matrix A(z) be upper triangular for any z ∈ Ω(r). Then under conditions (3.2) and (2.8), system (3.1) has at least two solutions: the trivial solution and a nontrivial one belonging to Ω(r). The similar result is valid if A(z) is lower triangular. Example 13.3.4 Let us consider the system φj (x1 , x2 )(aj1 (x)x1 +aj2 (x)x2 −fj ) = 0 (j = 1, 2; x = (x1 , x2 ) ∈ C2 ) (3.3) where functions φj : Ω(r) → C have the property φj (0, 0) = 0. In addition, fj , ajk (j, k = 1, 2) are the same as in Example 13.2.4. Assume that at least one of the numbers f1 , f2 is non-zero. Then due to Theorem 13.3.1, under conditions (2.3) system (3.3). has in Ω(r) at least two solutions.

164

13. Existence of Steady States. Positive and Nontrivial Steady States

13.4 Positive Steady States Consider the coupled system n

uj −

ajk (u)uk = Fj (u) (j = 1, ..., n),

(4.1)

k=1, k=j

where

ajk , Fj : Ω(r) → R (j = k; j, k = 1, ..., n)

are continuous functions. For instance, the coupled system n

wjk (u)uk = fj (j = 1, ..., n)

(4.2)

k=1

where fj are given real numbers, wjk : Ω(r) → R are continuous functions, can be reduced to (4.1) with ajk (u) ≡ − and Fj (u) ≡ provided

wjk (u) wjj (u)

fj , wj (u)

wjj (z) = 0 (z ∈ Ω(r); j = 1, ..., n).

(4.3)

We can write out system (4.1) in the form (1.1) with A(u) = (ajk (u))nj,k=1 , F (u) = column (Fj (u))nj=1 . Put

cr (F ) = sup

z∈Ω(r)

F (z) .

Let V+ (z) and V− (z) be the upper triangular, lower triangular parts of matrix A(z), respectively: 

0  0 V+ (z) =   . 0

a12 (z) 0 ... 0

... ... . ...

  0 a1n (z)  a21 (z) a2n (z)   , V− (z) =    . . 0 an1 (z)

... ... ... ...

0 0 .

an,n−1 (z)

Recall that N (A) is the Frobenius norm of a matrix A. So N 2 (V+ (z)) =

n−1

n

a2jk (z), N 2 (V− (z)) =

j=1 k=2

j=1 k=j+1

Put ˜ ± (z)) ≡ J(V

n j−1

n−1 k=0

N k (V± (z)) √ . k!

a2jk (z).

 0 0  .  0

165

13.5 Systems with Differentiable Entries

Theorem 13.4.1 Let the conditions αr ≡ max{ inf ( z∈Ωr

inf (

z∈Ωr

and

1 − V+ (z) ), ˜ J(V− (z))

1 − V− (z) )} > 0 ˜ + (z)) J(V cr (F ) < rαr (Ωr )

(4.4) (4.5)

hold. Then system (4.1) has at least one solution u ∈ Ω(r) satisfying the inequaly αr (Ωr ) u ≤ cr (F ). (4.6) In addition, let

ajk (z) ≥ 0 and Fj (z) ≥ vj

cr (F ) ; j = k; j, k = 1, ..., n) αr (Ωr ) for some non-negative constants vj . Then u is non-negative (z ∈ Rn : z ≤

(4.7)

For the proof see (Gil’, 2003c).

13.5 Systems with Differentiable Entries Consider the system fk (y1 , ..., yn ) = hk ∈ C1 , (k = 1, ...., n) where fj (x) = fj (x1 , x2 , ..., xn ), fj (0) = 0 (j = 1, ..., n) are scalar-valued differentiable functions defined and continuous with their derivatives on Cn . Put F (x) = (fj (x))nj=1 and F (x) ≡ (

∂fi (x) n ) . ∂xj i,j=1

That is, F (x) is the Jacobian matrix. Rewrite the considered system as F (y) = h ∈ Cn ,

(5.1)

For a positive number r ≤ ∞ assume that ρ0 (r) ≡ min d0 (F (x)) = min x∈Ω(r)

and

x∈Ω(r)

min |λk (F (x))| > 0, k

g˜0 (r) = max g(F (x)) < ∞. x∈Ω(r)

Finally put

n−1

√

p(F, r) ≡ k=0

g˜0k (r) . k!ρk+1 (r) 0

(5.2) (5.3)

166

13. Existence of Steady States. Positive and Nontrivial Steady States

Theorem 13.5.1 Let fj (x) = fj (x1 , x2 , ..., xn ), fj (0) = 0 (j = 1, ..., n) be scalar-valued differentiable functions, defined and continuous with their derivatives in Ω(r). Assume that conditions (5.2) and (5.3) hold. Then for any h ∈ Cn with the property h ≤

r , p(F, r)

there is a solution y ∈ Cn of system (5.1) which subordinates to the inequality y ≤

h . p(F, r)

For the proof of this result see (Gil’, 1996).

Appendix A. Bounds for Eigenvalues of Matrices

Although excellent computer software are now available for eigenvalue computation, analytical results on spectrum inclusion regions for finite matrices are still important, since computers are not very useful, in particular, for analysis of matrices dependent on parameters. Here we present some well-known bounds for eigenvalues. Let A = (ajk ) be a real n × n-matrix (n ≥ 2) with the nonzero diagonal: akk = 0 (k = 1, ..., n). Put n

Pj =

|ajk |. k=1, k=j

A1. In (Marcus and Minc, 1964, Section 3.2.2) it is shown that the eigenvalues of A lie in the union of the sets {λ ∈ C : |λ − ajj | ≤ Pj } (j = 1, ..., n). A2. In (Marcus and Minc, 1964, Section 3.2.4), it is shown that the spectrum of A lies in the sets {λ ∈ C : |aii − λ||ajj − λ| ≤ Pj Pi } (i, j = 1, ..., n). A3. Let V+ , V− and D be the upper nilpotent diagonal of A, respectively:    0 a12 . . . a1n  0 0 . . . a2n    , V− =  V+ =   . ...  . .  0 0 ... 0 and

part, lower nilpotent part and 0 a21 . an1

... ... ... ...

0 0 .

an,n−1

D = diag (a11 , a22 , ..., ann ).

Put

n−1

wn =

γn,j , j=0

where γn,j are defined in Section 1.5. One can replace wn by , M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 167–169, 2005. © Springer-Verlag Berlin Heidelberg 2005

 0 0    0

168

Appendix A. Bounds for Eigenvalues of Matrices n−1

w ˜n = j=0

1 √ . k!

Let V+ = 0, V− = 0 and assume that wn min {

V− V+ , } ≤ 1. N (V+ ) N (V− )

Here . is the Euclidean norm and N (.) is the Frobenius (Hilbert-Schmidt) norm. Put δ2 (A) = wn1/n min {N 1−1/n (V+ ) V−

1/n

, N 1−1/n (V− ) V+

1/n

}.

Then all the eigenvalues of matrix A = (ajk )nj,k=1 lie in the union of the discs {λ ∈ C : |λ − akk | ≤ δ2 (A)}, k = 1, ..., n.

(1)

For the proof see (Gil’, 2003, Theorem 4.5.1). Thus A is a Hurwitz matrix, provided akk + δ2 (A) < 0, k = 1, ..., n. A4. Recall that g(A) is defined in Section 1.5. Denote by z(ν) the unique positive root of the equation z n (A) = ν(A)

n−1

g k (A)γn,k z n−k−1 ,

k=0

where

ν(A) = min{ V− , V+ }.

Here one can replace γn,k by eigenvalue µ0 of A, such that

√1 . k!

Then for any k = 1, ..., n, there is an

|µ0 − akk | ≤ z(ν).

(2)

Moreover, the following inequalities are true: α(A) ≥ max

k=1,...,n

akk − z(ν).

and z(ν) ≤ ∆(A), where ∆(A) =

ν(A)wn g 1−1/n (A)[ν(A)wn ]1/n

For the proofs see (Gil’, 2003, Section 4.6).

if ν(A)wn ≥ g(A), . if ν(A)wn ≤ g(A)

(3)

Appendix A. Bounds for Eigenvalues of Matrices

169

A5. Let us point bounds of the other type. Denote j−1

qup = max

j=2,...,n

v˜k := w ˜k :=

k=1

max

j=1,...,k−1

max

j=k+1,...,n

n

mup (A) :=

n

|ajk |, qlow =

(1 + k=2

max

j=1,...,n−1

|ajk |, j+1

|ajk | (k = 2, ..., n),

|ajk | (k = 1, ..., n − 1),

v˜k ) and mlow (A) := |akk |

n−1

(1 + k=1

w ˜k ). |akk |

Without losing of generality, assume that min {qlow mup (A), qup mlow (A)} ≤ 1. Then all the eigenvalues of A lie in the union of the discs {λ ∈ C : |λ − akk | ≤ δ∞ (A)}, k = 1, ..., n, where δ∞ (A) :=

n

min{qlow mup (A), qup mlow (A)}.

For the proof see (Gil’, 2003, Theorem 4.7.1). Under (4), let maxj ajj + δ∞ (A) < 0. Then A is a Hurwitz matrix.

(4)

Appendix B. Positivity of the Green Function

Equations with Positive “Characteristic” Roots In this appendix the Cauchy problem for a higher order linear nonautonomous ODE on the positive half-line is considered. Explicit conditions for the positivity of the Green function and its derivatives are derived. Moreover, lower estimates for the Green function are established. Let ak (t) (t ≥ 0; k = 1, ..., n) be real continuous scalar-valued functions defined and bounded on [0, ∞), and a0 ≡ 1. Consider the equation n

(−1)k ak (t)

k=0

dn−k u(t) = 0 (t > 0). dtn−k

(1.1)

A solution of (1.1) is a function x(.) defined on [0, ∞), having continuous derivatives up to the n-th order. In addition, x(.) satisfies (1.1) for all t > 0 and the corresponding initial conditions. A scalar valued function W (t, τ ) defined for t ≥ τ ≥ 0 is the Green function to equation (1.1) if it satisfies that equation for t > τ and the initial conditions lim t↓τ

Put

∂ n−1 W (t, τ ) ∂ k W (t, τ ) = 1. = 0 (k = 0, ..., n − 2); lim k t↓τ ∂tn−1 ∂t

c2k = sup a2k (t), c2k−1 = inf a2k−1 (t) (k = 1, ..., [n/2]), t≥0

t≥0

(1.2)

where [x] is the integer part of x > 0. Theorem 15.1.1 Let all the roots of the polynomial n

Q(z) =

(−1)k ck z n−k (c0 = 1, z ∈ C)

k=0

be real and nonnegative. Then the Green function to equation (1.1) and its derivatives up to (n − 1)-th order are nonnegative. Moreover, ∂ j W (t, τ ) ≥ er1 (t−τ ) ∂tj

j k=0

Cjk

r1j−k (t − τ )n−1−k ≥0 (n − 1 − k)!

, M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 171–181, 2005. © Springer-Verlag Berlin Heidelberg 2005

172

Appendix B. Positivity of the Green Function

(j = 0, ..., n − 1; t > τ ≥ 0), where r1 ≥ 0 is the smallest root of Q(z) and

Cjk

=

(1.3)

j! (j−k)!k! .

Below we prove this theorem and remove the condition r1 ≥ 0.

Proof of Theorem 15.1.1 Lemma 15.2.1 Let all the roots of polynomial Q(z) be real and nonnegative. Then a solution u of (1.1) with the initial conditions u(j) (0) = 0, j = 0, ..., n − 2; u(n−1) (0) = 1

(2.1)

satisfies the inequalities u(j) (t) ≥ er1 t

j k=0

Proof:

Ckj

r1j−k tn−1−k ≥ 0 (j = 0, ..., n − 1; t > 0). (n − 1 − k)!

We have bk (t) := (−1)k (ck − ak (t)) ≥ 0 (k = 1, ..., n).

Rewrite equation (1.1) in the form n

(−1)k ck

k=0

dn−k u = dtn−k

Denote G(t) =

1 2iπ

n

bk (t) k=1

C

dn−k u . dtn−k

(2.2)

ezt dz , Q(z)

where C is a smooth contour surrounding all the zeros of Q(z). That is, G is the Green functions to the autonomous equation n

(−1)k ck

k=0

Put

n

v(t) ≡

dn−k w(t) = 0. dtn−k

(−1)k ck

k=0

dn−k u(t) . dtn−k

Then thanks to the variation of constants formula, u(t) = w(t) +

t 0

G(t − s)v(s)ds,

where w(t) is a solution of equation (2.3). Since

(2.3)

(2.4)

Appendix B. Positivity of the Green Function

G(j) (t) =

1 2iπ

C

173

z j ezt dz , (z − r1 )...(z − rn )

where r1 ≤ ... ≤ rn are the roots of Q(z) with their multiplicities, due to Lemma 1.11.2 of the present book, we get G(j) (t) =

dn−1 z j ezt 1 [ ]z=θ dz n−1 (n − 1)!

with a θ ∈ [r1 , rn ]. Hence, G

(j)

j

(t) = k=0

j!eθt θj−k tn−1−k ≥ (j − k)!(n − 1 − k)!k!

j k=0

j!er1 t r1j−k tn−1−k ≥ 0. (2.5) (j − k)!(n − 1 − k)!k!

According to the initial conditions (2.1), we can write out w(t) = G(t). So t

u(t) = G(t) +

0

G(t − s)v(s)ds.

(2.6)

Substitute this relation into (2.2). For j ≤ n − 2 we have G(j) (0) = 0 and dj dtj

t 0

G(t − s)v(s)ds = t 0

d dt

t 0

G(j−1) (t − s)v(s)ds =

G(j) (t − s)v(s)ds (j = 1, ..., n − 1).

Hence thanks to (2.2) and (2.4), n

v(t) =

bk (t)[G(n−k) (t) +

k=1 t

K(t, t) + where

n

K(t, τ ) =

0

t 0

G(n−k) (t − s)v(s)ds] =

K(t, t − s)v(s)ds,

(2.7)

bk (t)G(n−k) (τ ) (t, τ ≥ 0).

k=1

According to (2.5), K(t, τ ) ≥ 0 (t, τ ≥ 0). Put h(t) = K(t, t). Let V be the Volterra operator with the kernel K(t, t − s). Then thanks to (2.7) and the Neumann series, ∞

v(t) = h(t) + k=1

(V k h)(t) ≥ h(t) ≥ 0.

174

Appendix B. Positivity of the Green Function

Hence (2.6) yields, u(j) (t) = G(j) (t) + G(j) (t) +

t 0

t 0

G(j) (t − s)v(s)ds ≥

G(j) (t − s)K(s, s)ds ≥ G(j) (t) (j = 1, ..., n − 1).

(2.8)

This and (2.5) prove the lemma. ✷ Proof of Theorem 15.1.1: For a τ > 0, take the initial conditions u(j) (τ ) = 0, j = 0, ..., n − 2; u(n−1) (τ ) = 1. Then the corresponding solution u(t) to (1.1) is equal to W (t, τ ). Repeat the arguments of the proof of Lemma 15.2.1. Then instead of (2.8) we have ∂ j W (t, τ ) ≥ G(j) (t − τ ) + ∂tj

t τ

G(j) (t − τ − s)K(s, s − τ )ds ≥ G(j) (t − τ ).

According to (2.5) this proves the theorem. ✷

Equations with Arbitrary Real “Characteristic” Roots In this section we do not assume that the roots of Q(z) are nonnegative. Namely, let pk (t) (k = 1, ..., n) be real continuous functions bounded on [0, ∞), and p0 ≡ 1. Consider the equation n

pk (t) k=0

dn−k x = 0 (t > 0). dtn−k

(3.1)

Let the polynomial n

P (t, z) =

pn−k (t)z k (z ∈ C)

k=0

have the purely real roots ρk (t) (k = 1, ..., n) with the property ρk (t) ≥ −µ (t ≥ 0; k = 1, ..., n) with some µ > 0. Put x(t) = e−µt u(t) in (3.1). Then 0 = eµt

n

pk (t) k=0

dn−k e−µt u = dtn−k

n

pk (t)( k=0

d − µ)n−k u. dt

(3.2)

Appendix B. Positivity of the Green Function

175

That is, equation (3.1) is reduced to the equation P (t,

d − µ)u ≡ dt

n

pk (t)( k=0

d − µ)n−k u = 0. dt

(3.3)

But n

P (t, z − µ) = n

pk (t) k=0

m=k

n−k

pk (t) j=0

k=0

k=0 n

n

pk (t)(z − µ)n−k =

m−k Cn−k (−µ)m−k z n−m =

That is,

n

n

m

z n−m

m=0

j Cn−k (−µ)j z n−k−j =

k=0

m−k pk (t)Cn−k (−µ)k−m .

(−1)m qm (t)z n−m ,

P (t, z − µ) = m=0

where m

qm (t) = k=0

m−k pk (t)Cn−k (−1)k µm−k (m = 1, ..., n), q0 ≡ 1.

(3.4)

Take into account that n

P (t, z − µ) =

n

(z − ρk (t) − µ) = k=1

(z − ρ˜k (t)), k=1

where according to (3.2), ρ˜k (t) ≡ ρk (t) + µ ≥ 0. Hence it follows that qm (t) are nonnegative and we can apply Theorem 15.1.1 to equation (3.3). To this end put d0 = 1, d2k = sup q2k (t) and d2k−1 = inf q2k−1 (t) (k = 1, ..., [n/2]), t≥0

t≥0

(3.5)

where qk (t) are defined by (3.4). Due to Theorem 15.1.1 and the substitution x(t) = e−µt u(t), we get Theorem 15.3.1 Under (3.2), let all the roots of the polynomial ˜ Q(z) :=

n

(−1)k dk z n−k

k=0

˜ (t, τ ) to equation (3.1) be real and nonnegative. Then the Green function W is nonnegative and ˜ (t, τ ) ∂ j eµ(t−τ ) W ≥ er˜1 (t−τ ) j ∂t

j k=0

Cjk

r˜1j−k (t − τ )n−1−k ≥0 (n − 1 − k)!

176

Appendix B. Positivity of the Green Function

(j = 0, ..., n − 1; t > τ ≥ 0), ˜ In particular, where r˜1 ≥ 0 is the smallest root of Q(z). n−1 ˜ (t, τ ) ≥ e(−µ+˜r1 )(t−τ ) (t − τ ) (t > τ ≥ 0). W (n − 1)!

(3.6)

Example 15.3.2 Let us consider the equation d2 x dx + p1 (t) + p2 (t)x = 0 (t > 0). dt2 dt

(3.7)

Assume that p1 (t), p2 (t) ≥ 0 and p21 (t) > 4p2 (t) (t ≥ 0). Put p+ 1 = sup p1 (t). t≥0

Under consideration ρ1 (t) + ρ2 (t) = −p1 (t). So we can take µ = p+ 1 . Hence, + + q1 (t) = 2p+ 1 − p1 (t), q2 (t) = p1 (p1 − p1 (t)) + p2 (t)

and

d1 = inf q1 (t) = p+ 1 , d2 = sup q2 (t). t

t

2 (p+ 1)

If, in addition, > 4d2 , then due to Theorem 15.3.1, the Green function ˜ (t, τ ) to equation (3.7) is nonnegative and inequality (3.6) is valid with W n = 2 and 2 −µ + r˜1 = −p+ (p+ 1 /2 − 1 ) /4 − d2 .

Third Order Equations with Nonreal Unstable “Characteristic” Roots Let ak (t) (t ≥ 0; k = 1, 2, 3) be real continuous scalar-valued functions bounded on [0, ∞) and a0 ≡ 1. Consider the equation 3

(−1)k ak (t)

k=0

d3−k u(t) = 0 (t > 0). dt3−k

(4.1)

The function W (t, τ ) defined for t ≥ τ ≥ 0 is the Green function to equation (4.1) if it satisfies (4.1) for t > τ and the conditions lim t↓τ

Put

∂ k W (t, τ ) ∂ 2 W (t, τ ) = 0 (k = 0, 1); lim = 1. k t↓τ ∂t ∂t2

c1 = inf a1 (t), c2 = sup a2 (t), c3 = inf a3 (t). t≥0

t≥0

t≥0

(4.2)

Appendix B. Positivity of the Green Function

177

Let the polynomial 3

Q3 (z) =

(−1)k ck z 3−k (c0 = 1, z ∈ C)

k=0

have a pair of complex conjugate roots: γ ± iω (γ, ω > 0), and a positive root z0 . In addition, let z0 > γ + ω. (4.3) Denote

K(t) := c0 [ez0 t − eγt (cos (ωt) + b0 sin (ωt))],

where b0 :=

z0 − γ 1 and c0 := . ω (z0 − γ)2 + ω 2

Below we prove that K(t) and its first and second derivatives are nonnegative on the half-line. Theorem 15.4.1 Let polynomial Q3 (z) have a pair of complex conjugate roots: γ ± iω (γ, ω > 0), and a positive root z0 satisfying condition (4.3). Then the Green function W (t, τ ) to equation (4.1) and its first, and second derivatives are nonnegative. Moreover, ∂ k W (t, τ ) ∂ k K(t − τ ) ≥ (k = 0, 1, 2; t > τ ≥ 0). ∂tk ∂tk

(4.4)

To prove this theorem we need the following Lemma 15.4.2 Let Q3 (λ) have a pair of complex conjugate roots: γ ± iω (γ, ω > 0), and a positive root z0 . In addition, let condition (4.3) hold. Then K (j) (t) ≥ 0 (j = 0, 1, 2; t ≥ 0). Proof:

Put b1 = z0 − γ. So b1 = ωb0 . Since K(t) = c0 eγt f (t) where f (t) = eb1 t − cos(ω t) − b0 sin(ωt),

it is enough to check that f and its derivatives are positive. By virtue of the Taylor series eb1 t = 1 + b1 t + g(t) (g(t) > 0). Since |cos s| ≤ 1, |sin s| ≤ s (s ≥ 0), we have f (t) = 1 + b1 t + g(t) − cos ωt − b0 sin ωt (g(t) > 0). Thus, one can assert that f ≥ 0 for all t ≥ 0. Furthermore, due to (4.3), b0 ≥ 1 and b1 ≥ ω. Thus f˙(t) = b1 eb1 t + ωsin(ωt) − b0 ω cos(ωt) =

Appendix B. Positivity of the Green Function

178

b1 (1 + b1 t + g(t)) + ωsin(ωt) − ωb0 cos(ωt) ≥ 0. Moreover,

f¨(t) = b21 eb1 t + ω 2 cos(ωt)+

ω 2 b0 sin(ωt) = b21 (1 + b1 t + g(t)) + ω 2 cos(tω) + b0 ω 2 sin(tω) ≥ 0. As claimed. ✷ Lemma 15.4.3 Under the conditions of Theorem 4.1, a solution u of (4.1) with the initial conditions u(0) = u(0) ˙ = 0; u ¨(0) = 1

(4.5)

satisfies the inequalities u(j) (t) ≥ K (j) (t) (j = 0, 1, 2; t > 0). Proof:

Simple calculations show that K(t) is a solution of the equation Q3 (D)x(t) = 0 (D = d/dt),

(4.6)

˙ ¨ with the initial condition K(0) = K(0) = 0, K(0) = 1. So K is the Green function to (4.6). Furthermore, we have mk (t) := (−1)k (ck − ak (t)) ≥ 0 (k = 1, 2, 3). Rewrite equation (4.1) in the form 3

(−1)k ck

k=0

d3−k u = dt3−k

Put

3

v(t) :=

ck k=0

3

mk (t) k=1

d3−k u . dt3−k

d3−k u(t) . dt3−k

(4.7)

(4.8)

Since, K is the Green functions to (4.6), thanks to the Variation of Constants Formula and condition (4.5), we have u(t) = K(t) +

t 0

K(t − s)v(s)ds.

˙ But K(0) = K(0) = 0. Consequently, dj dtj

t 0

K(t − s)v(s)ds = t 0

d dt

t 0

K (j−1) (t − s)v(s)ds =

K (j) (t − s)v(s)ds (j = 1, 2).

(4.9)

Appendix B. Positivity of the Green Function

179

Hence thanks to (4.7) and (4.8), 3

t

mk (t)[K (3−k) (t) +

v(t) =

0

k=1 t

K0 (t, t) + where

0

K0 (t, t − s)v(s)ds,

3

K0 (t, τ ) =

K (3−k) (t − s)v(s)ds] =

(3−k)

mk (t)K0

(4.10)

(τ ) (t, τ ≥ 0).

k=1

According to the previous lemma, K0 (t, τ ) ≥ 0 (t, τ ≥ 0). Put h(t) = K0 (t, t). Let V be the Volterra operator with the kernel K0 (t, t − s). Then thanks to (4.10) and the Neumann series, ∞

v(t) = h(t) +

(V k h)(t) ≥ h(t) ≥ 0.

k=1

Hence (4.9) and the previous lemma yield u(j) (t) = K (j) (t) +

t 0

K (j) (t − s)v(s)ds ≥

K (j) (t) ≥ 0 (j = 0, 1, 2),

(4.11)

as claimed. ✷ Proof of Theorem 15.4.1: For a τ > 0, take the initial conditions u(τ ) = u(τ ˙ ) = 0, j = 0, 1; u ¨(τ ) = 1. Then the corresponding solution u(t) to (4.1) is equal to W (t, τ ). Repeat the arguments of the proof of Lemma 15.4.3. Then according to (4.11) we have the required result. As claimed. ✷

Third Order Equations with General Nonreal “Characteristic” Roots Let pk (t) (k = 1, ..., 3) be real continuous functions bounded on [0, ∞), and p0 ≡ 1. Consider the equation 3

pk (t) k=0

d3−k x = 0 (t > 0). dt3−k

(5.1)

180

Appendix B. Positivity of the Green Function

Let the polynomial 3

P3 (t, z) =

p3−k (t)z k (z ∈ C)

k=0

have the roots ρk (t) (k = 1, 2, 3) with the property Re ρk (t) ≥ −µ (t ≥ 0; k = 1, 2, 3)

(5.2)

with some µ > 0. Denote q1 (t) = 3µ − p1 (t), q2 (t) = 3µ2 − 2µp1 (t) + p2 (t) and and

q3 (t) = µ3 − p1 (t)µ2 + p2 (t)µ − p3 (t), d0 = 1, d1 = inf q1 (t), d2 = sup q2 (t), and d3 = inf q3 (t). t≥0

t≥0

t≥0

Let the polynomial ˜ 3 (z) = Q

3

(−1)k dk z 3−k (d0 = 1, z ∈ C)

k=0

have a pair of complex conjugate roots: γ˜ ± i˜ ω (˜ γ, ω ˜ > 0), and a positive root z˜0 . In addition, let z˜0 > γ˜ + ω ˜. (5.3) Denote

1 ˜b0 := z˜0 − γ˜ and c˜0 := . ω ˜ (˜ z0 − γ˜ )2 + ω ˜2

˜ 3 (z) have a pair of complex conjugate Theorem 15.5.1 Let the polynomial Q roots: γ˜ ±i˜ ω (˜ γ, ω ˜ > 0), and a positive root z˜0 satisfying condition (5.3). Then ˜ (t, τ ) to equation (5.1) is nonnegative. Moreover, for the Green function W all t > τ ≥ 0, we have ˜ (t, τ ) ≥ c˜0 [e(˜z0 −µ)(t−τ ) − eγ˜ −µ)(t−τ ) (cos (˜ W ω (t − τ )) + ˜b0 sin(˜ ω (t − τ )))] ≥ 0 Proof:

Put x(t) = e−µt u(t) in (5.1). Then 0 = eµt

3

pk (t) k=0

d3−k e−µt u = dt3−k

3

pk (t)( k=0

d − µ)3−k u. dt

181

Appendix B. Positivity of the Green Function

That is, equation (5.1) is reduced the equation d − µ)u ≡ dt

P3 (t,

3

pk (t)( k=0

d − µ)3−k u = 0. dt

(5.4)

But 3

pk (t)(z − µ)3−k =

P3 (t, z − µ) = k=0 3

3

pk (t) k=0

where

m=k

Cnk

3

3−k

pk (t) j=0

k=0

m−k C3−k (−µ)m−k z 3−m =

3 m=0

z 3−m

m k=0

j (−µ)j z 3−k−j = C3−k

m−k pk (t)C3−k (−µ)k−m .

= n!/k!(n − k)!. That is, 3

(−1)m qm (t)z 3−m ,

P3 (t, z − µ) = m=0

where m

qm (t) = k=0

m−k pk (t)C3−k (−1)k µm−k (m = 1, 2, 3), q0 ≡ 1.

Take into account that 3

P3 (t, z − µ) =

(z − ρk (t) − µ), k=1

where according to (5.2), Re ρk (t)+µ ≥ 0 and apply Theorem 15.4.1 to equation (5.4). Due to the substitution x(t) = e−µt u(t), Theorem 15.4.1 proves the required result. ✷

Notes

Chapter 1: This book presupposes a knowledge of basic matrix theory, for which there are good introductory texts. The books (Gantmaher, 1967) and (Bellman, 1970) are classical. For more details about the notions presented in Chapter 1 also see (Stewart and Sun, 1990). The stability definitions presented in Section 1.2 are particularly taken from the books (Reissig et al., 1974), (Bellman, 1953), and (Vidyasagar, 1993). Books that provide a broader look at the subjects we cover include the books (Coppel, 1965), (Harris and Miles, 1980), (Khalil, 1992), (Kohan, 1994), (Nijmeijer and van der Schaft, 1990), (Rugh, 1996), (Slotine and Li, 1991), (Sontag, 1990), (Willems, 1971), etc. The material of Section 1.7 is based on Section 3.1 (Daleckii and Krein, 1974). The material of Sections 1.10-1.12 is adapted from (Gil’, 1995). Estimates for roots of algebraic equations can be found in the book (Ostrovski, 1973, p. 277). Chapter 2: The material of Sections 2.1 and 2.2 is based on Sections 3.1 and 3.2 of the book (Daleckii and Krein, 1974). The multiplicative representation for solutions of linear ordinary differential equations is well-known cf. (Dollard and Friedman, 1979), (Gantmaher, 1967). Theorems 2.6.1 and 2.7.1 are proved in (Gil’, 1995). About other very interesting results on stability of linear systems see (Blondel, 1993), (Liu and Michel, 1994). Chapter 3: The results of Sections 3.1 and 3.2 are adapted from the paper (Gil’, 1989b). They are based on the freezing method developed in the book (Bylov et al., 1966), and the papers (Izobov, 1974) and (Vinograd, 1983). Theorem 3.3.1 is taken from the paper (Gil’, 2004b). A survey of results on uniform exponential stability under the hypothesis that pointwise eigenvalues of the slowly-varying A(t) have negative real parts is in (Ilchmann et al., 1987). Other very interesting results on stability and instability of slowly varying systems can be found in the papers (Skoog and Lau, 1972) and (Desoer, 1969). Chapter 4: Theorems 4.1.1 is a corollary of the multiplicative representation. Corollary 4.1.2 was derived by Lozinskii, Corollary 4.1.6 is due to Wazewski (see (Izobov, 1974)). Similar result was established by Winter (1946). For additional results see also (Wu, 1984). Chapter 5: Theorem 5.1.1 is adapted from (Gil’ and Ailon, 1999). It was generalized to retarded systems in (Gil’, 1994a). Stabilizability of nonlinear , M.I. Gil : Explicit Stability Conditions, LNCIS 314, pp. 183–184, 2005. © Springer-Verlag Berlin Heidelberg 2005

184

Notes

systems with leading autonomous parts have been discussed by many authors (see (Tsinias, 1991), (Vidyasagar, 1993), and references given therein). About some other analytical methods of estimating the domain of attraction for differential equations see for instance (Levin, 1994), (Khalil, 1992), (Siljak, 1978). As it was above mentioned the basic method for the stability analysis of autonomous continuous systems is the Lyapunov functions one, cf. (Rajalaksmy and Sivasundaram, 1992), (Lakshmikantham et al., 1989), (Lakshmikantham et al., 1991), etc. About the well-known relevant results see also (Gelig, Leonov and Yakubovich, 1978), (Leonov, 2001), (Yakubovich, 1998), etc. Chapter 6: As it was above mentioned, in 1966 N. Truchan has showed that the Aizerman’s hypothesis is satisfied by systems having linear parts in the form of single loop circuits with up to five stable aperiodic links connected in tandem, cf. (Voronov, 1979). Theorem 6.1.2 includes Truchan’s one. It was announced in (Gil’, 1983a) and proved in (1983b). The contents of Section 6.3 are based on the papers (Gil’, 1985) and (Gil’, 1994b). The paper (Gil’ and Shargorodsky, 1986) deals with applications of Theorem 6.1.2 to limit cycles and oscillations of autogenerators. The results presented in Sections 6.1-6.3 supplement the well-known absolute stability criteria cf. (Xiao-Xin, 1993), (Harris and Valenca, 1983), (Vidyasgar, 1993), (Krasnosel’skii and Pokrovskii, 1977), etc. The Aizerman type problem for retarded systems was considered in (Gil’, 2000a). Chapter 7: Theorems 7.1.1 and 7.2.1 are adapted from (Gil’ and Ailon, 1999). The contents of Sections 7.3 and 7.4 are taken from the papers (Gil’, 2004b). Chapter 8 is based on the papers (Gil’, 1989a) and (Gil’, 1989b). The contents of Section 7.8 is adapted from the paper (Levin, 1969). About the theory of perturbations of nonlinear systems see for instance (Bellman, 1953, Section 2.5), (Bellman et al., 1985), (Ladde et al., 1977), (Rajalaksmy and Sivasundaram, 1992), etc. About other interesting relevant results see the papers (Aeyels and Peuteman, 1997 and 1998), (Peuteman and Aeyels, 2002), etc. The material of Chapter 9 is probably new. The contents of Chapter 10 are taken from the paper (Gil’, 2004a). The results of Chapter 11 are based on the paper (Gil’ and Ailon, 1998). In the papers (Gil’, 2000b) and (Gil’ and Ailon, 2000) the similar result were derived for retarded systems. About the well-known input-to-state stability results see (Sontag, 1990), (Angeli, Sontag and Wang, 2000), (Arcak and Teel, 2002), (Krichman, Sontag and Wang, 2000), (Neˇsi´c and Teel, 2001), (Tsinias, 1999). The notion of input-state stability is closely connected with input-output stability, cf. (Vidyasagar, 1993), (Rugh, 1996, Chapter 12), etc. The material of Chapter 12 is based on the paper (Gil’, 2001). The results of Chapter 13 are taken from the papers (Gil’, 1996) and (Gil’, 2003d).

References

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Aeyels, D. and Peuteman, J. (1997). New criteria for exponential and uniform asymptotic stability of nonlinear time-variant differential equations. Proceedings of IEEE Conference on Decision and Control, San Diego.

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Aeyels, D. and Peuteman, J. (1998). A new asymptotic stability criterion for nonlinear time-variant differential equations. IEEE Trans. Autom. Control 43, No. 7, 968–971.

[3]

Aizerman, M.A. (1949). On a problem concerning global stability of dynamical systems, Uspekhi Matematicheskikh Nauk, 4(4), 187–188. In Russian.

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Angeli, D., Sontag, E.D. and Wang, Y. (2000). Further equivalences and semiglobal versions of integral input to state stability. Dyn. Control 10, No.2, 127–149.

[5]

Arcak, M. and Teel, A. (2002). Input-to-state stability for a class of Lur’e systems. Automatica 38, No.11, 1945–1949.

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Bellman, R.E. (1953). Stability Theory of Differential Equations. McGraw-Hill, New York.

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Bellman R.E., Bentsmann, J., and Meerkov, S.M. (1985). Stability of fast periodic systems, IEEE Transactions on Automatic Control, 30, No 3, 289– 291.

[9]

Blondel, V. (1993). Simulteneous Stabilization of Linear Systems, Lecture Notes in Control and Information Sciences, Vol. 191, Springer, London.

[10] Bylov, B.F., Grobman, B. M., Nemyckii V. V. and Vinograd R.E. (1966). The Theory of Lyapunov Exponents. Nauka, Moscow. In Russian. [11] Coppel, W.A. (1978). Dichotomies in Stability Theory. Lectures Notes in Mathematics 629, Springer-Verlag, New York. [12] Daleckii, Yu L. and Krein, M. G. (1974). Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I. [13] Desoer, C.A. (1969). Slowly varying systems dotx = A(t)x, IEEE Transactions on Automatic Control, 14, 780–781. [14] D¨ oetsch, G. (1961). Anleitung zum Praktischen Gebrauch der Laplace-transformation. Oldenburg, Munchen. [15] Dollard, J.D. and Friedman, Ch. N. (1979). Product Integration with Applications to Differential Equations. Encyclopedia of Mathematics and its applications; v.10., London, Addison-Wesley Publ. Company.

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List of Main Symbols

I

.

A α(A) ≡ maxk Re λk (A) g(A) λk (A) N (A) β(A) ≡ mink Re λk (A) Rλ (A) γn,p A∗ T r A = T race A σ(A) ρ(A, λ) det(A) Cn Rn (., .) Ω(r) R+ = [0, ∞) . Lp = . Lp (R+ ,C n ) . C = . C(R+ ,C n ) Lp (R+ , Cn ) C(R+ , Cn )

Euclidean norm identity operator operator norm of A eigenvalue of A Frobenius norm of A resolvent of A operator conjugate to A trace of A spectrum of A distance between σ(A) and λ determinant of A complex Euclidean space real Euclidean space scalar product

5 5 8 10 5 12 11

71 26 26 26 26

Index

absolute stability Banach space

multiplicative representation for solutions 32

85

26

dissipative matrices

62

eigenvalues 8 estimate for absolute values of matrix function 23 estimate for norm of – matrix-valued functions 10 – resolvent of matrix 12, 13 evolution operator 27 Euclidean norm 5 Euclidean space 5 exponential stability 7 freezing method for – linear systems 39 – nonlinear systems 111 Frobenius norm 5 global asymptotic stability Green function 86, 135 Hurwitz matrix 8 Hurwitz polynomial

9

impulse function 86 input-to-state stability

145

Lozinskii inequality 62 Lyapunov equation 18 matrix – adjoint 8 – dissipative 62 – Hermitian 8 – Hurwitz 8 – positive definite 8 matrix-valued functions

9

7

norm of – matrix 5 – vector 5 orbital stability

153

positive solutions region of attraction

144 7

spectrum 8 stability – absolute 85 – exponential 7 – global asymptotic 7 – input-to-state 145 – in the first approximation – orbital 153 Wazewski inequality 63

73

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