Qualitative Analysis of Large Scale Dynamical Systems (Mathematics in Science & Engineering, 134)

QuaZitative Analysis of Large ScaZe Dynamical Systems

This is Volume 134 in MATHEMATICS I N SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD RELLMAN, Unh?ersity of Southern California The complete listing of books in this series is available from the Publisher upcw request.

UALITATIVE ANALYSIS OF LARGE SCALE DYNAMICAL SYSTEMS Anthony N . MicheZ DEPARTMENT OF ELECTRICAL ENGlNEERlNG IOWA STATE UNIVERSITY AMES. IOWA

Richard K . MiZZer DEPARTMENT OF MATHEMATICS lOWA STATE UNIVERSITY AMtS, IOWA

ACADEMIC PRESS

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1977

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Library of Congress Cataloging in Publication Data Michel, Anthony N Qualitative analysis of large scale dynaniical sy s te 111s. (Mathematics in science and engineering series ; vol. 000) Bibliography: p. Includes index. I. System analysis. I. Miller, Richard K., joint author. 11. Title. 111. Series. QA402.M48 515l.7 76-50399 ISBN 0 12-493850-7

PRINTED I N T H E UNITED STATES O F AMERICA

T o Wolfgang Hahn and Irwin W .Sandberg

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Contents

Preface Acknowledgments

xi xv

Chapter I. Introduction 1.1 1.2 1.3 1.4

Introduction Qualitative Analysis of Dynarnical Systems: General Remarks Qualitative Analysis of Large Scale Systems: General Remarks An Overview of the Subsequent Chapters

Chapter 11. Systems Described by Ordinary Differential Equations 2.1 Notation 2.2 Lyapunov Stability and Related Results 2.3 Large Scale Systems 2.4 Analysis by Scalar Lyapunov Functions

12 13 14 20

26

...

CONTENTS

Ulll

2.5 2.6 2.7

Application of M-Matrices Application of the Comparison Principle to Vector Lyapunov Functions Estimates of Trajectory Behavior and Trajectory Bounds 2.8 Applications 2.9 Notes and References

46 56 63 69 89

Chapter 111. Discrete Time Systems and Sampled Data Systems

91

3.1 3.2 3.3 3.4 3.5 3.6

Systems Described by Difference Equations Large Scale Systems Stability and Instability of Large Scale Systems Examples Sampled Data Systems Notes and References

Chapter IV. 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Notation Systems Described by Stochastic Differential Equations Composite Systems Analysis by Scalar Lyapunov Functions Analysis by Vector Lyapunov Functions Examples Notes and References

Chapter V. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Systems Described by Stochastic Differential Equations

Infinite-DimensionalSystems

Notation C,-Semigroups Nonlinear Semigroups Lyapunov Stability of Dynamical Systems Examples of Semigroups Stability of Large Scale Systems Described on Banach Spaces Some Examples and Applications Functional Differential Equations-Special Results Application of Comparison Theorems to Vector Lyapunov Functions Notes and References

Chapter V1. Input-Output Stability of Large Scale Systems 6.1 6.2 6.3

Preliminaries Stability of Large Scale Systems: Results Involving Gains Instability of Large Scale Systems

92 93 94 101 104 108

109

110 110 113 115 122 127 132

134 135 136 138 141 145 155 173 183 187 190

193

194 208 216

CONTENTS

ix

6.4 Stability of Large Scale Systems: Results Involving Sector and Conicity Conditions 6.5 Stability of Large Scale Systems : Popov-Like Conditions 6.6 L,-Stability and I,-Stability of Large Scale Systems 6.7 Analysis and Design Procedure 6.8 Notes and References

220 237 249 255 257

Chapter VII.

259

7.1 7.2 7.3 7.4 7.5

Integrodifferential Systems

Preliminary Results &Stability and Instability of Interconnected Systems Linear Equations and Linearized Equations An Example Notes and References

259 261 263 267 269

References

27 1

Index

285

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Preface

The demands of today’s technology have resulted in the planning, design, and realization of sophisticated systems that have become increasingly large in scope and complex in structure. It is therefore not surprising that over the past decade or more, many researchers have directed their attention to various problems that arise in connection with systems of this type, which are called large scale systems. Although it is reasonable to assume that in the near future there will evolve a well-defined body of knowledge on large systems, the directions of such a discipline have not been entirely resolved at this time. However, there are several well-established areas that have reached a reasonable degree of maturity. One is concerned with the qualitative analysis of large scale dynamical systems, the topic of this monograph. There are numerous examples of large dynamical systems that provide great challenges to engineers of all disciplines, physical scientists, life scientists, economists, social scientists, and of course, applied mathematicians. Obvious examples of large scale dynamical systems include electric power systems, nuclear reactors, aerospace systems, large electric networks, economic systems, process control systems in the chemical and petroleum industries, difxi

xii

PREFACE

ferent types of societal systems, and ecological systems. Most systems of this type have several general properties in common. They may often be viewed as a n interconnection of several subsystems. (For this reason, such systems are often also called interconnected systems or composite systems.) In addition, such systems are usually endowed with a complex interconnecting structure and are frequently of high dimension. I n order that this monograph be applicable to many diverse areas and disciplines, we have endeavored to consider several important classes of equations that can be used in the modeling of a great variety of large scale dynamical systems. Specifically, we consider systems that may be represented by ordinary differential equations, ordinary difference equations, stochastic differential equations, functional differential equations, Volterra integrodifferential equations, and certain classes of partial differential equations. In addition, we consider hybrid dynamical systems, which are appropriately modeled by a mixture of different types of equations. Qualitative aspects of large scale dynamical systems that we consider include Lyapunov stability (stability, asymptotic stability, exponential stability, instability, and complete instability), Lagrange stability (boundedness and ultimate boundedness of solutions), estimates of trajectory behavior and trajectory bounds, input-output properties of dynamical systems (input-output stability, i.e., boundedness and continuity of the input-output relations that characterize dynamical systems), and questions concerning the well-posedness of large scale dynamical systems. The qualitative analysis of large scale systems can be accomplished in a variety of ways. We present a unified approach of analyzing such systems at different hierarchical levels, namely, at the subsystem structure and interconnecting structure levels. This method of analysis offers several advantages. As will be shown, the method of analyzing complex systems in terms of lower order and simpler subsystems and in terms of system interconnecting structure often makes it possible to circumvent difficulties that usually arise in the analysis of high-dimensional systems with intricate structure. We shall also see that this method of analysis is somewhat universal in the sense that it may be applied to all the types ofequations enumerated above. It will be seen that this approach is especially well suited for the qualitative analysis of hybrid dynamical systems (i.e., systems described by a mixture of different types of equations). In addition, analysis by this procedure yields trade-off information between qualitative effects of subsystems and interconnection components. This method ofanalysis also makes it possible to compensate and stabilize large systems at different hierarchical levels, making use of local feedback techniques. Furthermore, this method can be used as a guide in the planning of decentralized systems endowed with built-in reliability (i.e., safety) features. Because of these advantages, this method of analysis should be considered as being more important than the individual results presented. Indeed, all the subsequent results should

PREFACE

xiii

be viewed as models; in particular applications, one should tailor the present method of analysis to specific problems. This book consists of seven chapters. In the first chapter we provide an overview of the subject. Chapters 11-V are concerned with Lyapunov stability, Lagrange stability, estimates of trajectory behavior and trajectory bounds, and questions of well-posedness of large scale systems. In Chapter I1 we consider systems described by ordinary differential equations, in Chapter 111 systems that can be represented by ordinary difference equations and also sampled data systems, and in Chapter IV systems that can be modeled by stochastic differential equations. In Chapter V we address ourselves to infinitedimensional systems that can be represented by differential equations defined on Banach and Hilbert spaces. Such systems include those that can appropriately be described by functional differential equations, Volterra integrodifferential equations, certain classes of partial differential equations, and infinite-dimensional hybrid systems described by a mixture of equations. Chapters VI and VII are devoted to input-output stability properties of large scale dynamical systems. The results in Chapter V l are rather general, while Chapter VIl is confined to systems described by integrodifferential equations. To demonstrate the usefulness of the method of analysis advanced and to point to various advantages and disadvantages, we have included several specific examples from diverse areas, such as problems from control theory, circuit theory, nuclear reactor dynamics, and economics. Because of their importance in applications, we have emphasized frequency domain techniques in several examples. In order to make this book reasonably self-contained, we have included necessary background material on the following topics : the principal results from the Lyapunov stability theory (for finite-dimensional systems, infinitedimensional systems, and systems described by stochastic differential equations), the main results for boundedness and ultimate boundedness of solutions, the principal comparison theorems (the comparison principle), results from the theory of M-matrices, selected results from semigroup theory, and pertinent results from systems theory (relating to input-output stability). In addition, we have provided numerous references for this background material.

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AcknowZedgments

We would like to thank Professor Richard Bellman for encouraging us to undertake this project, and we feel privileged to have this book published in his distinguished series. Likewise, thanks are due to the staff of Academic Press for advice and assistance. A great part of this monograph is based on research conducted at Iowa State University by the authors and by former students of the first author, Drs. Eric L. Lasley, David W. Porter, and Robert D. Rasmussen. The work of both authors was supported in part by the National Science Foundation, and the first author was also supported by the Engineering Research Institute, Iowa State University. During 1972-1973, the first author’s research was accomplished on a sabbatical leave at the Technical University of Graz, Austria, where he had the privilege of being associated with Professor Wolfgang Hahn. We are particularly appreciative of the efforts of Mrs. Betty A. Carter in typing the manuscript. Above all, we would like to thank our wives, Leone and Pat, for their patience and understanding.

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CHAPTER I

Introdzlction

In this chapter we first discuss somewhat informally the motivation for the method of analysis advanced in this monograph. We then briefly indicate the type of qualitative analysis with which we concern ourselves. This is followed by an overview of qualitative results for large scale systems which will be of interest to us. Finally, we give an indication of the contents of the subsequent chapters. 1 .I

Introduction

In recent years many researchers have addressed themselves to various problems concerned with large systems. This is evidenced by an increasing number of publications in scientific journals and conference proceedings. At this time there also have appeared a number of monographs dealing with various aspects of large scale systems. For example, there is the fundamental work by Kron [I] on diakoptics, Tewarson [ l ] considers the theory of sparse matrices arising naturally in large systems, Lasdon [l] addresses 1

I

2

INTRODUCTION

himself to optimization theory of large systems, MesaroviC, Macko, and Takahara [l] and Mesarovii. and Takahara [l] develop a general systems theory for hierarchical multilevel systems, and so forth. Although the state of the art in the area of qualitative analysis of large systems has reached a reasonable degree of maturity, no text summarizing the important results of this topic has appeared. We address ourselves in this book to this problem. As was pointed out in an editorial by Bellman [4], problems associated with large systems offer new and interesting challenges to researchers. We hope that the present monograph will in a small way further stimulate work in this new and exciting field. It must be stated at the outset that no precise definition of large scale system can be given since this term has a different meaning to different workers. In this book we consider a dynamical system to be large if it possesses a certain degree of complexity in terms of structure and dimensionality. More specifically, we will be interested in dynamical systems which may be viewed as an interconnection of several lower order subsystems. This point of view motivates also the terms “composite system,” “interconnected system,” “multiloop system,” and the like. In certain applications, the term “decentralized system” has also been used. Roughly speaking, problems concerned with large scale systems may be divided into two broad areas: static problems (e.g., graph theoretic problems, routing problems) and dynamical problems. The latter may in turn be separated into quantitative problems (e.g., numerical solution of equations describing large systems) and into qualitative problems. AII topics of this book are concerned with qualitative analysis of large scale dynamical systems. The traditional approach in systems theory is to represent systems in certain “standard” or canonical forms. For example, the usual approach in classical as well as in modern control theory is to transform the equations describing a given system in such a fashion that the system in question may be represented, for example, in the familiar block diagram form of Fig. 1.1. Once this is accomplished, long-established and well-tested methods are
,

I

COMPENSATORS

Figure 1.1 Typical feedback control system.

SYSTEM OUTPUT,

1.1

3

INTRODUCTION

employed to treat the problem on hand. Frequently, however, certain complications arise with this approach. For example, often difficulties are encountered in analysis and synthesis procedures which can be attributed to the dimensionality and to structural complexity of a system. Another disadvantage of the traditional approach is that after the system in question has been cast into a standard mold (such as Fig. 1. l), the effects of individual components, subloops, etc., are often no longer explicitly apparent because several transformations had to be performed to put the system into a desired canonical form (e.g., Fig. 1.1). In the subsequent chapters we will develop a unified qualitative theory for large systems, in which the objective will always be the same: to analyze (and synthesize) large scale systems in terms of their lower order (and hopefully simpler) subsystems and in terms of their interconnecting structure. In this way, complications which usually arise in the qualitative analysis of high order systems with complex interconnecting structure may often be circumvented. Furthermore, such an approach provides insight into system structure in its original form, yielding information on effects of individual system components, subsystems or subloops, trade-off information between various subsystems and interconnecting structure, and the like. This type of information is usually of great value to the designer and to the analyst. In addition, the viewpoint advanced herein makes it often possible to compensate large systems (i.e., improve their performance) at several hierarchical levels (i.e., at the subsystem level and interconnecting structure level), using local feedback methods. A typical large scale system, in its original form, may conceptually be represented as shown in the block diagram of Fig. 1.2. The method of analysis advanced herein is of course not without disadvantages. Thus, if a system is decomposed into too many subsystems, one may obtain overly conservative results. However, we will demonstrate by means of several specific examples that this need not be the case, provided that the method is applied properly. INPUT TO

I

SUBSYSTEM Si

SUBSYSTEM

S;

OUTPUT OF SUBSYSTEM Si

4

Y

i

INTERCONNECTING STRUCTURE FOR SUBSYSTEM Si

Figure 1.2 Typical large scale dynamical system where i = 1 , ...,1.

4

1

INTRODUCTION

1.2 Qualitative Analysis of Dynamical Systems: General Remarks

Qualitative aspects of dynamical systems which we will primarily address ourselves to include the following: stability and instability in the sense of Lyapunov; boundedness of solutions (i.e., Lagrange stability); estimates of trajectory behavior and trajectory bounds; and input-output properties of dynamical systems. The direct method of Lyapunov has found a wide range of applications in engineering and in the physical sciences. Although most of these applications were originally concerned with the stability analysis of systems described by ordinary differential equations (see, e.g., Hahn [I, 21, LaSalle and Lefschetz [l], Kalman and Bertram [l]), the direct method of Lyapunov has also been extended to systems described by difference equations (see, e.g., Hahn [l, 21, Kalman and Bertram [2]), and more recently to systems represented by partial differential equations (see, e.g., Zubov [11, Wang [l, 51, Sirazetdinov 111, Chaffee [I], Hahn [2]), differential difference equations (see, e.g., Bellman and Cooke [l], Yoshizawa [l], Krasovskii [I], Hahn [2]), functional differential equations (see, e.g., Hale [2] Yoshizawa [I], Krasovskii [I], Hahn [2], Halanay [l]), integrodifferential equations (see, e.g., Driver [l], Levin [I], Miller [l], Suhadloc [l], Bronikovski, Hall and Nohel [l]), systems of countably infinite many ordinary differential equations (see, e.g., Bellman [I], Shaw [l]), stochastic differential equations (see, e.g., Kushner [I], Arnold [l], Kozin [l], Kats and Krasovskii [I], Bertram and Sarachik [l]), stochastic difference equations (see, e.g., Kushner [3]), and the like. The stability theory of general dynamical systems is still of current interest (see, e.g., Bhatia and Szego [I], Hale [I], Krein [l], Ladas and Lakshmikantham [l], Walter [I], Slemrod [l]). In the case of many systems (e.g., systems exhibiting nonlinear oscillations) it is not the Lyapunov stability or instability that is of interest. Yoshizawa and others (see, e.g., Yoshizawa [l], Hahn [2], LaSalle and Lefschetz [l]) have extended the direct method of Lyapunov to establish conditions for boundedness (Lagrange stability), ultimate boundedness, unboundedness, and the like, of solutions of dynamical systems. I n practice one is not only interested i n the qualitative type of information obtainable from the Lyapunov stability and the Lagrange stability of a dynamical system, but also in specific estimates of trajectory behavior and trajectory bounds. A system could for example be stable or bounded and still be completely useless because it may exhibit undesirable transient characteristics (e.g., its solutions may exceed certain limits or specifications imposed by the designer on the trajectory bounds). Estimates of trajectory

1.3

QUALITATIVE ANALYSIS OF LARGE SCALE SYSTEMS

5

behavior and trajectory bounds have been obtained using the Lyapunov-type approach by defining stability with respect to time-varying subsets (of the state space) which are prespeciJied in a given problem (this is not the case in Lyapunov and Lagrange stability). The boundaries of these sets yield estimates of system trajectory behavior and trajectory bounds (see, e.g., Matrosov [l, 21, Michel [l-31, Michel and Heinen [l-41). This concept includes the notions of practical stability and finite time stability (see, e.g., LaSalle and Lefschetz [l], Weiss and Infante [l], Michel and Porter [4]). In a radical departure from the classical approach described thus far, it is possible to view dynamical systems in a “black box” sense as relations mapping system inputs into system outputs on an extended function space. In the context of this formulation, the qualitative analysis of such systems is accomplished in terms of system input-output properties (see, e.g., Sandberg [2, 81, Zames [3, 41, Willems [l], Desoer and Vidyasagar [I], Holtzman [l]). Many important results have been obtained along these lines and for many important problems a connection between input-output stability and Lyapunov stability has been established.

1.3 Qualitative Analysis of Large Scale Systems: General Remarks

Despite its elegance and generality, the usefulness of the Lyapunov approach is severely limited when applied to problems of high dimension and complex interconnecting structure. For this reason it is frequently advantageous to view high order systems as being composed of several lower order subsystems which when interconnected in an appropriate fashion, yield the original composite or interconnected system. The stability analysis of such systems can often then be accomplished in terms of the simpler subsystems and in terms of the interconnecting structure of such composite systems. In this way, complications which usually arise when the direct method is applied to high order systems may often be avoided. This is precisely the approach which we will employ. Indeed, since statements similar to the above apply equally as well to problems involving Lagrange stability, estimates of trajectory behavior and trajectory bounds, input-output stability, and the like, throughout this book we will pursue a method of qualitative analysis of large systems at different hierarchical levels. As will be pointed out repeatedly, in a certain sense, the method of analysis advanced in this book should be viewed as being more important than the individual results presented. There is a sizable body of literature concerned with various aspects of

6

1 INTRODUCTION

qualitative analysis of large dynamical systems. Since this subject is rather new, and since the literature is still growing, it is difficult and quite pointless to make an attempt at citing every reference. Indeed, if we were to list all sources concerned with this topic, such a list would be too long and not justified. On the other hand, if we were to confine the listings only to immediate references used, such a list would be too short and would hardly reflect the importance of this subject. For this reason we shall compromise and mention only some selected results which are in the spirit of the method of analysis proposed herein. 1. Systems Described by Ordinary Differential Equations (Lyapunov Stability). Using vector Lyapunovfunctions, originally introduced by Bellman [2], Bailey [l, 21 invoked the comparison principle (see, e.g., Walter [l]) to establish sufficient conditions for exponential stability of interconnected systems described by nonlinear nonautonomous ordinary differential equations with exponentially stable subsystems and with linear timeinvariant interconnecting structure. Subsequently, results for exponential stability, uniform asymptotic stability, instability, and complete instability of composite systems represented by nonlinear nonautonomous ordinary differential equations with nonlinear and time varying interconnections and with subsystems that may be exponentially stable, uniformly asymptotically stable (and sometimes unstable) were obtained by Piontkovskii and Rutkovskaya [l], Thompson [l, 21, Porter and Michel [l, 21, Michel and Porter [3], Thompson and Koenig [l], Matrosov [l], Araki [l, 41, Araki and Kondo 111, Michel (5-7, 91, Matzer [l], GrujiC and Siljak [2], Weissenberger [l], Bose [I], Bose and Michel [l, 21, and others. In some of these results, vector Lyapunov functions as well as scalar Lyapunov functions consisting of a weighted sum of Lyapunov functions for the free or isolated subsystems are employed. In addition, absolute stability results for interconnected systems endowed with several nonlinearities were established by several authors (see, e.g., McClamroch and Ianculescu [I], Bose [I], Bose and Michel [l, 21, and Blight and McClamroch [l]). Additional related Lyapunov stability results for large systems are contained in the paper by Tokumaru, Adachi, and Amemiya [2] and in the survey paper by Athans, Sandell, and Varaiya [l]. 2. Systems Described by DifSerence Equations and Sampled Data Systems (Lyapunov Stability). Sufficient conditions for uniform asymptotic stability, exponential stability, and instability of composite systems described by nonautonomous nonlinear difference equations were established by Araki, Ando, and Kondo [I], Michel [S, 71, and GrujiC and Siljak [I, 31. Results for uniform asymptotic stability of sampled data composite systems (i.e., hybrid finite-dimensional systems) were obtained by Michel [S, 71.

1.3

QUALITATIVE ANALYSIS OF LARGE SCALE SYSTEMS

7

3. Systems Described by Stochastic Differential Equations and Stochastic Difference Equations (Lyapunov Stability). The Lyapunov stability of large scale systems described by stochastic differential equations (It0 equations as well as other types of stochastic differential equations) and stochastic difference equations are treated in papers by Michel [S, 10, 111, Michel and Rasmussen [l-31, Rasmussen and Michel [I, 31, and Rasmussen [l]. Scalar and vector Lyapunov functions are used in the analysis. In addition to Wiener processes, disturbances modeled by Poisson step processes, jump Markov processes, and the like, are considered. The approach in these references is general enough to allow disturbances to enter into the subsystem structure and into the interconnecting structure. Composite systems with stable as well as unstable subsystems are treated. The obtained results yield sufficient conditions for asymptotic stability with probability one and in probability and exponential stability with probability one, in probability and in the quadratic mean. 4. ZnJnite-Dimensional Systems (Lyapunov Stability). Matrosov [2] uses vector Lyapunov functions while Michel [5, 71, Rasmussen and Michel [2, 41, and Rasmussen [I] use scalar Lyapunov functions (consisting of a weighted sum of Lyapunov functions for the free or isolated subsystems) to obtain conditions for uniform asymptotic stability and exponential stability of composite systems described by differential equations which are defined on Hilbert and Banach spaces. These results are general enough to allow analysis of large systems which can be represented by ordinary differential equations, differential difference equations, functional differential equations, integrodifferential equations, certain classes of partial differential equations, as well as other types of evolutionary systems. In addition, these results provide a systematic procedure for the stability analysis of hybrid dynamical systems, i.e., dynamical systems described by a mixture of different types of equations. 5. Lagrange Stability and Estimates of Trajectory Behavior and Trajectory Bounds. Sufficient conditions for uniform boundedness and uniform ultimate boundedness of solutions of interconnected systems described by ordinary differential equations were obtained by Bose and Michel [ l , 21 and Bose [I]. Estimates of trajectory behavior and trajectory bounds of composite systems described by ordinary differential equations were established in references by Michel [3, 4, 91, Matrosov [ l , 21, and Michel and Porter [ l , 21. These results include earlier ones for finite time stability of systems described on product spaces (see Weiss and Infante [13). 6. Input-Output Stability. Conditions for the input-output stability (input-output boundedness and continuity) for large classes of interconnected dynamical systems are established in Porter [I], Porter and Michel [3-51, Tokumaru, Adachi, and Amemiya [l], Lasley and Michel [I-61,

8

1

INTRODUCTION

Lasley [I], Cook [2], Miller and Michel [I, 21, Vidyasagar [I], and Araki [3]. Input-output instability results are obtained by Vidyasagar [l] and Miller and Michel [l]. In the majority of these references large scale systems are analyzed i n terms of their lower order subsystems and interconnecting structure. Overall stability conditions for the systems considered are phrased in terms of figures of merit which characterize the degree of stability of each subsystem and in terms of parameters expressing the degree of coupling of the interconnecting structure. Although the majority of these results are phrased in terms of general input-output relations, graphical frequency domain interpretations are emphasized when appropriate (e.g., circle criteria, Popov conditions). In this connection, stability on L,- and I,-spaces, 15p I co, are of special interest. Closely related to the above, are recent results by McClamroch [l] and Callier, Chan, and Desoer [I, 2). 7. Stabilization. Most, but not all, of the results cited above are in a form which make stabilization and compensation procedures of dynamical systems at different hierarchical levels feasible. Such procedures can in principle be implemented by the use of local compensating feedback at the subsystem and interconnecting structure levels. Significant case studies worthy of mention, where the method of analysis of the above references has been applied, have apparently not been conducted at this time. Nevertheless, great potential for applications of results of this type to significant problems in many diversified disciplines is apparent. Such disciplines include most branches of engineering (electrical, mechanical, aerospace, chemical, nuclear, biomedical, etc.), the physical sciences (physics, etc.), the life sciences (biology, etc.), economics, numerical analysis (e.g., iteration theory), and others. Typical examples of large scale systems with great promise for applications include electric power systems, nuclear power systems, aerospace systems, processes in the chemical and petroleum industry, circuits, economic systems, transportation systems, ecological systems, societal systems, and many others. Solutions to problems of this type will have to be effected by specialists with expertise in these areas, for in the case of truly large scale systems, great insight and understanding of modeling is crucial. 1.4 An Overview of the Subsequent Chapters

In the subsequent chapters we develop a unified approach to qualitative analysis of large scale systems described by many diversified types of equations. While our presentation constitutes an expansion and adaptation

1.4

AN OVERVIEW OF THE SUBSEQUENT CHAPTERS

9

of most of the references cited in the previous section, we emphasize that many of the results presented here are new and have not appeared elsewhere. A brief overview of the contents of the remaining chapters follows. In Chapter I1 we concern ourselves with the analysis of finite-dimensional large scale systems described by ordinary differential equations. Items of interest are uniform stability, uniform asymptotic stability, exponential stability, instability, and complete instability (all in the sense of Lyapunov), as well as uniform boundedness and uniform ultimate boundedness of solutions, and estimates of trajectory behavior and trajectory bounds. Both scalar Lyapunov functions and vector Lyapunov functions (involving suitable comparison theorems) are employed in our approach. In addition, extensive use of the properties of Minkowski matrices is made, when applicable, to simplify some of the results. Our exposition in Chapter I1 is intentionally somewhat extensive and detailed. One of the reasons for this is to make many extensions and results which are not explicitly stated in later chapters apparent to the reader. In Chapter 111 we modify some of the results of Chapter I1 to treat finitedimensional discrete time systems described by difference equations. In this chapter we also consider a class of (continuous time) sampled data systems (i.e., finite-dimensional hybrid systems). It will become apparent that the results of this chapter do not always involve obvious modifications of corresponding results of Chapter 11. In all of the subsequent chapters we concern ourselves with deterministic dynamical systems, except in Chapter IV, where large scale systems described by stochastic differential equations are considered. Because of the presence of second order effects which need to be taken into account, such systems must be treated separately. In Chapter IV we primarily consider Ito differential equations and we use scalar as well as vector Lyapunov functions to obtain sufficient conditions for asymptotic stability with probability one, exponential stability with probability one, and exponential stability in the quadratic mean. In Chapter V we generalize most of the results of Chapter I1 to infinitedimensional dynamical systems. Using the theory of C,-semigroups and nonlinear semigroups as a basis to characterize dynamical systems, we consider systems which may be described by differential equations defined on Hilbert and Banach spaces. In addition to Lyapunov stability and boundedness of solutions of large scale systems, we also address ourselves to questions of well posedness of such systems. The results of this chapter are general enough to be applicable to large systems described by ordinary differential equations, functional differential equations, Volterra integrodifferential equations, certain types of partial differential equations, and the like. Furthermore, these results make it possible to analyze hybrid dynamical

10

1

INTRODUCTION

systems (i.e., systems described by a mixture of equations) in a systematic manner. We use scalar as well as vector Lyapunov functions in our approach. Chapter VI is concerned with input-output properties of large dynamical systems. Of primary interest is the input-output stability and instability of such systems. By input-output stability we mean boundedness and sometimes also continuity of relations (and operators) which connect system inputs to system outputs in the system description. Continuous time systems as well as discrete time systems are considered. Of particular interest are results in the setting of L,- and I,-spaces and L,- and 1,-spaces. Whenever appropriate, we emphasize results involving frequency domain methods (e.g., Nyquist plots, gain sector conditions, circle criterion, Popov-type results, and the like). Because of their special structure and properties, large scale systems described by integrodifferential equations are treated in Chapter VII, where we further apply the stability and instability results of Chapter VI. To establish the required notation and to make our presentation reasonably self contained, we shall introduce extensive background material from several diverse areas of applied mathematics. Since most of these results can be found in several excellent books, we will omit all proofs involving background material. On the other hand, we will prove in detail all of the results presented which are concerned with the qualitative analysis of large scale systems. Background material will include the following: Lyapunov results and Lyapunov related results for the types of systems considered in Chapters 11-V; results from the theory of Minkowski matrices, which are used throughout; selected results from the theory of semigroups and dynamical systems; and appropriate material from system theory involving input-ou tput properties of dynamical systems. To demonstrate the usefulness of the method of analysis advanced, we consider in each chapter several specific examples and applications. The choice of these examples was in part dictated by the following objectives : to demonstrate the applicability of the method presented to problems in a variety of diverse disciplines; to illustrate the ease with which the results can be applied to complex problems; and to show that when used properly and applied to the right kinds of problems, the present method of analysis yields results which are not overly conservative and in fact, are not necessarily more conservative than corresponding results obtained by other methods. Specific examples which we consider include problems which arise in automatic control theory (e.g., direct control problem, indirect control problem), in circuit theory (e.g., networks with linear resistors and time varying capacitors, networks with linear resistors and nonlinear transistors), in the theory of nuclear reactors (point kinetics model of a coupled multicore nuclear reactor), in economics (Hicks conditions for the model of a multi-

1.4

AN OVERVIEW OF THE SUBSEQUENT CHAPTERS

11

market economic system), and so forth. In several of the examples considered (e.g., systems endowed with several troublesome elements, such as nonlinear gains, time varying gains, hysteresis elements, time delays, etc.) graphical frequency domain techniques are employed. Furthermore, to make comparisons possible, some of these examples are analyzed by several of the results presented. For example, the model of the coupled multicore reactor is treated in Chapters V and VII, deterministic and stochastic versions of the indirect control problem are considered in Chapters I1 and IV, respectively, continuous time and discrete time versions of a system with two nonlinearities are analyzed in Chapters I1 and 111, respectively, and so forth. Before proceeding further, some comments concerning the labeling of items is in order. Sections are assigned numerals which reflect the chapter and section numbers. For example, Section 2.3 signifies the third section in the second chapter. Extensive sections are sometimes divided into subsections identified by upper case common letters A, B, C, etc. Equations, definitions, theorems, corollaries, lemmas, examples, and special remarks are assigned monotonically increasing numerals which identify the chapter, section, and item number. For example, Theorem 3.3.6 denotes the sixth identified item in the third section of Chapter 111. This theorem is followed by Eq. (3.3.7), the seventh identified item in the same section. Figures are identified separately. Thus, Fig. 1.2 denotes the second figure in Chapter 1. Finally, the end of a proof is signified by the symbol ..

C H A P T E R I1

Systems Described Ly Ordinmy Dzfferentiul Eputions

In this chapter we consider the qualitative analysis of large scale systems described by ordinary differential equations. Our exposition will intentionally be detailed because (a) most of the existing results on large dynamical systems involve systems described by such equations, (b) ordinary differential equations play a crucial role in technology, and (c) many extensions and results not explicitly stated in the subsequent chapters (for systems described by other types of equations) will become apparent from results of the present chapter. The necessary notation is established in the first section while in the second section we present selected results from the Lyapunov theory which serve as essential background material. In the third section we introduce several classes of large scale systems which are analyzed in the fourth section by the use of scalar Lyapunov functions. This is followed by a presentation of several results from the theory of Minkowski matrices (M-matrices) which constitute background material used throughout this book. We then apply M-matrices in the fifth section to the qualitative analysis of large systems. We begin the sixth section with a summary of several standard comparison theorems which usually go under the heading of “comparison principle.” Next, we show how I2

2.1 NOTATION

13

this principle can be applied to vector Lyapunov functions in the analysis of large scale systems. In the seventh section we present a method of determining estimates of trajectory behavior and trajectory bounds. To illustrate how the results can be applied and to demonstrate the usefulness of the method of analysis advanced herein, we consider several specific examples in the eighth section. To point out relative advantages and disadvantages of the present procedure, examples from diverse areas are included which were treated by previous workers using methods that differ significantly from the present approach. In the last section a brief discussion of the literature cited is presented. 2.1

Notation

Let V and W be arbitrary sets. Then V u W, V n W , V - W , and V x W denote the union, intersection, difference, and Cartesian product of V and W , respectively. If V is a subset of W we write V c W and if x is an element of V we write x E V . Iff is a function or mapping of V into W we writefi V - + W and we identify the domain o f f and the range o f f by D ( f ) and Ra(f), respectively. Let @ denote the empty set, let R denote the real numbers, let R + = [0, a), and let J = [to,a)where to 2 0. We let R" be the Euclidean n-space and we let I . I represent the Euclidean norm. If x E R",then xT= (xl, .. ., x,,) denotes the transpose of x. If x, y E R", then x 5 y signifies xi< yi,x < y signifies xi < y,, and x > 0 signifies xi > 0 for all i = 1, ...,n. Let Y c R". Then Yand dY represent the closure and the boundary of Y , respectively. Also, B ( r ) = {x E R" : 1x1< r } and B(r) = {x E R" : 1x1 r } for some r > 0. Unless otherwise specified, matrices are usually assumed to be real. If A = [aii] is an arbitrary matrix, then A T denotes the transpose of A , A > 0 indicates that aii > 0, and A 2 0 signifies that a y 2 0 for all i , j . Now let A be a square matrix. If A is nonsingular, then A - ' denotes the inverse of A . An eigenvalue of A is identified as A(A) and ReA(A) denotes the'real part of I ( A ) . If all eigenvalues of A happen to be real we write I , ( A ) and I , " ( A ) to denote the largest and smallest eigenvalues of A , respectively. Matrix A is said to be stable if all its eigenvalues have negative real parts and unstable if at least one of its eigenvalues has positive real part. The determinant of an n x n matrix A is denoted by

detA

=

a,,

.'.

.

...

.

14

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

If A is a diagonal matrix we write A = diagca,, ...,a,,]. The identity matrix is denoted by I. Finally, the norm of an arbitrary matrix A , induced by the Euclidean norm, is given by

/ / ~ j j= m i n { c c E ~ +: c c ~ x ~ r ~X~E xR l" ,) = J L , ( A ~ A ) . 2.2

Lyapunov Stability and Related Results

In this section we present essential background material concerned with the stability analysis of dynamical systems described by ordinary differential equations. Since this material can be found in several standard texts dealing with the direct method of Lyapunov (also called the second method of Lyapunov) and related topics, we will not present proofs for any of the results presented in this section. We consider systems which can appropriately be described by ordinary differential equations of the form

P = g(x,t)

(1)

where x E R", t E J , P = dxldt, and g : B ( r ) x J + R" for some r > 0. Henceforth we assume that g is sufficiently smooth so that Eq. (I) possesses for every xo E B ( r ) and for every to E R + one and only one solution x ( t ; x o , t o ) for all t E J , where xo = x ( t o ;xo,to). We call xo an initial point, we refer to t as "time," and we call to initial time. Henceforth we also assume that Eq. (I) admits the trivial solution x = 0 so that g(0, t ) = 0 for all t E J . This solution is also called an equilibrium point or a singular point of (I). In addition, we also assume that x = 0 is an isolated equilibrium, i.e., there exists r' > 0 so that g(x', t ) = 0 for all t E J holds for no nonzero x' E B(r'). The preceding formulation pertains to local results. When discussing global results, we always assume that g : R" x J R" and that g is sufficiently smooth so that Eq. ( I ) possesses for every xo E R" and for every to E R+ a unique solution x ( t ; xo,t o ) for all t E J . In this case we also assume that x = 0 is the only equilibrium of Eq. (I). Since Eq. (I) can generally not be solved analytically in closed form, the qualitative properties of the equilibrium are of great practical interest. This motivates the following stability definitions in the sense of Lyapunov. --f

2.2.1. Definition. The equilibrium x = 0 of Eq. (I) is stable if for every 8 > 0 and any to E R + there exists a d ( 8 , t o ) > 0 such that / x ( t ;xo,r o ) /

whenever

/xol< d ( 8 , to).

< 8 for all t 2 to

2.2

15

LYAPUNOV STABILITY AND RELATED RESULTS

In the above definition, 6 depends on & and to. If 6 is independent of to, i.e., 6 = a(&), then the equilibrium x = 0 of Eq. (I) is said to be uniformly stable. 2.2.2. Definition. The equilibrium x = 0 of Eq. (I) is asymptotically stable if (i) it is stable, and (ii) there exists an ? ( t o )> 0 such that lim,+mx(t; x,, to) = 0 whenever IxoI < q . The set of all x, E R" such that condition (ii) of Definition 2.2.2 is satisfied is called the domain of attraction of the equilibrium x = 0 of Eq. (I). 2.2.3. Definition. The equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable if (i) it is uniformly stable, and (ii) for every & > 0 and any to E R + there exists a 6, > 0, independent of to and &, and a T ( € )> 0, independent of to, such that Ix(t; xo, ?,)I < & for all r 2 to+ T(&)whenever IxoJ< 6,. Of special interest in applications is the following special case of uniform asymptotic stability. 2.2.4. Definition. The equilibrium x = 0 of Eq. (I) is exponentially stable if there exists an ci > 0, and for every & > 0 there exists a 6(&) > 0 such that whenever Ix, 1 < 6 (E). 2.2.5. Definition. The equilibrium x = 0 of Eq. (I) is unstable if it is not stable. (In this case there exists a sequence {x,,} of initial points and a sequence { t , } such that Ix(t,+ t,;xOn,t,)l> & for all n.) When g : R" x J + R" and Eq. (I) possesses unique solutions for all x, and every to E R + , the following global characterizations are of interest.

E

R"

2.2.6. Definition. A solution x(t;x,, to) of Eq. (I) is bounded if there exists a I x ( t ;x,, to)l < p for all t 2 to, where fi may depend on each solution.

p > 0 such that

2.2.7. Definition. The solutions of Eq. (I) are uniformly bounded if for any ci > 0 and to E R', there exists a p = P(ci) > 0 (independent of t o ) such that if Ixol < ci, then Ix(t;x,, to)(< p for all t 2 to. 2.2.8. Definition. The solutions of Eq. (I) are uniformly ultimately bounded (with bound B ) if there exists a B > 0 and if corresponding to any CI > 0 and to E R', there exists a T = T(a) > 0 (independent of to) such that lx,l to+T. 2.2.9. Definition. The equilibrium x = 0 of Eq. (I) is asymptotically stable in the large if it is stable and if every solution of Eq. (I) tends to zero as t + co. (In this case the domain of attraction of the equilibrium of Eq. (I) is all of R".)

16

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.2.10. Definition. The equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable in the large if (i) it is uniformly stable, (ii) the solutions of Eq. (I) are uniformly bounded, and (iii) for any a > 0, any d > 0, and to E R + , there exists T(F,a) > 0, independent of to,such that if lxol< a then Ix(t;xo,to)l< & for all t 2 to+ T(b,a). 2.2.11. Definition. The equilibrium x = 0 of Eq. (I) is exponentially stable in the large if there exists ci > 0 and for any p > 0, there exists k(') > 0 such that lx(t;x,, to)l I k ( a ) Ixo(e-"('-'O) for all t 2 to

whenever

IxoJ< p.

Results which yield conditions for stability, instability and boundedness in the sense of the above definitions involve the existence of functions u : D + R, where in the case of local results D = B(r) x J for some r > 0, while in the case of global results D = R" x J . Henceforth we always assume that such v-functions are continuous on their respective domains of deJinition and that they satisfy locally a Lipschitz condition with respect to x. Also, unless otherwise stated, we assume henceforth that u(0, t ) = 0 for all t E J . The upper right-hand derivative of v with respect t o t along solutions of Eq. (I) is given by Dv(,)(x,t)= lim s u p ( l / h ) { u [ x ( t + h ; x , t )t+h] , - u(x,t)} h+O+

=

lim s u p ( l / h ) { u [ x + h . g ( x , t ) ,t+h] - u ( x , t ) } .

(2.2.12)

h+0+

If u is continuously differentiable with respect to all of its arguments, then the total derivative of u with respect t o t along solutions of Eq. (I) is given by DU(,)(X,t ) = V u ( x , t)'g(x, t )

+ h ( x ,tyat,

(2.2.13)

where V v ( x ,t ) denotes the gradient vector of the scalar function v and &/at represents the partial derivative of u with respect to t. Whether u is continuous or continuously differentiable will either be clear from context or it will be specified. In the former case Du(,, is specified by Eq. (2.2.12) while in the latter case Do(,) is given by Eq. (2.2.13). We now characterize several properties of u-functions in terms of special types of comparison functions. 2.2.14. Definition. A continuous function cp: [0, r l ] -+ R' (or a continuous function c p : [0, co) 4 R') is said to belong t o class K , i.e., cp E K , if cp(0) = 0 and if cp is strictly increasing on [O,r,] (or on [0, m)). If cp: R + -+ R', if cp E K , and if lim,+m cp(r) = co,then cp is said t o belong t o class KR. 2.2.15. Definition. Two functions cpl, cp2 E K defined on [0, r l ] (or on [0,a)) are said to be of the same order of magnitude if there exist positive constants

2.2

LYAPUNOV STABILITY AND RELATED RESULTS

17

k , and k , such that k , rpl(r) 2 rp2(r) I k,rp,(r) for all r E [O,r,] (or for all r E co, a>>. 2.2.16. Definition. A function u is said to be positive definite if there exists rp E K such that u(x, t ) 2 (~(1x1)for all t E J and for all x E B ( r ) for some r > 0. (Recall the assumption u(0, t ) = 0 for all t E J.) 2.2.17. Definition. A function v is said to be negative definite if - u is positive definite. 2.2.18. Definition. A function u, defined on R ” x J , is said to be radially unbounded if there exists cp E K R such that u(x, t ) 2 cp(lx1) for all x E R” and t E J . (Recall the assumption u(0, t ) = 0 for all t E J . ) 2.2.19. Definition. A function u is said to be decrescent if there exists cp E K such that Iu(x, t)l I cp(lxl) for all t E J and for all x E B ( r ) for some r > 0. In this case u is aiso said “to admit an infinitely small upper bound” or “to become uniformly small.” 2.2.20. Definition. A function u is said to be positive (negative) semidefinite if u(x, t ) 2 0 ( u ( x , t ) 5 0) for all t E J and x E B ( r ) for some r > 0. (Recall the assumption v(0, t ) = 0 for a11 t E J . ) For alternate equivalent definitions of the above concepts (positive definite, negative definite, etc.) the reader is referred to several standard texts dealing with the Lyapunov theory cited in Section 2.9. We are now in a position to summarize several well-known stability and instability results. In the first four of these, which are local results, we assume that u is defined and continuous on B ( r ) x J for some r > 0. 2.2.21. Theorem. If there exists a positive definite function u with a negative semidefinite derivative Do(,,, then the equilibrium x = 0 of Eq. (1) is stable. 2.2.22. Theorem. If there exists a positive definite, decrescent function u with a negative semidefinite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly stable. 2.2.23. Theorem. If there exists a positive definite, decrescent function u with a negative definite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable. In this case there exist cpl, cpz, (p3 E K such that cpt(lxl>

for all x E B ( r ) and t

u(x,t> I cpz(lxl), E

Du(l,(x,t)5

-cp3(IxI)

J.

2.2.24, Theorem, If in Theorem 2.2.23 cpl, cpz, cp3 E K are of the same order of magnitude, then the equilibrium x = 0 of Eq. (1) is exponentially stable.

18

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Theorem 2.2.24 is true if in particular there exist three constants c1 > 0,

c2 > 0, and c3 > 0 such that c1

lxI2 I u ( x , t ) I c2lxl

2

,

DU(,)(X,t) 5 -c31x12

for all x E B ( r ) and t E J. In the next three theorems, which are of great practical importance, we assume that u is defined and continuous over R" x J and that Eq. (I) possesses unique solutions for all xo E R" and all to E R'. 2.2.25. Theorem. If there exists a positive definite, decrescent, and radially unbounded function u with a negative definite derivative D U ( ~ then ), the equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable in the large. In this case there exist 'pl, q 2 E K R and q3 E K such that CPi(IXI)

5 u(x,t) 5

q2(IXI),

D ~ c I5) -4O3(IXI)

for all x E R" and for all f E J . 2.2.26. Theorem. If in Theorem 2.2.25 q 1 , q 2 , q E3 K P and if cp1,q2,q3 are of the same order of magnitude, then the equilibrium x = 0 of Eq. (I) is exponentially stable in the large. Theorem 2.2.26 is true if in particular there exist three constants c1 > 0, c2 > 0, and c3 > 0 such that C]

for all x

E

lx12 I u(x,t) I c21xI2,

DU(,)(X,t)I -c3lxI2

R" and t E 1.

2.2.27. Theorem. If there exists a function u defined on 1x1 2 R (where R may be large) and 0 I t < m, and if there exist J/,,I,!J2 E K R such that u ( x , t ) I I,!Jz(lxl), (i) $ 1 (Ixl) I (ii) D q , , ( x ,t ) I 0, for all 1x12 R and 0 I t < co, then the solutions of Eq. (I) are uniformly bounded. If i n addition there exists q3 E K (defined on Rf)and if (ii) is replaced by (iii) Du(,,(x,t) I -J/3(IxI) then the solutions of Eq. ( I ) are uniformly ultimately bounded. The next theorem, which yields conditions for instability, is a local result. We assume once more that for some r > 0, u is defined and continuous on B(r) x J . 2.2.28. Theorem. Assume there exists a function u having the following properties. (i) For every W > 0 and for every t 2 to there exist points x' such that u ( x ' , t ) < 0 and such that Ix'/ < 8.The set of all points (x,t ) such that 1x1 < r

2.2

LYAPUNOV STABILITY AND RELATED RESULTS

19

and v(x, t ) < 0 will be called the "domain v < 0." It is bounded by the hypersurfaces 1x1 = r and u = 0 and may consist of several component domains. (ii) In at least one of the component domains D of the domain u < 0, u is bounded from below and 0 E BD. (iii) In the domain D,Du(,)I - cp (v) where q E K . Then the equilibrium x = 0 of Eq. (I) is unstable. If in particular there exists a positive definite function u (a negative definite function v) such that Du(,,(x,t ) is positive definite (negative definite), then the equilibrium x = 0 of Eq. (I) is unstable. In fact, in this case the equilibrium is said to be completely unstable. Equation (I) is called a nonautonomous ordinary differential equation. Many systems of practical interest can appropriately be described by autonomous ordinary differential equations given by f = g(x)

(1')

where x E R", f = dxldt, and g: B ( r ) -+ R" for some r > 0, or in the case of a global setting, g: R"-+ R". For the unique solutions of Eq. (1') we have x ( t ; xo, to) = x ( t - to;xo,0) which allows us to assume to = 0 without loss of generality. Furthermore, since Eq. (1') is a special case of Eq. (I), all preceding statements made for (I) hold equally as well for (I'), with obvious modifications. In particular, since Eq. (1') is invariant under translations of time, it makes sense to consider only uniform stability, uniform asymptotic stability, uniform asymptotic stability in the large, and so forth. Conditions for stability, instability, boundedness, and the like, involve in this case the existence of functions v : B ( r ) -+ R for some r > 0 or u : R" -+ R . Such functions are characterized as being positive definite, negative definite, or radially unbounded as was done before using functions of class K and class K R and deleting all reference to t E J . For Eq. (If), the preceding theorems are modified by replacing u ( x , t ) by u(x). In addition, in Theorems 2.2.22, 2.2.23, and 2.2.25, all references to the word "decrescent" are deleted. For system (1') there is a significant extension for asymptotic stability, which we want to consider. First we require the following concept, given here in a global setting. 2.2.29. Definition. A set I-' of points in R" is invariant with respect to Eq. (1') if every solution of Eq. (1') starting in remains in I' for all time, i.e., if xo E r then x ( t ; x , , O ) E r for all t E R .

This concept has been used to prove the results summarized in the next theorem. 2.2.30. Theorem. If there exists a continuously differentiable function u : R" -+ R (recall the assumption u(0) = 0) and $ E K R such that

20

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(i) u(x) 2 $(lxl) for all x E R", (ii) Du(,.,(x)I 0 for all x E R", (iii) the origin x = 0 is the only invariant subset of the set E

= {X E

R" : Du(,.,(x)= 0},

then the equilibrium x = 0 is asymptotically stable in the large. If hypothesis (iii) is deleted then all solutions x ( t ; x,,O) of Eq. (1') are bounded for t 2 to. The preceding results are phrased in terms of the Euclidean norm 1.1. They are also true with respect to a n y other equivalent norm defined on R". In this case, convergence needs to be interpreted relative to the particular norm that is used. Henceforth we refer to results such as those given in this section as Lyapunov-type theorems and we call a function u satisfying any theorem of this type a Lyapunov function. The Lyapunov theorems are very powerful. However, in general, great difficulties arise in applying these results to high-dimensional systems with complicated structure. The reason for this lies in the fact that there is no universal and systematic procedure available which tells us how to find the required Lyapunov functions. Although converse Lyapunov theorems have been established, these results provide no clue (except in the case of linear equations) for the construction of Lyapunov functions. For this reason we will pursue an approach which allows us to analyze the stability of highdimensional systems with intricate structure in terms of simpler system components, which we shall call subsystems and interconnecting structure. This viewpoint makes it often possible to circumvent many of the difficulties associated with the Lyapunov method. First we need to consider the description of large scale systems, also called interconnected systems or composite systems. This is the topic of the next section. 2.3

Large Scale Systems

Since it simplifies matters considerably, our exposition in the present section is in a global setting. A development in a local setting involves obvious modifications. To fix some of the subsequent ideas, we begin with a specific example. In particular, we consider systems described by the set of equations ii = AiZi

y. =

+ DiU,

H.Z. 1 1

(2.3. I ) (2.3.2)

2.3

LARGE SCALE SYSTEMS

21

where zi E Rni,ii = dzi/dt,Ai is an n,x ni matrix, Di is an ni x mimatrix, ui E R"1, Hi is a p i x ni matrix, and y i E RPL.Equations (2.3.1) and (2.3.2) describe the input-output characteristics of a linear time-invariant system described by ordinary differential equations. We call this system the ith transfer system, where ui is interpreted as the input and y i as the output. Associated with Eq. (2.3.1) is the system described by the linear equation 2.

=

A I. z .

17

(2.3.3)

which we call the ith isolated subsystem or the ith free subsystem. Next, let us consider I transfer systems (i.e., i = 1 , ...,1) and let us interconnect these by means of the equations 1

ui =

C BYyj + G i u ,

j= 1

(2.3.4)

i = 1, ...,I, to form a composite system or interconnected system. Here B, is an mi x p j matrix, Giis an m i x q matrix and u E Rq. In this case B, represents a linear, time-invariant connection from the output of the j t h transfer system to the input of the ith transfer system. Frequently we may assume Bii to be zero, since feedback around any transfer system can often (for purposes of analysis) be combined with the matrix A i . Combining Eqs. (2.3.1), (2.3.2), and (2.3.4), we obtain ii = A i z i +

I

1C , z j + K i u ,

j= 1

(2.3.5)

i = I , ...,I, where C , = Di B, H j and Ki = Di Gi are n, x n and ni x q matrices, respectively. Letting ni = n, xT = ( z I T..., , zrT)E R",

cf=

we can rewrite Eq. (2.3.5) as f = AX

+ CX + Ku.

(2.3.6)

Finally, let the output of this interconnected system be represented by y = HX

(2.3.7)

where y E RP and H is a p x n matrix. Equations (2.3.6) and (2.3.7) describe the input-output characteristics of the composite system considered, where u denotes the input and y the output. This system may be viewed as a linear interconnection of I isolated subsystems

22

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

described by Eq. (2.3.3) with interconnecting structure specified by Eqs. (2.3.1), (2.3.2), and (2.3.4) and output characterized by Eq. (2.3.7). Systems described by equations of the form (2.3.6) and (2.3.7) are examples of large scale systems. For obvious reasons, the terms interconnected system and composite system are more descriptive. In control theory, such systems are sometimes also called multi-loop feedback systems, decentralized systems, and the like. In the Lyapunov stability analysis of the equilibrium x = 0 of the above system, the output equation (2.3.7) is of no concern whatsoever. Furthermore, in such an analysis it is the free or unforced dynamical system that is of interest. Thus, when investigating the Lyapunov stability of the above system, we consider the set of differential equations ii = Aizi+

I

C C,zj,

j= 1

(2.3.8)

i = 1, . ..,I, or equivalently, the differential equation i= AX

+ CX A Fx,

(2.3.9)

which is a special case of Eq. (1) of Section 2.2. On the other hand, when studying the qualitative input-output properties of the above system, Eqs. (2.3.6) and (2.3.7) are of interest. We will devote Chapters VI and VII in their entirety to such investigations. Meanwhile, we primarily concern ourselves with Lyapunov stability and related concepts for large scale systems. Many systems of practical interest (e.g., power systems, aerospace systems, circuits, economic systems, etc.) are appropriately described by nonlinear time varying ordinary differential equations and may often be viewed as interconnected or composite systems. In the following we consider several classes of ordinary differential equations which may be used to characterize such systems. We consider systems described by equations of the form 5, = f i ( z i , t )+gi(zl,..',z*,f),

(xi)

(2.3.10)

i = l , ..., I, w h er e z iERni ,t E J , f i : R n i x J + R n i ,a n d g , : R " ' x . . . x R " ' x J - , R"'. Henceforth we assume thatfi(zi, t ) = 0 for all t E J if and only if z, = 0. Letting Cf= n j = n,

xT = (zlT,...,)z:

f(x,t IT

=

g(X>t)T=

E R"

CfI(Zl? t IT, ...,f;(z1, tY1 CS1(Z1,...,Z~,t)T,...,9*(Z1,...,ZI,t)T1

and gi(z,,

...,zl, t )

g i ( x ,t ) ,

i = 1, ..., I,

2.3

LARGE SCALE SYSTEMS

23

so that g ( x , t)T = [g,(x, t ) T ,...,gl(x,t)T],we can represent Eq. (2.3.10) equivalently as (9) f = f ( x , t ) + g ( x , t ) Li h ( x , t ) . (2.3.1 1) Clearly, f : R" x J - + R",g: R" x J + R", and h : R" x J - + R". We always assume that h ( x , t ) = 0 for all t E J if and only if x = 0. A system described by Eq. (2.3.1 1) may be viewed as a nonlinear and time varying interconnection of I systems represented by equations of the form

(m

(2.3.12)

ii = f i ( Z i , t ) .

Henceforth we assume that for every to E R + and every xo E R",Eq. (2.3.11) possesses a unique solution x ( t ; xo, to) for t 2 to with xo = x ( t o ; xo,to). Also, we assume that for every to E R + and every zio E Rni,Eq. (2.3.12) has a unique solution z,(t;zio,to) for t 2 to with zi(to;zio,to) = zio. Subsequently we refer to Eq. (2.3.11) as composite system ( Y ) ,or interconnected system (Y), or large scale system (9')with decomposition (&) (described by Eq. (2.3.10)). We refer to Eq. (2.3.12) as the ith isolated subsystem (q)or as the ith free subsystem (q.) or as the ith unforced subsystem (8). We call x in Eq. (2.3.11) a hyper vector. Finally note that Eqs. (2.3.1 1) and (2.3.12) are of the same form as Eq. (I) and as such, all results of Section 2.2 are applicable to composite system (9) and to isolated subsystem (q). In specific applications, more information concerning system structure is usually available than indicated in Eq. (2.3.10). The following two classes of systems are examples of special cases of Eq. (2.3.10) encountered in practice. Let (2.3.13) where C, is a constant n i x n j matrix. Then ( X i ) assumes the form

ii = f i ( z i , t )

+ 2

j=l.i#j

cqzj,

(2.3.14)

i = 1, ..., I. This equation represents a system consisting of 1 isolated subsystems ($) which are linearly interconnected. Its structural properties are depicted in the block diagram of Fig. 2.1. Next let (2.3.15) where gl:R"j x J - + Rni.Then ( C i ) assumes the form (2.3.16)

i = 1, ..., 1.

24

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Figure 2.1 Interconnected system (2.3.14) where i = 1 , ..., 1.

Equations (2.3. lo), (2.3.1 I ) , and (2.3.12) characterize a class of composite systems which can appropriately be described by nonautonomous ordinary differential equations. Deleting t E J in these equations, we also consider time-invariant interconnected systems described by the corresponding autonomous ordinary differential equations given by

(xi')

ii = ,fi(zi)+ gi(zl, . .., z,),

(Y')

.i =f(x)

(XI)

i i

+ g(x) A h(x),

i

=

1 , ..., I,

(2.3.17) (2.3.18) (2.3.19)

= fi(zi>,

respectively. Note that in all cases considered thus far, the interconnecting structure enters additatively into the system description. As such, composite system (9) with decomposition ( X i ) is actually a special case of systems described by equations of the form (C:C)

ii = .fi(zi,gi(zl, ..., z I , t ) , t ) = f i ( z i ,gi(x,t ) , t )

A

h i ( x , t),

(2.3.20)

i = I , ..., I , where zi E R"', t E J , I:.=, n j = n, xT= (zIT, ..., z,') E R", g i :R" x J + R r i , f , : Rntx Rrcx J + R"', and hi:R" x J + R"'. Letting hT(x,t ) = [ h , ( x , t ) T ,... , h l ( x ,r)T], Eq. (2.3.20) can equivalently be represented by

(9")

i= h ( x , r ) .

(2.3.21)

2.3

LARGE SCALE SYSTEMS

25

We assume that Eq. (2.3.21) possesses only one equilibrium, x = 0, and that for every to E R+ and every xo E R”, Eq. (2.3.21) has a unique solution x ( t ; x o ,1 , ) for all t 2 to. In the present case we speak of composite system (9”) with decomposition (C;) and leave the notion of isolated subsystem undefined. In the present chapter, as well as in subsequent ones, we advance a method of qualitative analysis at different hierarchical levels, involving the following general steps. Step 1. A large scale system (9) (see Eq. (2.3.11)) is decomposed into I isolated subsystems (q) (see Eq. (2.3.12)) which when interconnected in an appropriate fashion (see Eq. (2.3.10)) yield the original composite or interconnected system. Step 2. The qualitative properties of the lower order (and hopefully simpler) free subsystems (3)are characterized in terms of Lyapunov functions u i ,using standard and well-established techniques involving the results of Section 2.2.

Step 3. Qualitative properties of the overall system (9) are deduced from the qualitative properties of the interconnecting structure and the individual free subsystems.

For composite system (9”) with decomposition (C;), the above procedure has to be modified somewhat, for in this case free subsystems (3) are not defined. Nevertheless, as will be demonstrated, even i n this case a method of analysis at different hierarchical levels is still possible. The process of decomposing a large system (Step 1) into an appropriate form is by no means a trivial task. We will not address ourselves explicitly to the problem of “tearing,” pioneered by Kron [I]. However, we would like to point to two general approaches. In the first of these, the structural properties of the process being modeled usually dictate a natural decomposition. In the second approach, the decomposition is usually influenced by mathematical convenience to overcome technical difficulties. In connection with Step 2 we note that for isolated subsystems of sufficiently low order and simplicity, many well-known results from the Lyapunov theory are available. Note also that in this step the converse Lyapunov theorems can play a crucial role. Thus, if the stability properties of the free subsystems are known a priori, we are usually in a position to search for Lyapunov functions with certain general properties. We will use two methods of implementing Step 3. In the first approach, which is developed in Sections 2.4 and 2.5, scalar Lyapunov functions (consisting of weighted sums of Lyapunov functions for the isolated subsystems) are constructed and applied to the results of Section 2.2. In the second approach, vector Lyapunov functions (i.e., I-vectors whose components are

26

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Lyapunov functions) are employed and stability is deduced from an Zth-order differential inequality involving these vector Lyapunov functions by invoking an appropriate comparison principle. The method outlined above frequently enables us to circumvent difficulties which arise when the Lyapunov approach is applied to high-dimensional systems with complex structure. 2.4 Analysis by Scalar Lyapunov Functions

I n this section, which consists of five parts, we develop qualitative results for the systems considered in the previous section, making use of scalar Lyapunov functions. First we concern ourselves with uniform stability and uniform asymptotic stability. Next, we consider exponential stability. This is followed by instability and complete instability results. Uniform boundedness and uniform ultimate boundedness are treated in the fourth part. The section is concluded with a discussion of the results. A. Uniform Stability and Uniform Asymptotic Stability

In characterizing the qualitative properties of the free subsystem (Yi), we will find it convenient to use the following convention. 2.4.1. Definition. Isolated subsystem (3.) possesses Property A if there exists a continuously differentiable function u i : R”, x J + R,functions $ i , , $i2 E K R , $i3 E K , and a constant oi E R such that the inequalities iil(Izi1)

5

ci(zi, t >

5

$i2(lzil)>

hold for all zi E Rniand for all t E J .

DUi(yi)(Zi,t)

ci$i3(/zil)

(x)

Clearly, if oi < 0, the equilibrium zi = 0 of is uniformly asymptotically stable in the large. If oi = 0, the equilibrium is uniformly stable. If oi> 0, the equilibrium of (Sq) may be unstable.

2.4.2. Theorem. The equilibrium x = 0 of composite system (9) with decomposition ( C i ) is uniformly asymptotically stable in the large if the following conditions are satisfied. possesses Property A ; (i) Each isolated subsystem given ui and $ i 3 of hypothesis (i), there exist constants aUE R such (ii) that

(z)

I

Vui(zi, tITgi(ZI,. . . , z [ , t > 5

C

C$i3(I~il>I”~

for all ziE R“<,i = 1 , ..., I, and t E J ; and

j =1

~ijC~+j3(Izjl)I”~

2.4

27

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

(iii) given oi of hypothesis (i) there exists an h e c t o r aT = (a1,..., al) > 0 such that the test matrix S = [s,] specified by

is negative definite. Proof. For composite system (9) we choose a Lyapunov function

c CliVi(Zi,t) 1

u(x,t) =

(2.4.3)

i= 1

where the functions vi are given in hypothesis (i) and where ai > 0, i = 1, . .., I are constants given in hypothesis (iii). Clearly, v ( x , t ) is continuously differentiable and u(0, t ) = 0 for all t E J , since each vi(zi, t ) satisfies these conditions. Since each isolated subsystem (q)possesses Property A, it follows that

for all x E R",t E J, and zi E R"', i = 1, . ..,1. Since by assumption $il, t+hi2 E KR, it follows that u(x, t ) is positive definite, decrescent, and radially unbounded. Indeed, there exist $1, 11/2 E KR such that

so that

C ai$il(IziI)

i= 1

c mi+iz(JziJ), 1

1

$l(lXl) 5

and

~lrz(ixI)2

i= 1

28

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Now let wT

and let R

=

= [4!'13(1Zi"'''>

(2.4.6)

...,4!'13(1Zil)1'21

[re] be the I x 1 matrix specified by y..

We now have

=

Lag,

aj[uj+uij],

Dv(,,(x,t) I wTRw

=

i =.j

i#j.

wT((R+RT)/2)w

=

wTSw,

where S = [sJ is the test matrix given in hypothesis (iii). Since S i s symmetric, all its eigenvalues are real. Also since by hypothesis (iii), S is negative definite, all its eigenvalues are negative so that A,(S) < 0. We thus have 1

Dv(,,(x,t) 5 ',(S)wTw

=

',(S)

1

1=

1

(2.4.7)

+i3(IziO.

Therefore, Dv(,)(x, t ) is negative definite for all x E R" and t E J. In fact, there exists a function tj3 E K such that ~ , h ~ ( l x5l ) so that

x:=l

Dv(,)(x,t)

' M ( S ) $ ~ ( I X I ) ~ AM(^> < 0,

(2.4.8)

for all x E R" and t E J. Inequalities (2.4.4) and (2.4.8) show that all hypotheses of Theorem 2.2.25 are satisfied. Therefore, the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. Before proceeding further, we note that the test matrix S above theorem is negative definite if and only if

=

[sJ of the

2.4.10. Corollary. If the test matrix S in Theorem 2.4.2 is negative semidefinite, then the equilibrium x = 0 of composite system (9) with decomposition (Xi)is uniformlystable. (In this case we require that the main determinants (2.4.9) be nonnegative.) Proof. Choose the positive definite, decrescent, and radially unbounded Lyapunov function given in Eq. (2.4.3). Since by assumption matrix S is negative semidefinite, we have ' ( S ) I 0. Thus, inequality (2.4.7) yields D u ( , , ( x ,t ) 2 0 for all x E R" and t E J . Therefore, the hypotheses of Theorem 2.2.22 are satisfied and the equilibrium of system (9') is uniformly stable.

2.4

29

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

We can generalize Therorem 2.4.2 somewhat.

2.4.11. Theorem. The equilibrium x = 0 of composite system (9) with decomposition (Ci) is uniformly asymptotically stable in the large if the following conditions hold. (i) Each isolated subsystem possesses Property A ; (ii) given ui and t+bi3 of hypothesis (i), there exist continuous functions a y :R” x J - t R such that

(z)

I

Vui(Zi,t)Tgi(Z1,

C alj.(x,t)Ct,bj3(I~jI)I”’

. . . , ~ i , t )I C $ i 3 ( I ~ i l ) I ~ / ~

j= 1

for all zi E R”’,i = 1 , ...,I , x E R”, and t E J ; (iii) there exists an I-vector aT = ( a l , ...,aI) > 0 and d > 0, such that for all x E R“ and t E J , the test matrix S(x, [ ) + & I is negative definite, where I denotes the I x I identity matrix and S(x, t ) = [slj.(x,t)] is defined by

Proof. Choose the positive definite, decrescent, and radially unbounded Lyapunov function given in Eq. (2.4.3). Let R ( x , t ) = [ r l j . ( x ,t ) ] be the I x I matrix specified by . . YG(X,t

)

=/

ai[ai+aii(X,t)],

I

aiaij(x, f ) ,

i # j

=

and let w be defined as in Eq. (2.4.6). Using Eq. (2.4.5) and hypotheses (i), (ii), and (iii), we obtain Dv(,,(x, t ) 2 wTR(x,t ) w

=

w T [ ( R ( x ,t ) + R ( x , t ) T ) / 2 ] ~

= WTS(X,t)W

I -&WTW

= -8

c I

Iji3(IZil).

i= 1

Therefore, D v ( , ) ( x , t ) is negative definite for all x E R” and t E J . Indeed there exists a function $3 E K such that $3(IxI) i t,bi3(Izil),so that

xi=,

D v ( , , ( x , t ) 5 -8$3(IxI). Thus, the hypotheses of Theorem 2.2.25 are satisfied and the equilibrium of system (9) is uniformly asymptotically stable in the large. W Deleting all references to t E J , Theorems 2.4.2, 2.4.1 1, and Corollary with de2.4.10 are also applicable to autonomous composite system (9’) composition (Xi’).For system (9’) an extension involving invariant sets is possible.

30

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.4.12. Definition. Isolated subsystem (8’) possesses Property A‘ if there exists a continuously differentiable function vi: R”’ -+ R, t+hil E KR, $i3 E K , and a constant rsi E R such that $il(Izil)

2 ui(zi),

vi(0) = 0,

DUi(yi*)(Zi)

~i$i3(I~il)

for all zi E R”‘. If oi< 0, the equilibrium zi = 0 of system (8’) is asymptotically stable in the large and if oi> 0, the equilibrium may be unstable. If oi = 0, all solutions of (q.‘) are bounded for all t 2 to = 0, and also the equilibrium of (8’) is stable. 2.4.13. Theorem. Assume that for composite system (9’) with decomposition (Xi’) the following conditions hold. (i) Each isolated subsystem (Yi’)possesses Property A ’ ; (ii) given ui and $ i 3 of hypothesis (i), there exist constants aii E R such that Vvi ( Z J T g i (

. .., ZJ 5

~ 1 3

I

(Izil)I

C$i3

”’ C

j= 1

C$ j 3 (Izj1)I ’”

for all Z ~ Rni, E i = 1, ..., 1; (iii) given rsi of hypothesis (i), there exists an I-vector aT = (a1, ..., al) > 0 such that the matrix S = [se] specified by . . s.. =

(ai (ci

+4,

j(“iaq+ajaji),2,

1 =J

i#j

is negative semidefinite. Then the following statements are true. (a) All solutions of (9’) are bounded; (b) the equilibrium x = 0 of (9”) is stable; and (c) if the origin x = 0 is the only invariant subset of the set E = {x E R” : Du(,.,(x) = 0}, where u ( x ) = aivi(zi), then the equilibrium x = 0 of composite system (9”) is asymptotically stable in the large.

xi=,

Proof. For composite system (9”)we choose the Lyapunov function

v(x) =

c I

UiUi(Zi)

i= 1

where the functions vi are given in hypothesis (i) and where mi > 0, i = 1, ..., I are constants given in hypothesis (iii). Clearly v ( x ) is continuously differentiable, v ( 0 ) = 0, and u ( x ) is positive definite and radially unbounded. Thus, there exists E K R such that 2

*l(lXl>

(2.4.14)

2.4

31

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

for all x E R”. Modifying the proof of Theorem 2.4.2 in an obvious way, we see that 1

Ct + i s ( l Z i l )

Dv(,,)(x) I J+,(s) i=

for all x E R”. Since S is by assumption negative semidefinite, we have J+,(S) I 0, so that DV(,,)(X) I 0 (2.4.15) for all x E R”. Clearly, the equilibrium x = 0 of system (9’) is stable. Furthermore, it follows from Theorem 2.2.30 that all solutions of (9’) are bounded for t 2 to = 0. If in addition the origin x = 0 is the only invariant subset of the set E = {x E R” : Du(,,,(x) = 0}, then it follows from Theorem 2.2.30 that the equilibrium of (9’) is asymptotically stable in the large. W Next we consider composite system (9”’) with decomposition (Cy). 2.4.16. Theorem. Assume that for composite system (9”)with decomposition (C;) the following conditions hold. (i) There exist continuously differentiable functions u i : R”‘ x J + R and + i l , J/iz E K R , i = 1, ..., I, such that

+i1(lzil) 5 ui(zi,t) I +iz(Izil) for all zi E R“( and t E J ; (ii) given vi in hypothesis (i), there exist constants ug E R and i, j = 1, ..., I, such that V~i(zi,tIThi(-~, t ) + aui(zi, t > / a t I +i4(Izil)

Ic/i4

E

K,

1

C1 a”+j4(I~jI)

j=

for all zi E R”’, x E R”, t E J , and i = 1 , ..., I ; and ...,a,) > 0 such that the test matrix (iii) there exists an I-vector aT = (al, S = [sg] specified by ai aii > i=j s.. = ((aiui+ajuji)/2, i#j is either negative semidefinite or negative definite. (a) If S is negative semidefinite, the equilibrium x = 0 of composite system (9”’) is uniformly stable. (b) If S is negative definite, the equilibrium of (9”’)is uniformly asymptotically stable in the large. Proof. Given the functions vi of hypothesis (i), we choose for composite system (9”’) the Lyapunov function 1

v(x,t )

=

C CliVi(Zi, i= 1

t)

(2.4.17)

32

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where aT = (aI, ..., a,) > 0 is given in hypothesis (iii). Clearly u ( x , t ) is continuously differentiable and 1

1

i= 1

1

Mi$il(lzil)

5 v(x,t>5

1

i= 1

ai$iz(lZil>

for all zi E R"', i = 1 , . . .,I, x E R",and t E J . Since by assumption $il, E KR, it follows that v ( x , t ) is positive definite, decrescent, and radially unbounded, and there exist $z E K R such that 1

$1

(1x1) 5

Hence,

C Mi $i1

i= I

and

(Izil)

$l(lXl>

5 u(x, t ) 5

I

$Z(IxI) 2

i= 1

ai $iz(IziI). (2.4.18)

$z(lXl>

for all x E R" and t E J . Along solutions of (9") we have I

DV,.,,,(X,

t) =

1

i= I

cli

(vui(zi,t)Thi(~, t)+(dvi(zi, t ) / d t ) )

= W T R W = W T ( ( R + R T ) / 2 ) W = WTSW

I

c $i4(1zil)2, 1

i= 1

where wT = [$14(Izll), ..., $,4(\zJ)], R = [ r V ] [aia,], and S is given in hypothesis (iii). Therefore, if S is negative semidefinite, then so is Du(,.,,(x, t ) and the equilibrium x = 0 of composite system (9") is uniformly stable. Similarly, if S is negative definite then D u ( , . , ) ( x , t ) is negative definite for all x E R" and t E J and the equilibrium of (9") is uniformly asymptotically stable in the large. w

B. Exponential Stability In studying the exponential stability of composite system (9) we will find

it useful to employ the following convention.

2.4.19. Definition. Isolated subsystem (q)possesses Property B if there exists a continuously differentiable function ui: Rnix J + R, $ i l , Icli2, $i3 E K R u.hich are of the same order of magnitude, and a constant oi E R , such that $il(lzil> 5 ui(zi,t) $i2(Izil)> D u i ( y i ) ( Z i r t ) 5 oi$i3(IziO for all zi E R". and t E J . Isolated subsystem (Sq.) possesses Property B' if it possesses Property B with $ i l (IZiI) = C ~ Izi12, I $iz(lzil) = ci2 IziI2, and $i3(Izil) = JziI2, where ci2 2 c i I > 0 are constants.

2.4

33

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

If isolated subsystem (8) possesses either Property B or Property B', and if oi < 0, then the equilibrium zi = 0 is exponentially stable in the large. If cri = 0, the equilibrium of (q) is uniformly stable and if oi > 0, the equilibrium of may be unstable.

(x)

2.4.20. Theorem. The equilibrium x = 0 of composite system (9') with decomposition (Xi) is exponentially stable in the large if the following conditions are satisfied. possesses Property B ; (i) Each isolated subsystem (q) (ii) all comparison functions t,hG, i = 1, . .., I, j = 1,2,3 of hypothesis (i) are of the same order of magnitude; (iii) given vi and ~ i of3 hypothesis (i), there exist constants ag E R such that

for all zi E R"', zj E R"j, i, j = 1, ..., I, and t E J ; and (iv) given oi in hypothesis (i), there exists an I-vector such that the matrix S = [sG] specified by (ai
is negative definite.

j(ctiag+ajaji),2,

ctT =

(ul, ..., a,)

>0

i #j

Proof. First we note that if the above hypotheses are satisfied, then all hypotheses of Theorem 2.4.2 are also satisfied. It follows that the equilibrium x = 0 of composite system (9') is uniformly asymptotically stable in the large. As in the proof of Theorem 2.4.2, we choose a Lyapunov function I

v(x,t) =

and conclude that

1 a,u,(z,,t)

i= 1

and

for all x E R", t E J , and zi E R"', i = I , ..., I, where , l M ( S )< 0 is the largest eigenvalue of test matrix S. $ 2 , $3 E K R which To complete the proof we first show that there are are of the same order of magnitude (see Definition 2.2.15) such that and

11

34

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

for all x E R“ and t

E

J , where

llxlla

=

maxIzil, i

i = 1, ...,I.

By hypothesis (ii), there is a function $ E KR, e.g., $ = +hIi, and positive constants k, such that

and

1

1

Thus, inequalities (2.4.21) and (2.4.22) are true. Since the Lyapunov theorems of Section 2.2 are valid for any norm on R”, such as 11. /Irn, it follows from Theorem 2.2.26 (with 1 . I replaced by 11. lim) that the equilibrium x = 0 of composite system ( Y )is exponentially stable in the large in the norm 11 . /Im. Furthermore, since f E J, /x(t;xo,to)/ I [“zllx(t;xo,to)IIm, it follows that the equilibrium x = 0 of composite system (9’is ) also exponentially stable in the large in the norm 1 . (.

=

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

35

In practice it is often convenient to use quadratic comparison functions $ij(r), i = 1, ...,I, j = 1,2,3. In this case, the above theorem assumes the following form. 2.4.23. Corollary. The equilibrium x = 0 of composite system (9') with decomposition (Xi) is exponentially stable in the large if the following conditions are satisfied. (i) Each isolated subsystem possesses Property B'; (ii) given ui of hypothesis (i), there exist constants a, E R such that

(z)

for all zi E Rni,z j E R"J,i, j = 1, ..., I, f E J ; and >0 (iii) given oiin hypothesis (i), there exists an I-vector aT = (ctl, ...,al) such that the matrix S = [s,] specified by s.. =

ai("i+

4,

i=j

+

y [ (ai uij ctj aji)/2, i # j is negative definite. Proof. Since each isolated subsystem (8) possesses Property B', we have $ i l ( l z i l ) =cilIziI2,t,bi2(lzil)= ci21zi12,and $i3(lzil)=/zi12,i = 1, ..., 1. Each of these comparison functions is of the same order of magnitude. Thus, all hypotheses of Theorem 2.4.20 are satisfied. Hence, the equilibrium x = 0 of composite system (9) is exponentially stable in the large. If composite system ( Y )is decomposed into n scalar subsystems (q), it is possible to reduce the conservatism of Corollary 2.4.23 by eliminating norms in hypothesis (ii). The price paid for this improvement is an increase of the order of the test matrix S . We have 2.4.24. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (Xi)is exponentially stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (X) possesses Property B' with n, = 1 so that zi = x i ; (ii) given ui of hypothesis (i), there exist constants ay E R such that

n

C

Ca~i(~i,t)/axiICgi(~~,...,~,,f)I 5 xi aexj j =1

for all xi E R,i = 1, ..., n, and t E J ; and (iii) hypothesis (iii) of Corollary 2.4.23 holds with 1 = n. Proof. Choose as a Lyapunov function u(x, t ) =

c n

i= 1

UiUi(Xi,t).

36

IT

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Following along the lines of the proof of Theorem 2.4.2, we see that c j lxI2 I u(x, t ) I c2 1x12 and

Du(,)(x, t ) 5

c3

1xI2

for all

X E R" and f E J, where el = mini{aicil}, c2 = maxi(aici2], and c3 = A,(S) < 0. The conclusion of the corollary follows from the remark following

Theorem 2.2.26.

For composite system (9'with ) decomposition specified by Eq. (2.3.16), we have the following result.

2.4.25. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (2.3.16) is exponentially stable in the large if the following conditions are satisfied. possesses Property B'; (i) Each isolated subsystem (3) given ui of hypothesis (i), there exists a positive constant ci4 such that (ii) IVui(zi, t>l I ci4 lzi( for all zi E R"I; (iii) for each i, j

=

1, . . . ,I, i # j , there exists a constant k, > 0 such that

Igi(zj,t>lI kijlzjl for all z j E R"' and t E J ; and (iv) given ci in hypothesis (i), there exists an !-vector such that the test matrix S = [sJ specified by s.. =

(L:ri:4

tlT =

..., aJT > 0

i=j k , + a j cj4 kji)/2,

i #j

is negative definite. Proof. Given zii in hypothesis (i) and for system (9' the ) Lyapunov function

CI

> 0 in hypothesis (iv), we choose

c CI,v,(z,, t ) . 1

u(x,t )

=

i= 1

From hypothesis (i) it follows that c1 /XI2

I e(x, t ) I

c2

1x12

(2.4.26)

for all x E R" and t E J , where c, = mini{aici,} and c2 = maxi{ccici,}. we have, taking hypotheses (i)-(iv) into Along solutions of system (9')

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

37

account

=

wTRw = wT((R+RT)/2)w = wTSw 2 1M(S)1x12 (2.4.27)

for all x E R” and t E J. Here, wT = ( 1 ~ ~ 1 ..., , lzll), A,(S) < 0 is the largest eigenvalue of test matrix S given in hypothesis (iv), and R = [re] is specified by r.. = y

[

Ui cj 9

i=j

aici4k , ,

i #j.

The conclusion of the theorem follows now from Theorem 2.2.26 and inequalities (2.4.26) and (2.4.27).

=

For composite system (9’with ) decomposition specified by Eq. (2.3.14), the above result assumes the following form. 2.4.28. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (2.3.14) is exponentially stable in the large if the following conditions are satisfied. (i) Hypotheses (i) and (ii) of Corollary 2.4.25 hold; (ii) given ai of hypothesis (i), there exists an I-vector aT = ( a l , ...,a!) > 0 such that the test matrix S = [s,] specified by

is negative definite.

To prove Corollary 2.4.28, replace in hypothesis (iii) of Corollary 2.4.25 g , ( z j , t ) by C e z j and k, by IIC,II. We now consider system (9”’). 2.4.29. Theorem. The equilibrium x = 0 of composite system (9”) with decomposition (C;) is exponentially stable in the large if the following conditions hold. (i) There exist continuously differentiable functions ui : R“‘ x J + R and constants ci2 2 cil > 0,i = 1, ..., I, such that Cil IZiI2 I Ui(Zi,t) I ci2 lZil 2

for all zi E Rniand t

E

J;

38

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(ii) given ui of hypothesis (i), there exist constants uii E R, i, j = 1, ..., I, such that I

+

v ~ , ( z , , r ) ~ h , ( x , ta)u i ( z i , t ) / a tI lzil

C aGlzj(

j= 1

for all zi E R“’,i = 1, . .., 1, x E R”, and t E J ; ...,a[) > 0 such that the matrix S = [se] (iii) there exists a vector aT = (al, specified by ai aii i=j s.. = ((aiuG+ajuji),2, i#j 3

is negative definite. The proof of this theorem follows along similar lines as the proof of Theorem 2.4.16.

C. Instability and Complete Instability In our next definition we let Bi(ri)= { z i E Rni : lzil < r i } for some ri > 0.

(z),

2.4.30. Definition. For isolated subsystem let there exist a continuously differentiable function ui:Bi(ri)x J + R, three functions t+ki2,$ i 3 E K , and constants ail,6 i z ,ci E R such that dil$il(tzil)

u i ( z i , t ) 5 6iz$iz(Izil)~

ouicyi,(zitf)

ei+ij(lzil>

for all zi E Bi(ri)and r E J . If hi, = diz = - 1 we say that (q.) possesses Property C. If di, = di2 = 1 we say that (Sq.)possesses Property A”.

If (Sq) has Property C with o i < 0, then the equilibrium zi = 0 is completely unstable. If (Sq) has Property A“ and ci < 0, the equilibrium of (Yi) is uniformly asymptotically stable. 2.4.31. Theorem. Let N # @, N c L = { I , ..., E ) . Assume that for composite system (9’) with decomposition ( X i ) the following conditions hold. (i) If ic N , then isolated subsystem (q) has Property C and if iQ N , i E L , then (Yi)has Property A“; (ii) given ui of hypothesis (i), there exist constants uGE R such that

for all zi E Bi(ri),z j E B j ( r j ) ,i, j E L , and t E J ; (iii) given ciof hypothesis (i), there exists an I-vector aT = (a,, . ..,a,) > 0

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

39

such that the test matrix S = [sg] specified by

is negative definite. (a) If N # L, then the equilibrium x = 0 of composite system (9’is) unstable. (b) If N = L, the equilibrium of ( Y )is completely unstable. Proof. Given ui of hypothesis (i) and CI of hypothesis (iii) we choose u(x,t) =

C aiui(zi,t). i= I

1

Proceeding as in the proof of Theorem 2.4.2, we obtain 1

Du(,)(x,t) 5

C $i3(IziI) i= 1

for all zi E Bi(ri), i E L, and t E J . Since test matrix S is negative definite, we have A M ( S )< 0 and Du(,,(x, t ) is negative definite. Now consider the set D

R” x J : zi E B i ( r ) whenever i E N , where r < min r i , and zi = 0 whenever i .$ N , and t E J } .

= { ( x ,t ) E

For (x, t ) E D we have

i.e., in every neighborhood of the origin x = 0, there is at least one point x’ # 0 for which u(x’, t ) < 0 for all t E J . Furthermore, on the set D, u ( x , t ) is bounded from below. Thus, all conditions of Theorem 2.2.28 are satisfied. If N # L , then the equilibrium x = 0 of composite system (9) is unstable. If N = L, then u(x, t ) is negative definite and the equilibrium of (9’) is completely unstable. In the next result we consider composite system (9”’). 2.4.32. Theorem. Let L = { I , ..., r } , N c L , and N # 0. Assume that for composite system (9”’) with decomposition (C;) the following conditions hold. (i) There exist continuously differentiable functions u i : Bi(ri)x J - , R and $il, t+kiZ E K , i E L, such that $il

(Izil) 5 ui(zi, t ) 5 $i~(lzil)~ i .$ N ,

i E L,

40

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and

(lzil) 5 Ui(zi,t) -+iz(lzil)~ i E N, for all zi E Bi(ri) for some ri > 0, i E L, and all t E J ; (ii) given ui of hypothesis (i), there exist a# E R and t,bi4 E K , i , j E L, such that -$ij

/

for all zi E Bi(ri),zj E B j ( r j ) ,i, j E L , and t E J ; and (iii) there exists a vector aT = (al,..., q) > 0 such that the matrix S = [sJ given by mi aii i= j s.. ‘I == [ ( a i u G + a j a j i ) / 2 , i j 9

+

is negative definite. (a) If N # L , then the equilibrium x = 0 of composite system ( Y )is unstable. is completely unstable. (b) If N = L, the equilibrium of (9”) The proof of this theorem involves obvious modifications of the proofs of Theorems 2.4.16 and 2.4.31.

D. Uniform Boundedness and Uniform Ultimate Boundedness And now we consider the boundedness of solutions of system (Y), 2.4.33. Definition. Isolated subsystem (Sq) possesses Property D if there exists a continuously differentiable function u j : R”‘x J + R, $ i l , t,bjZ, 4hi3 E K R and a constant ci E R such that $ii(Izil)

5

ui(zi,f)

5 $iz(lziI)j

Dui(Y,)(zi,t ) 5

~i+i3(lzil)

for all t E J and for all (zi12Ri (where Ri may be large) and such that ( z i ,t ) are bounded on Bj(Ri) x J . ui (zi ,t ) and If ci 5 0, then the solutions of subsystem (Sq) are uniformly bounded and if ci < 0, then the solutions of (Sq) are uniformly ultimately bounded.

2.4.34. Theorem. The solutions of composite system (9) with decomposition ( X i ) are uniformly bounded, in fact, uniformly ultimately bounded if the following conditions are satisfied. (i) Each isolated subsystem (Yi)possesses Property D; (ii) given ui and t,hi3 of hypothesis (i), there exist constants a# E R such that

2.4

41

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

for all zi E Rni,z j E R"], i, j = 1, ..., I, and t E J ; and (iii) given vi of hypothesis (i), there exists an I-vector aT = (a1, ..., a,) > 0 such that the matrix S = [SJ specified by

is negative definite. ProoJ Given the functions ui of hypothesis (i) and the vector hypothesis (iii), we choose

tl

of

c tliUi(Z,,t). I

o(x,t) =

i= 1

Following the proof of Theorem 2.4.2 it is clear that there exist E K R such that

$1,$z,$3

$1

(1x1) 5 o(x, t ) 5 *z(lxI),

Dq,,(x, t ) I n,(s)

(1x1)

$3

where A,(S) < 0, whenever x E R"- B, ( R , ) x ... x B,(R,) and t E J. To complete the proof we need to consider the situation where some of the ziare such that (zi( < Ri. First consider the case where lzil 2 Ri, i = 1, ..., r , and lzil < Ri, i = r + l , ..., 1. Then I

1

i=r+ 1

aioi(zi,t) +

,

r

1 ai$il(lZil) 5 u ( x , t > 5 C i= 1

i= 1

1

.i$i2(l~il)

+ 1 1 aiui(zi,t). ,=r+

Since ui is continuous on R", x J and bounded on Bi(Ri)x J , i = 1, ..., 1, it follows that there are q l ,q 2 E K R such that q 1(1x1) I o(x, t ) 5 (p2(lxl) f o r a l l t E J a n d forall x ~ R " s u c h t h a tl z i l < R i , i = r + l , ..., 1, and such that Izil, i = 1 , ..., r , are sufficiently large. Along solutions of (9) we have r

Do(,,(x, t ) =

1 ai{Dui(y,)(Zi,t)+Vui(zi, t)Tgi(Z1,...,zl, t > ) i= 1

11

42

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Now for all Izi(< Ri, i = r + l , ..., I , there exist constants K,, K2, K3, and Mi such that I

K1 2

Let

M,’ =

.

C

J=r+l

IaijIC$j3(IzjI>I”2,

[$,3((zll), . . . , $ r 3 ( z) , \ ) ] , l e t t h e r x r m a t r i x R = [ri] bespecifiedby

cci(ai+aii), and let

Mi 2 I ~ ~ i ~ ~ ~ ~ ( z i ~ ~ ) I

s” = ( R + RT)/2. Then

i =j

i#j,

I

Therefore, if the Izi/ for i = 1, ..., r are sufficiently large, the sign of D u ( , , ( x , r ) is determined by L,(s”)C:=,$i3(1zil), and Du(,)(x,t) < 0 for all t E J . Above we have assumed that lzil> R i , i = 1, ..., r, and lzil< Ri, i = r + I , ..., 1. For any other combination of indices, the proof is similar. E. Discussion

At this point, a few remarks concerning the preceding results are in order. In all results for composite systems (9’) and (Y),the analysis is accomplished in terms of the qualitative properties of the lower order isolated sub-

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

(x’),

43

systems (q) and respectively, and in terms of the properties of the interconnecting structure expressed by gi (x, t ) and gi( x ) , respectively. These results involve generally a reduction in dimension (from n to I ) and are in such a form as to reveal qualitative trade-off effects between subsystems and interconnecting structure. In the case of composite system (Y”), a reduction in dimension is also realized, however, the results do not yield qualitative trade-off information between isolated subsystems and interconnecting structure, for in this case the notion of isolated subsystem is undefined. For systems with a fine decomposition (i.e., a decomposition into many subsystems) it is as a rule easy to find Lyapunov functions for the lower order subsystems; however, the resulting stability (or instability or boundedness) conditions tend to be conservative. (Note however, that in the case of Corollary 2.4.24 involving the finest possible decomposition, the conservatism can be reduced appreciably.) On the other hand, in the case of systems with a coarse decomposition (i.e., a decomposition into few subsystems) it is usually more difficult to find Lyapunov functions for the subsystems (which may no longer be of low order); however, the resulting stability conditions are usually less conservative. This points to the following advantages (items (a) and (b) below) and disadvantages (item (c)) of the present method. (a) It is often possible to circumvent difficulties that arise when the Lyapunov method is applied to systems of high dimension and intricate interconnecting structure. (b) In the case of systems ( Y )and (9’) the analysis is accomplished in terms of system components and system structure. (c) If a system is decomposed into many subsystems, the results may be conservative. Theorems 2.4.16, 2.4.29, and 2.4.32 are of course also applicable to composite system (Y),since this system is a special case of system (9”). This demonstrates that the decomposition of large scale systems into isolated subsystems is a conceptual convenience and not a necessity. Indeed, Theorems 2.4.16, 2.4.29, and 2.4.32 may in principle yield less conservative conditions than corresponding results involving isolated subsystems, when applied to a given problem, for the former set of results involves fewer majorizations (estimates by comparison functions) than the latter set. However, it is usually more difficult to apply the former than the latter. Of all the preceding results, Corollaries 2.4.25 and 2.4.28 are perhaps the easiest to apply. The reason being that these results pertain to systems for which a great deal of information of the interconnecting structure is available. This suggests that the method of analysis advanced herein is in a sense much more important than the specific results presented. That is to say, given a specific system with special structure, it may often be more desirable to arrive at a result tailored to this particular system, using the present method of

44

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

analysis, rather than try to force this system into one of the specific forms considered in Section 2.3. We will demonstrate this further in Section 2.8. In all results presented thus far we require the existence of a vector ciT = (al, ...,ciJ > 0 which arises i n the choice of Lyapunov functions employed. These u-functions are always of the form

The motivation for choosing functions of this type is as follows. We may think of u as representing a measure of energy associated with a given composite system and we may view uias providing a measure of energy for isolated subsystem (9, Once ).the isolated subsystems are interconnected t o form the composite system, it is reasonable to assume that the qualitative effects of different subsystems on the overall system may vary in importance. Therefore it seems reasonable to assign different weights cii > 0 t o different functions u i , thus forming the indicated Mteighted sum of Lyapunov functions which serves as a Lyapunov function for the overall interconnected system. It is of course possible to combine the weighting factors cii with the respective functions ui . In this case the test matrices S are modified, replacing each cii by 1. However, there are the following two good reasons for not doing this. (a) In applications t o specific problems, the presence of a provides us with an added degree of flexibility, and in addition, judicious choice of a enables us frequently to reduce the conservatism of the results. (b) In the special case when the off-diagonal elements of S are nonnegative, the presence of ci will enable us to obtain results which are equivalent t o the present ones, however, are much easier t o apply. This will be accomplished in the next section. In physical large systems, it is usually true that a given subsystem (9, is) i = 1, ..., 1, i # k . Thus, in not connected to every other subsystem applications, matrix S is usually sparse and it may be possible t o apply linear programming methods (see Dantzig [I]) and sparse matrix results (see Tewarson [ 11) t o determine the optimal choice of ci. On the other hand, when the order of S is small, the optimal choice of CI is usually obvious. The concept which gave rise t o the qualitative analysis of dynamical systems at different hierarchical levels is the notion of vector Lyapunov function, introduced by Bellman [2]. In a certain sense, the results of this section may be viewed as analysis via vector Lyapunov functions. Specifically, if VT = ( u l , ..., u l ) E R' and ciT = (ul, ..., CI') E R',ai > 0, i = 1, ...,I, then

(z),

c aiui(.). 1

u(.)

=

ciTV =

i= 1

Nevertheless, u ( . ) is a scalar Lyapunov function. We shall reserve the term

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

45

“vector Lyapunov approach” to cases when an appropriate comparison principle is applied to a differential inequality (of order I ) involving vector Lyapunov functions. This type of approach will be considered in Section 2.6. To simplify matters, we consider in the following discussion only Theorem 2.4.2. However, similar statements apply to the remaining results as well. We call the parameter oi, introduced in Definition 2.4.1, a degree or a margin of stability. In order to satisfy hypothesis (iii) of Theorem 2.4.2, it is necessary that ( o i + a i i )< 0, i = 1, ..., 1. Thus, if ci > 0, so that (Yi) may be unstable, we require that a,, < 0 and laiil > oi. In other words, this theorem is applicable to composite systems with unstable subsystems, provided that there exists a sufficient amount of stabilizing feedback (i.e., local negative feedback) asStability ). results of the sociated with (YJ,which is not an integral part of (9, type presented herein for composite systems with unstable subsystems and without local stabilizing feedback have apparently not been established at this time. This problem is of great practical importance and needs to be pursued further. (Results for systems with unstable subsystems are given in Thompson [2]. However, they do not involve a reduction in dimension, and as such, this problem may be regarded as being essentially unsolved.) Generally speaking, the greater the margin of stability associated with each subsystem (i.e., the more negative the terms ( c T ~ + Q ~ ~ )i, = I , ..., /), the easier it is to satisfy the negative definiteness requirement of matrix S. Note that as in the case of weights cli, it is of course possible to combine cri with replacing oi by - 1, 0, or 1, when is uniformly asymptotically stable, uniformly stable, or possibly unstable, respectively. We emphasize that the off-diagonal terms of test matrix Scan have arbitrary signs. Tn the next section we consider additional results for which the off-diagonal terms of the test matrix are required to possess the same sign. Note also that in this section the test matrices are always symmetric. This will in general not be required in the results of the next section. Finally note that inequalities of the type encountered in hypothesis (ii) of Theorem 2.4.2, which express restraints on the interaction among the subsystems, are more easily satisfied than may appear at first glance. For example, hypothesis (ii) of Theorem 2.4.2 can be satisfied for appropriately chosen aij if g i ( x ,t ) are linear functions and v i ( z i ,t ) are quadratic, i = 1, .. ., 1. The results of this section can often be utilized in compensation and stabilization procedures at different hierarchical levels. To simplify matters, we consider once more Theorem 2.4.2. Similar statements can be made for the remaining results. We choose in hypothesis ( 5 ) of this theorem cli = 1, i = 1, ..., 1. It can be shown (see, e.g., Taussky [l]) that all eigenvalues of matrix S are negative if the diagonal dominance conditions

(x)

46

I1

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

hold. Now suppose we are able to ascertain by some means (such as Theorem 2.4.31) that a specific system (9) is unstable. Or suppose we are able to determine by Theorem 2.4.2 that (9) is uniformly asymptotically stable, however, the degree of stability of (Y), as expressed by A M ( S ) ,is unsatisfactory. Using appropriate local feedback associated with the subsystems and the interconnecting structure, it is frequently possible to stabilize (Y),or to enhance the stability of (9). In this process, the local compensators are chosen so as to increase the terms -((aii+aji)and decrease the terms l a g + a j i [ .However, it must be noted that this procedure may often yield overly conservative results. Furthermore, it may physically be impossible or undesirable to use this type of an approach.. We conclude our discussion by noting that all preceding results involving test matrices with off-diagonal terms having arbitrary signs are applicable to strongly coupled systems as well as weakly coupled systems.

2.5 Application of M - M a t r i c e s In the special case when the off-diagonal elements of the test matrices S in the results of Section 2.4 are nonnegative, we can utilize the properties of Minkowski matrices, called M-matrices, to establish results which are easier to apply because they do not involve usage of weighting vectors a. However, we are obliged to emphasize at the outset that (a) in the case the off-diagonal elements of S are nonnegative, the results of the preceding section and corresponding results of this section are equivalent, (b) whereas it is always possible to obtain the results of this section from those of the preceding section, the converse is in general not true, and (c) the results of the present section are generally applicable only to weakly coupled systems. This section consists of six parts. In the first of these we give a summary of selected results from the theory of M-matrices. Since this material is well covered in several references, all proofs of M-matrix results are omitted. In the remaining parts we establish several Lyapunov results for composite systems (9’) and (Y”),most of which have corresponding counterparts in Section 2.4. A. M-Matrices We begin with the following definition.

2.5.1. Definition. A real I x l matrix D = Id,] is said to be an M-matrix if d, i 0, i # j (i.e., all off-diagonal elements of D are nonpositive), and if all principal minors of D are positive (i.e., all principal minor determinants of D are positive).

2.5 Let D

= [d,]

APPLICATION OF M-MATRICES

47

be an I x I matrix. If the determinants D,, given by

are all positive, we say that the ‘‘successive principal minors of D are positive,” or, the “leading principal minors of D are positive.”

2.5.2. Theorem. Let D = [d,] be a real I x I matrix such that d, I 0 for all i # j . Then the following statements are equivalent. (i) The principal minors of D are all positive (i.e., D is an M-matrix). (ii) The successive principal minors of D are all positive. (iii) There is a vector u E R‘ such that u > 0 and such that Du > 0. (iv) There is a vector u E R‘ such that u > 0 and such that DTu > 0. (v) D is nonsingular and all elements of D-’are nonnegative (in fact, all diagonal elements of D - l are positive). (vi) The real parts of all eigenvalues of D are positive.

A direct consequence of parts (iii) and (iv) of Theorem 2.5.2 is the following. 2.5.3. Corollary. Let D = [d,] be an I x I matrix with nonpositive off-diagonal elements. Then the following statements are true. (i) D is an M-matrix if and only if there exist positive constants ,Ij, j = 1, ..., I, such that (2.5.4) (ii) D is an M-matrix if and only if there exist positive constants q j , j = 1, .. .,I, such that

C q j d j i > 0, 1

j= 1

i

=

1, ..., I.

(2.5.5)

Additional useful properties of M-matrices which we will require and which are a consequence of Theorem 2.5.2 are the following.

2.5.6. Corollary. Let D = [d,] be an I x I matrix with nonpositive off-diagonal elements. Then D is an M-matrix if and only if there exists a diagonal matrix A = diag[a,, ..., a*], ai > 0, i = I, ..., I, such that the matrix

is positive definite.

B

=

AD

+ DTA

(2.5.7)

2.5.8. Corollary. Let C = diag [el, .. .,el] > 0 be a diagonal I x I matrix and let Q = [q,] 2 0 be an I x I matrix. Then C- Q is an M-matrix if and only if

48

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

there is a diagonal I x I matrix A B

=

= diagca,,

...,cc,]

> 0 such that the matrix

C A C - QTAQ

(2.5.9)

is positive definite. 2.5.10. Corollary. If D is an M-matrix, then D - p I is an M-matrix if and only if p < minitL Re[Ai(D)], L = { I , ..., I}, where I is the 1 x 1 identity matrix and Re[Ai(D)] denotes the real part of the ith eigenvalue of D.

B. Uniform Asymptotic Stability Our first result is as follows. 2.5.11. Theorem. Assume that for composite system (9’ with ) decomposition (C,) the following conditions hold. (i) Each isolated subsystem ($) possesses Property A (see Definition 2.4.1); and (ii) given v, and $ i 3 of hypothesis (i), there exist constants aii E R such that (a) at 2 0 for all i # j , and C:.=~ for all (b) VOi(Zi,t)Tgi(Z1, . . . , ~ l r t ) I C $ ~ ~ ( I Z ~ I ) I ~ ’ ~ac[$j3(I~jI)I”~ z i ~ R n i , z j ~ R “ ~ , i ,..., , j = 11 , a n d t ~ J . is uniformly asymptotically stable in The equilibrium x = 0 of system (9’) the large if any one of the following conditions hold. (iii) given (o, of hypothesis (i), the successive principal minors of the I x I test matrix D = [d,] are all positive, where - ((oi

+a,,),

i

=j

i # j. (iv) The real parts of the eigenvalues of D are all positive. (v) There exist positive constants Ai,i = 1, . . .,I, such that

C 1

-((oi+aii)-

j=l,i#j

(Aj/Ai)uy > 0,

(vi) There exist positive constants qi, i -((oi+aij)-

I j=l,i#j

=

i

= I , ..., 1.

(2.5.12)

1, ...,1, such that

(qj/qi)aji > 0,

i = 1, ..., 1.

(2.5.13)

Proof. Since by assumption ay 2 0 for all i # , j , and since the successive principal minors of matrix D are all positive, it follows from part (ii) of Theorem 2.5.2 that D is an M-matrix. In view of Corollary 2.5.6 there exists a

2.5

49

APPLICATION OF M-MATRICES

diagonal matrix A = diag[a,, ..., 4 > 0 such that the matrix -2s 4 AD

+ DTA

is positive definite. Here, - 2 s = - 2 [SJ is specified by sij =

+ aii),

i=j

(cciai+ajaji)/2,

i # j.

ai(0i

But if - 2 s is positive definite, then the matrix S = [ S J is negative definite. Therefore, if hypotheses (i)-(iii) of the present theorem are satisfied, then all hypotheses of Theorem 2.4.2 are also satisfied. Ijence, the equilibrium x = 0 of system (9’)is uniformly asymptotically stable in the large. The proof is now complete, for hypotheses (iii)-(vi) are equivalent statements (see Corollary 2.5.3 and part (vi) of Theorem 2.5.2). H Henceforth we shall refer to inequalities (2.5.12)and(2.5.13) as rowdominance condition and column dominance condition, respectively. In all results of the previous and present section, we have used Lyapunov functions which are continuously differentiable. In our next theorem, we make use of continuous Lyapunov functions which need not be continuously differentiable. In doing so, we employ the following convention.

(z)

2.5.14. Definition. Isolated subsystem possesses Property & . if there exists a continuous function vi: Rnix J - , R, Iclil, Ic/i2 E KR, t+hi3 E K , constants L i , oi E R , Li > 0, such that Ic/il(lzil) 5 u i ( z i , t ) 5 +iz(Izil),

+

(zi, t ) = lim sup(l/h) {vi [ z i ( t + h; z i , t ) , t h] - v i ( z i , t ) } h+O+ (Ti

and

Ic/i3(lziI),

IUi(Zi’,

for all z i , zi‘, z;

E

t)-vi(z;,t)l

5 LiIzi’-z;I

Rniand I E J.

2.5.15. Theorem. The equilibrium x = 0 of composite system (9’)with decomposition ( X i ) is uniformly asymptotically stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (Sp) possesses Property A ; (ii) given $i3 of hypothesis (i), there exist constants a8 2 0, i, j = 1, ..., I, such that

for all zi E R“’,i = 1 , ...,I, and

tE

J ; and

50

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATlONS

(iii) given ci and Li of hypothesis (i), the successive principal minors of the I x 1 test matrix D = [d,] are all positive, where

- (ui + Li qi),

i

=j

i#j. Proofi Given ui of hypothesis (i), let aT = ( a l , ..., a,)> 0 be a n arbitrary constant vector and choose as a Lyapunov function for system ( Y ) ,

c UiUi(Zi,t). 1

u(x,t) =

i= 1

From hypothesis (i) it follows that u is continuous, positive definite, decrescent, and radially unbounded. Along solutions of system (9’we ) have f k , : , , ( x , r ) = lim sup(l/h) h-O+

lim sup(l/h)

=

h+O+

- ui

C ai[ui(Zi(r+h;zi,t),t+h)-ui(zi,t)]

{ i =l 1

c ai{ui[zi+h.J;(zi,t ) + h . g , ( x , t>+o(h),t+h]

i= 1

(zi > t ) >

lim sup ( I / h )

=

h-O+

C ai {ui [zi+ h.f,(zi,t )+ o(h), t + h] - ui ( z i ,t ) 1

i= 1

+ ui[zi+h.f;.(Zi,t ) + h . g , ( x , r)+o(h), t+h] -

uiCZi+h.fi(Zi, t ) + o ( h ) , t + h ] }

where o(h) denotes higher order terms so that o(h)/h+O as h.0. In view of hypotheses (i) and (ii) we now have I

1

Du(,)(x,t) 5

1 aiDui(y,)(Ziyt)+ iC= a i L i J g i ( x , t ) (

i= 1

1

Now let wT = [t,hl3(Iz,I),..., I + ~ ~ ~ ( / Zand ~ \ ) ]note that w = 0 if and only if

x = 0. We have

D u , ~ , ( x , I~ )-aTDw

=

-yTw,

where aTD = yT and matrix D is defined in hypothesis (iii). Since D has positive successive principal minors and since d,, 5 0 for all i # , j , it follows from parts (ii) and (v) of Theorem 2.5.2 that D is an M-matrix, that D-’

2.5

51

APPLICATION OF M-MATRICES

exists and that D-' 2 0. Thus, c1

= (D-')Ty.

Since each row and column of D-' must contain at least one nonzero element (in fact, the diagonal elements of D-' are positive), we can always choose y in such a fashion that y > 0, so that a > 0. Therefore, DU(,,(X,t)

< - y T w < 0,

x # 0.

Hence, D U ( ~ ) (t)X is , negative definite for all x E R" and t E J . It follows that the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. By taking advantage of Corollary 2.5.3 and following the proof of Theorem 2.5.15, we can establish our next result. 2.5.16. Corollary. Assume that hypothesis (i) of Theorem 2.5.15 is satisfied with oi E R replaced by a continuous function o i : Rnix J - + R . Assume that hypothesis (ii) of Theorem 2.5.15 is satisfied with ayE R replaced b,y a continuous function ay: R"Jx J R with the property a y ( z j ,t ) 2 0 for all z j E R"J and all t E J . Let D ( x , t) = [d,(zj,t ) ] be determined by -+

- [ai ( z i , t)

d,(zj,t)=

+Lia ii( zi ,t)],

- Lia@(zj9 t>,

i

=j

i#j

If there exist constants l i > 0, i = 1, ..., I, and a constant y > 0 such that 1

d j j ( z j t, ) -

C (Ai/Aj)ldy(zj,t)l L y, i=l,i#j

j = 1, ..., 1,

for all z j E R"j, j = 1, ..., I, t E J, then the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. For system (9"') we have the following result. 2.5.17. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.16 are true. The equilibrium x = 0 of composite system (9'") with decomposition (C:) is uniformly asymptotically stable in the large if ay 2 0 for all i # j and if the successive principal minors of the test matrix D = [Id,], where dy = - a y , are all positive. Proof. Using an argument similar to that of Theorem 2.5.1 I , the proof follows from Theorem 2.4.16 and Corollary 2.5.6. C. Exponential Stability Next, we consider several exponential stability results for composite and (9'"). systems (9')

52

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.5.18. Theorem. Assume that hypotheses (i)-(iii) of Theorem 2.4.20 are true. Then the equilibrium x = 0 of composite system ( Y )with decomposition ( X i ) is exponentially stable in the large if a, 2 0 for all i # j and if the successive principal minors of the test matrix D = [d-,] are all positive, where d.. =

La,,

-(oi+aii),

i =j i # j.

Proof. The proof follows from Theorem 2.4.20 and Corollary 2.5.6, following a similar procedure as in the proof of Theorem 2.5.1 1.

2.5.19. Corollary. Assume that hypotheses (i) and (ii) of Corollary 2.4.23 are true and assume that a, 2 0 for all i # j . The equilibrium x = 0 of system (9) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where - (Oi

+U i i ) ,

i

=

;

i # .i. f r o q f The proof follows from Corollaries 2.4.23 and 2.5.6.

m

The next two results are concerned with composite system (9’) having decompositions determined by Eqs. (2.3.16) and (2.3.14), respectively. 2.5.20. Corollary. Assume that hypotheses (i)-(iii) of Corollary 2.4.25 are true. The equilibrium x = 0 of system (9’)with decomposition (2.3.16) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where

pro(^ The proof follows from Corollaries 2.4.25 and 2.5.6.

2.5.21. Corollary. Assume that hypothesis (i) of Corollary 2.4.28 is true. Then the equilibrium x = 0 of system (9’)with decomposition (2.3.14) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where

Pro($ This proof follows from Corollaries 2.4.28 and 2.5.6.

m

I n the next theorem we consider system (9”’). 2.5.22. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.29 are true. Then the equilibrium x = 0 of composite system (9”) with decompo-

2.5

APPLICATION OF M-MATRICES

53

sition (C;) is exponentially stable in the large if aY 2 0 for all i # j and if the successive principal minors of the test matrix D = [d,] are all positive, where d.. = - a.. . (f ProoJ The proof follows from Theorem 2.4.29 and Corollary 2.5.6.

The next result provides another interesting application of M-matrices. 2.5.23. Corollary. Let D denote the test matrix ofTheorem 2.5.1 1 (or Theorem 2.5.18). Assume that composite system (9’)with decomposition (Ci) has been shown to be uniformly asymptotically stable in the large (or exponentially stable in the large) using one of these results. Then any modification of the isolated subsystems or their feedback as expressed by a,i, i = 1, . .., I, which increases (ai+ aii),i = 1, .. .,I, by an amount less than p = mink Re [Ak(D)], k = 1, . .., I, will leave system (9’)uniformly asymptotically stable in the large (exponentially stable in the large).

(x),

Proof. The proof follows directly from Theorem 2.5.11 (or Theorem 2.5.18) and from Corollary 2.5.10. The parameter p in Corollary 2.5.23 may be interpreted as a margin of stability or as a degree of stability of the overall interconnected system (9) and may be used to judge how sensitive the qualitative properties of system ( Y )are with respect to structural changes. D. Instability and Complete Instability Next, we consider the instability of systems (9) and (9’”). 2.5.24. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.31 are true, that uv 2 0 for all i # J , and that all principal minors of the test matrix D = EdY]are positive, where d.. =

-(ai+aii), [-uY,

i

=j

i # j.

(a) If N # L, then the equilibrium x = 0 of composite system (9’)is unstable. (b) If N = L, the equilibrium of system (9’)is completely unstable. Proof. The proof follows from Theorem 2.4.31 and Corollary 2.5.6.

2.5.25. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.32 hold, that uv 2 0 for all i #j, and that all principal minors of the test matrix D = [dv] are positive, where dii = -as. If N # L, then the equilibrium x = 0 of

54

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

composite system (9”) is unstable and if N completely unstable.

= L,

the equilibrium of (9”) is

ProoJ The proof follows from Theorem 2.4.32 and Corollary 2.5.6.

w

E. Uniform Boundedness and Uniform Ultimate Boundedness Our last result yields sufficient conditions for boundedness of solutions of composite system (9’). 2.5.26. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.34 are true, that ay > 0 for all i # j , and that the successive principal minors of the test matrix D = are all positive, where

[4]

d.. =

Lag,

-(ai+aii),

i =j i # j.

Then the solutions of composite system (9) with decomposition (Xi) are uniformly bounded, in fact, uniformly ultimately bounded. Proof. The proof follows from Theorem 2.4.34 and Corollary 2.5.6.

F. Discussion We conclude this section with a few observations, phrased in the notation of Theorem 2.5.1 1. Similar statements can be made for the remaining results. To satisfy hypotheses (iii)-(vi) of this theorem, it is necessary that (ai+ aii)< 0, i = 1, .. .,1. Thus, if ai > 0, so that subsystem (q)may be unstable, we require that a,, < 0 and laii\> a i . In other words, as in the case of Theorem 2.4.2, Theorem 2.5.11 is applicable to composite systems with unstable subsystems, provided that a sufficient amount of stabilizing feedback is associated with each unstable subsystem, where the feedback is not an integral part of the subsystem structure. Clearly, the greater the degree of stability associated with each subsystem, as expressed by a i + a i i , i = 1, ..., I, the easier it is to satisfy hypotheses (iii)-(vi) of Theorem 2.5.11. Also, the weaker the interconnections, i.e., the smaller the terms ay > 0, i # j , the easier it is to satisfy these hypotheses. Thus, Theorem 2.5.11 constitutes a set of weak coupling conditions for uniform asymptotic stability of system (9). The same is true in the case of Theorem 2.4.2 when ag 2 0, i # j . These statements are of course not surprising, for if ag 2 0, i # j , then Theorems 2.4.2 and 2.5.1 1 are equivalent. However, it is emphasized that the test matrix S in Theorem 2.4.2 has no restrictions in sign for the off-diagonal elements (i.e., a y , i # j , can have any sign) and therefore Theorem 2.4.2 is a more general result than Theorem

2.5

APPLICATION OF M-MATRICES

55

2.5.1 1. Note however that the latter result is easier to apply than the former. Specifically, in Theorem 2.4.2 we utilize a symmetric test matrix S involving a weighting vector a > 0, while in Theorem 2.5.1 1 we employ a test matrix D which in general need not be symmetric and which does not involve a weighting vector. Note that hypothesis (iv) of Theorem 2.5.11 enables us to deduce the stability properties of the nonlinear, nonautonomous n-dimensional composite system (9’)from the linear autonomous system j = Dy, where I I n. Note also that in the row and column dominance conditions (2.5.12), (2.5.131, ‘there appear arbitrary sets of constants {Ai), {qi}, i = 1, ..., 1, respectively. Due to the simple form of these conditions, usage of linear programming methods appear to be attractive to determine the optimal choice of these constants in specific high-dimensional problems (to obtain the least conservative results). In the case of low-dimensional problems, this choice may often be determined by inspection. The row and column dominance conditions are especially well suited for systematic stabilization and compensation procedures of large scale systems. Conceptually, such procedures involve the following steps. (a) Enhance the degree of stability of specific subsystems, using local feedback at the subsystem level structure (i.e., increase -((ai+aii));or (b) weaken coupling effects, using local feedback at the interconnecting structure level (i.e., decrease appropriate choices of ay 2 0, i # j ) ; or (c) combine items (a) and (b). In this procedure Corollary 2.5.23 is useful in providing a measure for the margin of stability p of the overall composite system (9). The constants av 2 0, i # j , in Theorem 2.5.1 1 provide a measure of dynamic interaction among subsystems. If in particular aij = 0, we view (Yj)as not being connected to (YJand if aji = 0, we view as not being connected to (Yj).Suppose now that any one of the equivalent hypotheses (iii)-(vi) of Theorem 2.5.1 1 have been satisfied for system (9) for a given set of constants a” 2 0, i # j . Then it is clear that if any one (or all) of the aij 2 0, i # j , are decreased or set equal to zero, the stability conditions (iii)-(vi) are preserved. (This has motivated some authors to speak of so-called “connective stability.” Since results of the type considered above have this feature automatically built in, we decline to pursue this notion any further.) Notice further that if an increase of say a i k ,i # k , is accompanied by an appropriate decrease of a.IP p = 1, ..., 1, i # p # k , then stability condition (2.5.12) can be preserved. The preceding discussion suggests that in certain cases one may want to pursue the following approach in the planning of reliable large scale dynamical systems. (a) If applicable, a given system is viewed as an interconnection of subsystems, also called “areas”;

(x.)

7

56

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(b) each area is planned in such a fashion so that it is endowed with a sufficient margin of stability; (c) the dynamic interaction among various areas is limited in such a fashion that either the row or the column dominance conditions are satisfied; (d) the overall system is planned in such a way that it exhibits a sufficient degree of stability; (e) if at a future point in time it is necessary to increase the interaction among certain areas, this will have to be accompanied by an appropriate increase in the margin of stability of the affected areas and/or by a suitable decrease in the strength of coupling elsewhere. Note that a system planned in this fashion is reliable in the sense that subsystems can be disconnected and reconnected (intentionally or by accident) without affecting the basic stability properties of the main system and the disconnected subsystems. 2.6 Application of the Comparison Principle t o Vector Lyapunov Functions

In the present section we first give a brief overview of several comparison theorems which are the basis of the comparison principle. Next, we show how this principle can be applied in the analysis of large scale systems using vector Lyapunov functions. We will show that for most of the specific cases considered thus far in the literature, this method reduces to the scalar Lyapunov function approach considered in Sections 2.4 and 2.5, where usage of the comparison principle is not required. Furthermore, we will demonstrate that the method of the present section, when applied to interconnected systems, involves test matrices which are always required to be M-matrices. Since the comparison principle is treated in detail in several texts, proofs of the comparison theorems are omitted. We begin by considering a scalar ordinary differential equation of the form

where y E R , t E J , and G : B ( r ) x J + R for some r > 0. Assume that G is continuous on B ( r ) x J a n d that G(0, t ) = 0 for all t 2 to. Under these assumptions it is well known that Eq. (C) possesses solutions y ( t ; yo, to) for every y o = y ( t o ;y o , to) E B ( r ) , which are not necessarily unique. These solutions either exist for all t E [to,a)or else must leave the domain of definition of G at some finite time t , > to. Also, under the above assumptions, Eq. ( C ) admits the trivial solution y = 0 for all t 2 to. As before, we assume that y = 0 is an isolated equilibrium. Finally, for the sake of brevity, we frequently write y ( t ) in place of y ( t ; yo, to) to denote solutions, with y ( t o ) = y o .

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

57

2.6.1. Definition. Let p ( f ) be a solution of Eq. (C) in the interval [to,a). Then p ( t ) is called a maximal solution of (C) if for any other solution y ( t ) existing on [to,a), such that p ( t o ) = y(to) = yo, we have y ( t ) I p ( t ) for all t E [ t o , 4. 2.6.2. Definition. Let q ( t ) be a solution of Eq. (C) on the interval [to,a). Then q ( t ) is called a minimal solution of (C) if for any other solution y ( t ) existing on [to,a), such that q(to)= y ( t o )= yo, we have y ( t ) 2 q ( r ) for all t E [ t o , 4.

For maximal and minimal solutions of Eq. (C) we have the following existence theorem. 2.6.3. Theorem. If G is continuous on B(r) x J and if y o E B(r), then Eq. (C) has both a maximal solutionp(t) and a minimal solution q ( t ) for anyp(to) = q (t o )= yo. Each of these solutions either exists for all t E [to,a)or else must leave the domain of definition of G at some finite time t 1 > to. The following comparison theorem is fundamental to the theory. 2.6.4. Theorem. Assume that G is continuous on B(r) x J and that p ( t ) is the maximal solution of Eq. (C) on the interval [to,a) with p ( t o ) = yo. If r(t) is a continuous function such that r(to) 5 yo, if Dr(t) and if

=

lim sup[r(t+h) - r(t)]/h

h+O+

almost everywhere on

Dr(t) 5 G(r(t), t )

[to,a),

then r(t) 5 p ( t ) on [ro,a). We also have 2.6.5. Theorem. Assume that G is continuous on B(r) x J and that q ( t ) is the minimal solution of Eq. ( C ) on the interval [ t o ,a) with q(to) = yo. If s ( t ) is a continuous function such that s(to)2 yo, and if almost everywhere on

Ds(t) 2 G ( s ( t ) ,t ) then

s(t) 2

[to,a),

q ( t ) on [to,a).

Theorem 2.6.4 (as well as Theorem 2.6.5), referred to as a comparison principle, is a very important tool in applications because it can be used to reduce the problem of determining the behavior of solutions of Eq. (I),

1

= g(x,t),

(1)

x E R",g: B(r) x J + R", to the solution of a scalar equation (C). To be more

specific, we have in mind the application of Theorem 2.6.4 (as well as Theorem

58

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.6.5) to the case where r ( t ) = v ( x ( t ) ,t ) (or s ( t ) = v ( x ( t ) ,t ) ) , where v : B(r) x J + R is a Lyapunov function and x ( t ) is a solution of Eq. (I). (As in Section 2.2, we assume that g(0, t ) = 0 for all t 2 to.) In particular, applying Theorem 2.6.4 to u ( x , t ) , one can easily see that the following results are true. 2.6.6. Theorem. Let g and G be continuous on their respective domains of definition. Let v : B ( r ) x J - , R be a continuous, positive definite function such that DV(,)(X,

t > 5 G ( v ( x ,t ) , t ) .

(2.6.7)

Then the following statements are true. (i) If the trivial solution of Eq. ( C ) is stable, then the trivial solution of Eq. (I) is stable; (ii) if u is decrescent and if the trivial solution of Eq. (C) is uniformly stable, then the trivial solution of Eq. (I) is uniformly stable; (iii) if v is decrescent and if the trivial solution of Eq. (C) is uniformly asymptotically stable, then the trivial solution of Eq. (I) is uniformly asymptotically stable; (iv) if there are constants a > 0 and b > 0 such that a J x J 0 is most commonly used in parts (iii) and (iv) of the above theorem. Applying Theorem 2.6.5 to u( x, t ) , we can also see that the next result is true. 2.6.8. Theorem. Let g and G be continuous on their respective domains of definition. Let u : B ( r ) x J + R be a continuous, positive definite function such that Du,,,(x,t ) 2 G ( u ( x ,r ) , t ) . If the trivial solution of Eq. ( C ) is unstable, then the trivial solution of Eq. (1) is also unstable.

The generality and effectiveness of the preceding comparison technique can be improved by considering vector-valued comparison equations and vector L ~ ~ u ~ l m o z ~ ~ ~ (see n c t iSection o n s 2.4E). Here, the scalar case is included as a special case. Specifically, consider a system of 1 ordinary differential equations, j,

=

H(y,t),

Y ( h >= Yo

iVC)

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

59

where y E R', t E J, H : B(r) x J R' is continuous on B ( r ) x J, and H ( 0 , t ) = 0 for all t 2 to. Under these assumptions Eq. (VC) possesses solutions y ( t ) for every yo = y(to) E B(r) which again are not necessarily unique. These solutions either exist for all t E [ t o , co) or else must leave the domain of definition of H at some finite time t , > to. Furthermore, under the above assumptions Eq. (VC) admits the trivial solution y = 0 for all t 2 to. Once more we assume that y = 0 is an isolated equilibrium. I f w e l e t a s u s u a l y I z d e n o t e y i I z i , i = 1 , ...,I , a n d y < z d e n o t e y i < z i , i = 1, ...,1 then Definition 2.6.1 of maximal solution and Definition 2.6.2 of minimal solution still make sense. In order to extend Theorems 2.6.6 and 2.6.8 to the vector case, we require the following additional concept. --f

2.6.9. Definition. A function H ( y , 2 ) = ( H , ( y , t), ..., H , ( y , t))' is said to be quasimonotone if for each component Hi, j = 1, . ..,I, the inequality Hi ( y ,t ) I Hj(z, t ) is true whenever yi I zi for all i # j and y j = z j . The above property was used by Muller [I] and Kamke [l]. It is sometimes called the Wazewski condition with reference to the work by Wazewski [l]. 2.6.10. Theorem. If H(y, t ) is continuous and quasimonotone and if yo E B(r), then Eq. (VC) has a maximal and minimal solution. Each of these solutions must either be defined for all t E [to,co) or else leave the domain of definition of H at some finite time t , > to. Analogous to Theorem 2.6.4 we have the following comparison theorem. 2.6.11. Theorem. Assume that H is continuous on B ( r ) x J , that H is quasimonotone, and let p ( t ) be the maximal solution of Eq. (VC) on [to,a) with p ( t o ) = yo. If r ( t ) is a continuous function such that r ( t o ) 5 yo, if D r ( t ) = [Eh+ + ( r l ( t + h) - rl ( t))/A, .. ., hm,, + (rl( t + h) - r, (t))/hIT,and if

-

D r ( t ) I H ( r ( t ) ,t )

almost everywhere on

[to,a)

then r ( t ) I p ( t ) on [to,a). We also have the following theorem. 2.6.12. Theorem. Assume that H is continuous on B ( r ) x J, that H is quasimonotone, and let q ( t ) be the minimal solution of Eq. (VC) on [to,a) with q(to) = yo. If s ( t ) is a continuous function such that so,) 2 yo and if Ds(t) 2 H(s(t), t )

almost everywhere on

[to,a)

then s ( t ) 2 q ( t ) on [to,a). Now let ui (x, t ) , i = 1, .. .,I, denote I continuous Lyapunov functions and let w , t >=

(Ul(X,t),

...,u,(x,t))'.

60

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

We call V ( x , t ) a vector Lyapunov function. Such functions were originally introduced by Bellman [2]. Applying Theorem 2.6.1 1 to V ( x ,t ) the following result is easily established.

2.6.13. Theorem. Let g and H be continuous on their respective domains of definition and let H be quasimonotone. Let V(x,t) be a continuous nonnegative vector Lyapunov function (of dimension I ) such that IV(x,t)l is positive definite and such that

ov,,, (x, t ) I H ( V(x,t ) , t ) .

(2.6.14)

Then the following statements are true. (i) If the trivial solution of Eq. (VC) is stable, then the trivial solution of Eq. (I) is also stable; (ii) if J V ( x t, ) / is decrescent and if the trivial solution of Eq. (VC) is uniformly stable, then the trivial solution of Eq. (I) is also uniformly stable; (iii) if lV(x,r)l is decrescent and if the trivial solution of Eq. (VC) is uniformly asymptotically stable, then the trivial solution of Eq. (I) is also uniformly asymptotically stable; 1 IV(x, ~ t ) l , if (iv) if there are constants a > 0 and b > 0 such that ~ 1 x 5 iV(x, t)l is decrescent, and if the trivial solution of Eq. (VC) is exponentially stable, then the trivial solution of Eq. (I) is also exponentially stable; (v) if g : R" x J + R" and H : R' x J + R' and V: R" x J --+ R' and if (2.6.14) is true for all (x, t ) E R" x J , and if the solutions of Eq. (VC) are uniformly bounded (uniformly ultimately bounded), then the solutions of Eq. (I) are also uniformly bounded (uniformly ultimately bounded). Applying Theorem 2.6.12 to V ( x ,t ) we can also prove the following instability result. 2.6.15. Theorem. Let H and g be continuous on their respective domains of definition and let H be quasimonotone. Let V ( x , t ) be a continuous nonnegative vector Lyapunov function (of dimension 1 ) such that IV(x,t)l is positive definite and such that DV(,,(X, l ) 2 H ( V ( x ,t ) , 2 ) .

If the trivial solution of Eq. (VC) is unstable, then the trivial solution of Eq. ( I ) is also unstable. Proofs of the preceding results can be found in several standard references, some o f which we cite in Section 2.9. Let us now see how these results apply i n the qualitative analysis of large scale systems. As Matrosov [ I , 21 has pointed out, from the point of view of applications, the following special case of (VC), j, = Py+m(y,t) (VC')

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

61

is particularly important. Here P = [ps] is a real I x I matrix and the function m : B(r) x J + R’is assumed to consist of second or higher order terms, so that lim Irn(y, t ) l / l yI

lyI-0

=

uniformly in

0,

t 2 to.

Applying the principle of stability in the first approximation (see Hahn [2, p. 1221) to Eq. (VC’), we have the following result. If matrix P has either only eigenvalues with negative real parts or at least one eigenvalue with a positive real part, then the equilibrium of (VC’) shows the same stability behavior as that of the corresponding linearized system, jJ = Py.

Using the principle of stability in the first approximation and Theorems 2.6.13 and 2.6.15, we immediately obtain the following result. 2.6.16. Corollary. Let g be continuous and let V(x, t ) be a nonnegative continuous vector Lyapunov function (of dimension I ) such that IV(x, t ) l is positive definite and decrescent. Suppose there is an I x I matrix P = [ p G ] and a function m (V, t ) such that DV(,,(X, t ) I PV(X, t ) + m ( V ( x , t ) , t )

and

ps 2 0

if

(2.6.17) (2.6.18)

i # j,

and m(V, t ) is quasimonotone in V , and lim Im(y, t ) l / l yl

lyi-0

=

0,

uniformly in

t 2 to.

(2.6.19)

Then the following statements are true. (i) If matrix P has only eigenvalues with negative real part, the trivial solution of Eq. (I) is uniformly asymptotically stable; (ii) if matrix P has only eigenvalues with negative real part and if in addition IV(x, t ) l 2 6 lxI2 for some 6 > 0, then the trivial solution of Eq. (I) is exponentially stable; (iii) if the inequality (2.6.17) is reversed and if P has at least one eigenvalue with positive real part, then the trivial solution of Eq. (I) is unstable; (iv) if the inequality (2.6.17) is reversed and if the real parts of all eigenvalues of P are positive, then the trivial solution of Eq. (I) is completely unstable. To simplify matters, we consider only parts (i) and (ii) of Corollary 2.6.16 in the following discussion. Similar statements can be made for the remaining parts. First we note that in (i) and (ii) of Corollary 2.6.16, the quasimonoticity condition (2.6.18) means that - P is an M-matrix. Therefore, the condition that all eigenvalues of P have negative real parts, i.e., Re[1(P)] < 0, is

62

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(-

])k

Pll

...

Plk

.

...

.

Pkl

"'

>0,

k = l , ..., 1.

(2.6.20)

Pkk

where V ( x ,t ) = ( u , ( x , t ) , . . ., u,(x, t))'. Under the assumptions of Corollary 2.6.16, this function u(x, t ) is a positive definite and decrescent scalar Lyapunov function such that Do, , , ( x, t )

I uT[PV(x,t)+m(V(x,t),t)] =

(u'P) V ( x ,t ) + crTm(V(x,t ) , t ) .

Since (mTP)V ( x , t ) is negative definite and since u'm(V, t ) consists of terms of second or higher order in V ( x ,t ) , it follows that Do(,,(x,t ) is negative definite in a neighborhood of the origin x = 0. This argument shows that in the important special case of (2.6.17) the vector Lyapunov function V ( x ,t ) can be reduced to a scalar Lyapunov function v ( x , t ) and because of the quasimonoticity requirements, the seemingly more general approach of applying the comparison principle to vector Lyapunov functions is really equivalent to an approach utilizing scalar Lyapunov functions (as discussed, e.g., in Section 2.5). It would be interesting to know whether or not this same equivalence is true for systems of inequalities more general than (2.6.17). Furthermore, we note that whereas the present approach of applying the comparison principle to vector Lyapunov functions appears to require test matrices (e.g., matrix P) whose off-diagonal terms are of the same sign, this is not the case in the approach of Section 2.4. Turning our attention to interconnected systems, let us consider once more composite system (9') with decomposition (Xi) and isolated subsystems (q) (see Eqs. (2.3.10)-(2.3.12)). In particular, recall the equations for (Xi)and

(Xi)

ii = fi(zi,t)+ g i ( x , t ) ,

(sq)

ii = f , ( z i , t ) ,

i

(x),

=

i = 1, ..., I,

(2.6.21)

1, ..., 1.

(2.6.22)

For interconnected system ( Y ) ,the method originally proposed by Bailey

2.7

63

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

[1,2] has been the most successful procedure of constructing vector Lyapunov functions (see Matrosov [I, 21). Specifically, we consider isolated subsystems (q)for which we can find positive definite, decrescent, continuously differentiable Lyapunov functions ui such that the derivative of ui with respect to t along solutions of (q) satisfies

(2.6.23)

Dui(yi)(zi,t ) I -Piui(zi, t ) .

Systems for which (2.6.23) can be satisfied are often linear or nearly linear and the corresponding ui is usually a quadratic function in z i . Suppose now that we can find an I x I matrix P = [ p J satisfying pU 2 0,

i # .i,

(2.6.24)

and a quasimonotone function m : R' x J + Rf satisfying lim I m ( y , t ) l / l y l = 0,

uniformly in

(2.6.25)

t 2 to.

lYI+O

Assume also that the following inequalities hold, -Vui(zi,t)Tgi(X,t) +

i

C (pij+6,iPi)uj(zj,t) + m i ( V ( x , t ) , t ) 2 0,

j= 1

(2.6.26)

i = I , ...,I, where 6, is the Kronecker delta, mi(V,t ) is the ith component of m ( V , t ) , and V ( x , t ) = ( u l ( z l , t ) ,..., u f ( z , ,t))'. We now have

D V , , ) ( X ? t ) = CD~,(,,)(z,,t)+V~,(z,,t)'gl(X,t),

+ vuf(ZI

9

'..>D~f(Y,)(ZfJ)

t>'g, ( x , ?)IT

s I:- P I u1 (z1,t )+ vu,( z , ,tl'g, + VV,(Zf t>TgL(X,?)IT. 9

Combining (2.6.26) and (2.6.27) yields D V ( y ) ( x ,t ) 5 P V ( x , t )

( x , t ) , .. ., - Bf Of (Zf 1 ) (2.6.27)

+ m(V(x, t ) , t ) .

7

(2.6.28)

All conditions of Corollary 2.6.16 are now satisfied. I n particular, if matrix P has only eigenvalues with negative real parts, then the equilibrium x = 0 of composite system (9') is asymptotically stable. Parts (ii)-(iv) of Corollary 2.6. I6 apply equally as well, with obvious modifications.

2.7 Estimates of Trajectory Behavior and Trajectory Bounds In this section we obtain estimates of trajectory behavior and trajectory bounds of dynamical systems, using Lyapunov-type results. I n our approach we define stability in terms of subsets of R" which are prespecijied in a given

64

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

problem and which in general may be time varying. The properties of these sets yield information about the transient behavior and about trajectory bounds of such systems. Before considering interconnected systems, we need to develop essential preliminary results. We begin by considering systems described by equations of the form i = g(x,t)

(1)

where x E fz", t E J , and g: R" x J + R". We assume that Eq. (I) possesses for every x o E R" and for every to E R + a unique solution x ( t ;xo, to) for all r E J . However, in the present section we do not insist that x = 0 be an equilibrium of Eq. (1). Subsequently we employ time varying open subsets of R" defined for all t E J and denoted by S ( t ) , S o ( t ) , and D ( t ) , with appropriate subscripts or superscripts if necessary. The boundary of S ( t ) is denoted by a S ( t ) and its closure is denoted by S(t>. Henceforth we assume that all such sets possess the following "well-behaved'' property.

2.7.1. Definition. A time-varying open set S ( t ) is said to possess Property P if, whenever p : J - t R" is continuous and p ( t l ) E S ( t , ) for some t , E J , then either p ( t ) E S ( t ) for all t E [ t i ,a)or there exists a t , E ( t l ,a)for which p ( t z )E dS(t,) and p ( t ) E S ( t ) for all t E [ t l ,t2). For example, the set S , ( t ) = {x E R" : 1x1< keKa', k > 0, a > 0, t possesses Property P. On the other hand, the set

E

R'}

does not possess Property P. In the next definition we let x ( t ; x i , t i ) be the solution of Eq. (I) which satisfies x ( t i ; x i ,ti) = x i .

2.7.2. Definition. System (I) is called stable with respect to {So(to),S ( t ) ,to} ifx, E & , ( t o )implies that x ( t ; xo, to) E S ( t ) for all t E J . System (I) is uniformly stable with respect to { S o ( t ) S, ( t ) } , if for all ti E J , xiE So(ti)implies that x ( t ; x i , t i )E S ( t ) for all t E [ti,co). System (I) is unstable with respect to {S,,(t,),S ( t ) ,t o } , S,(to) c S ( t , ) , if there exists an .yo E So(to) and a t, E J such that x(t,; xo,to) E dS(t,). We emphasize that in Definition 2.7.2 the sets S o ( t ) and S ( t ) are prespecified in a given problem. In the literature the term practical stability is used for the special case S o ( t )= B(u) and S ( t ) = B ( p ) , a p. Also, the term ,finite time stability is used for the special case when J is replaced by the interval [to,to + T ) , T > 0.

2.7

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

65

2.7.3. Theorem. System (I) is stable with respect to {So(to), S ( t ) ,t o ) , S(to)3 So(to),if there exists a continuous function u : R" x J + R and a function G which satisfies the assumptions of Theorem 2.6.4 such that (i) Du(,)(x,t ) I G(u(x, t ) , t ) for all x E S ( t ) and t E J ; and ~ to) ( < ~ infxEas(,)u(x, ~ ) ~ ( X t, ) , for all t E J, where (ii) p ( t ; S U ~ ~ ~ ~to), p ( t ; yo, to) is defined in Theorem 2.6.4. System (I) is uniformly stable with respect to { S o ( t ) , S ( t ) }S, ( t ) 3 S o ( t ) for all t E J , if (i) and (ii) are replaced by (iii) Du(,)(x,t ) I G(u(x,r ) , t ) for all x E [S(t)-s,(t>] and t E J ; and (iv) p ( t 2 ;S U P , ~ ~ ~ ~ ( , , ) ~ ( X < ,infx.aS(,2) ~ , ) , ~ , )u(x,t,) for all t , > t,, t i , t 2 E J.

Proof. Since the proofs for stability and uniform stability are very similar, only the proof for stability is given. The proof is by contradiction. Let xo E So(to)and assume there exists t , E (to,00) for which x ( t 2 ;xo,to)$ S(t2). Since S ( t ) possesses Property P, there exists t , E (to,t , ] such that x ( t ; xo,to) E S ( r ) for all t E [to,t , ) and such that x(t,;x,, to) E dS(t,). Define r ( t ) as r ( t ) = u [ x ( t ; x o ,to),t ] (refer to Theorem 2.6.4) and write W t )

=

D v ( , , ( x ( t ;xo, to>,t ) I G [ u ( x ( t ;xo,to),t ) , t ] = G ( r ( t ) ,t ) .

Now r ( t o ) = u(x(to;xo,to),to) = u(xo,to) I S U ~ ~v(x, ~ to), ~ ~and( thus ~ ~ it, follows from Theorem 2.6.4 that u ( x ( t ; xo, to),t ) I p ( t ; S U ~ ~u ( x~, to), ~ to) ~ for all t E (to,t , ] . In view of hypothesis (ii) we can now write + ( t , ; x o , to),t l >

<

inf

xE

~ ( xt ,d .

But the above inequality implies that x(t,;xo,to) $ dS(t,), a contradiction. Hence, no r , as asserted exists and x ( t ; xo, t o ) E S ( t ) for all t E J . Note that from a computational point of view, the u-functions in Theorem 2.7.3 as well as in the remaining results of this section, may sometimes have less stringent requirements than those used in the usual Lyapunov theorems. Thus, in Theorem 2.7.3 there are in general no definiteness conditions on u and we do not require that u ( 0 , t ) = 0. 2.7.4. Theorem. System (I) is unstable with respect to {So(ro),S ( t ) ,t o } , S ( t o )3 So(to),if there exists a continuous function u : R" x J + R and a function G which satisfies the assumptions of Theorem 2.6.5 and a t , E (to,CO) such that

(i) Du(,,(x,t ) 2 G ( u ( x , t ) , t ) for all x E S ( t ) and t E J, (ii) q(t,; infx.SO(,O) u(x, to), t o ) 2 SUPxEas(tl)~ ( xt , A and (iii) v(x,t l ) < t , ) for all x E S ( t , ) where q ( t ;y o , to) is defined in Theorem 2.6.5.

(

~

~

)

66

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Proof, The proof of this theorem is similar to that of Theorem 2.7.3.

It is often convenient to view the right-hand side of Eq. (I) as the sum of two functions f :R" x J + R" and u : R" x J + R", i.e., i= f ( x , t )

+ u ( x , t ) 4k g(x, t ) .

(2.7.5)

Here we view u ( x , t ) as (deterministic) "persistent perturbation terms" and we call (2.7.6)

f =f(x,t)

the "unperturbed system." System (2.7.5) is a special case of systems described by equations of the form f =f(x,u(~,t),t)

h(x,t)

whereu:R"xJ+R', f:R"xR'xJ+R",and

(2.7.7)

h:R"xJ+R".

2.7.8. Definition. System (2.7.7) is totally stable with respect to { S O ( t O ) > S ( t ) ,D ( t > ,t o > ,

if the conditions (a) xo E So(to)and (b) u(x, t ) E D ( t ) whenever x E S ( t ) and t E J imply that x ( t ; xo,to) E S ( t ) for all t E J . System (2.7.7) is uniformly totally stable with respect to { S o ( t )S, ( t ) ,D ( t ) } if for all ti E J the conditions (a) x iE So(ti)and (b) u ( x , t ) E D ( t ) whenever x E S ( t ) and t E [ti,cn) imply that x ( t ; x i , t i ) E S ( t ) for all t E [ti, a). For system (2.7.5) we have the following result. 2.7.9. Theorem. System (2.7.5) is totally stable with respect to {So(to),S(f)?D ( t > ,t o > ,

S(r0) = S O ( t O ) ,

if there exists a continuously differentiable function u : R" x J + R and two integrable functions v : J - . R, q : J + R, such that (i) D v ( ~ , , , ~ )t () x<, v ( t ) for all x E S ( t ) and t E J ; (ii) Vo(x,t ) T 2 ~ q ( t ) for all u E D ( t ) , x E S ( t ) ,t E J; and Ciii) J:,"+~(~)I 5 infx.,,(,,u(x, t ) - s u P x t s o ( t o ) ~ (to). ~, System (2.7.5) is uniformly totally stable with respect to { S o ( t ) S, ( t ) ,D ( t ) } , & ( t ) c S ( t ) for all t E J , if (i)-(iii) are replaced by ~ )all X E [ S ( t ) - S O ( t ) ] and t E J ; (iv) D U ( ~ . , , ~<) v((Xt ),for c

_

2.7

67

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

(v) V u ( x ,t)'u I q ( t ) for all X . E [S(t)-s,(t)], u E D ( t ) , t E J ; and (vi) S:: Cv(7) + v (.r)I d7 I inf, as(t2)u(x, t 2 )- supx a s o ( t , ) v ( x , t l ) for t , > t,, t , , t , E J.

all

Proof. Since the proofs for total stability and uniform total stability are similar, only the proof for the former is given. The proof is by contradiction. Let x o E So(to)and assume that u ( x , t ) E D ( t ) whenever x E S ( t ) and t E J . Assume that there exists a t , E (to,m) for which x ( t , ; xo, to) 4 S(t,). Then, since S(t) possesses Property P, there exists t , E (to,t , ] such that x ( t ;x o , t o ) E S ( r ) for all t E [to,1 , ) and such that x ( t , ; x o , to)E a S ( t , ) . Now u(x(t1; xo, to), t l ) = 4 x 0 , to) +

Jr

u(x0,to)

=

+

lr

xo, to), t ) dt

Dq2.,.&(t;

~V~(~(t;xo,to),t)T~(X(t;Xo,to),t)

+ d u ( x ( t ; x 0 ,to),t ) / d t ] dr

In view of hypotheses (i)-(iii) we have u ( x ( t , : x , , to),t , ) <

E

x

E

< =

SUP

x

SO(l0)

l: +

u(x,t o ) +

sup u ( x , t o ) So(t0)

C v ( t > + W l dt

inf

u ( x , t , ) - sup u ( x , t o ) x E SO(t0)

x E as(t1)

inf u ( x , t , ) . asrf,)

But this inequality implies that x ( t l ;x o , to)4 dS(t,) which is a contradiction. Therefore no t , as asserted above exists and x ( t ; x o , to) E S ( t ) for all t E J . For the special case when u ( t ) 3 0, Theorem 2.7.9 yields a stability result with respect to {So(to), S ( t ) ,t o } ,and a uniform stability result with respect to { S o ( t )S, ( t ) } for the unperturbed system (2.7.6). In this case we take q ( t ) = 0 and delete hypothesis (ii) of Theorem 2.7.9 in its entirety. Turning our attention to composite systems, let us consider once more interconnected system (9") with decomposition (Cy), i.e.,

(m

ii

=fi(Zi,gi(zl,

...,zl,t),t)

= fi(zi, g i ( x ,t ) , t ) =

(9"")

2

=

h(x,t)

h i ( x ,t ) ,

i

=

1, ..., I,

(2.7.10) (2.7.11)

68

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where all symbols i n Eqs. (2.7.10) and (2.7.11) are as defined in Eqs. (2.3.20) and (2.3.21), respectively. However, in the present case we do not require that x = 0 be an equilibrium. In the following, all time varying sets possess Property P. The notation x E Xf= S j ( t ) signifies z j E S j ( t ) , j = 1, ..., I, and t E J . In particular, X;=, B j ( r j ) = {x E R” : lzjl < r j , j = 1, ..., I } . 2.7.12. Definition. Composite system (9”) with decomposition (Cy) is stable with respect to (X:= S o j ( f o )Xj= , S j ( t ) , t o ) if xo E Xi= Soi(to)implies that x ( t ; xo, t o ) E Xi= S j ( t ) for all t E J . Composite system (9”) with decomposition (Cy) is uniformly stable with respect to S,j(t),)(f=, S j ( t ) } if for all ti E J , x i E Soj(ti)implies that x ( t ; x i , ti) E S j ( t ) for all t E [ t i , m).

,

xf=

xi= ,

We are now in a position to prove the following result. 2.7.13. Theorem. Composite system (9”) with decomposition (Cy) is stable with respect to Soj(lo), S j ( t ) ,to>, S j ( t o )2 Soj(to),j = 1 , ..., I, if the following conditions hold. (i) g i ( x ,t ) E D i ( t )c Rri whenever x E Xi=, S j ( t ) and t E J ; (ii) each system described by

{xf=

xi= ,

i i =f i ( Z i , S i ( X ,

t),t )

(2.7.14)

is totally stable with respect to {Soi(to),S’(t),D i ( t ) ,t o } , Soi(to)c S’(to), i=

],...,I.

Composite system (9”‘) is uniformly stable with respect to

{

j = 1

Soj(t),

j= 1

I

Sj(t) ,

S j ( t ) 3 S O j ( t ) , j = 1, ...)I,

for all t E J, if (i) is true and if (ii) is replaced by the following. (iii) Each system described by Eq. (2.7.14) is uniformly totally stable with respect to { S o i ( t )S, i ( t ) ,D i ( t ) } So’(t) c S i ( t ) , i = 1, ...,I, and t E J . Proof. The proofs for stability and uniform stability are similar. Only the proof for stability is given. Sj(to). For purposes of contradicton, assume Let xo E Xi= Soj(to)c that at some t , E J , x ( t 2 ;xo,t o ) $ Sj(t,). Since x ( t ;xo,to) is continuous and since sets S j ( t ) , j = 1, ..., I, possess Property P, there exists a finite first time t , < t , such that

xi.=,

-

x;=

xf=,

Hence, x ( t ; xo,t o ) E S j ( t ) for all t E [to,t , ] . Now by hypothesis (i), gi(x,r ) E D i ( t ) whenever x E S i ( t ) and t E J. Thus, g , ( x ( t ;xo,to), t ) E

2.8 D i ( t ) , i = I , ..., 1, for all t

E

69

APPLICATIONS

[to,tl]. Next, define

i = 1, ..., I,andchooseu,(f)insuchafashionthatu,(t) E Di(t)forallt E (tl,m). Then clearly gi*(t) E D i ( t ) for all t E J . Corresponding to each equation describing (Ci), consider j i = .L(yi, gi*(t),t),

i

= 1,

...)1,

where the mappings fi, i = 1, ...,I, are defined as in (Ci). Let yio = zio. By hypothesis (ii), each system described by Eq. (2.7.14) is totally stable with respect to { S o i ( t o ) , S i ( t ) , D i ( tt)o, } , Soi(to)c S i ( t o ) , i = I , ...,I. Thus, y i ( t ;y i o ,to) E S ' ( t ) for all t E J . By causality (i.e., by uniqueness), z i ( t ;zio,to) = y i ( t ;yio,t o ) for all t E [ t o , t ! ] . Therefore, z i ( t ;zio,to) E S i ( t ) for all f E [ t o , t l ] and x ( t ; x , , t , ) E X ~ = , S ' ( r )for all t e [ t o , t l ] .But above we assumed that

Thus, we have arrived at a contradiction and there exists no t , E J such that x ( t , ; x o , to)E a Sj(tl)}. This in turn implies that there is no t , E J such that x ( t , ; x o , to) # Sj(t,), because each Sj( t) possesses Property P and Si(t) because of the continuity of x ( t ; xo, to). Therefore x ( t ;x o , to) E for all t E J.

{xi=

xf=

xi=

2.8 Applications

In order to demonstrate the usefulness of the method of analysis advanced in the preceding sections, we consider several specific examples. 2.8.1. Example. (Longitudinal Motion of an Aircraft.) The controlled longitudinal motion of an aircraft may be represented by the set of equations (see Piontkovskii and Rutkovskaya [l]) i k

=

& =

-pkxk 4

+

6,

k

=

1,2,3,4,

1 Pkxk- rpa -f ( o )

(2.8.2)

k= 1

where p k > 0, r > 0, p > 0, Pk are constants, where xk E R , o E R , and where + R has the following properties: (a) it is continuous on R , (b) f ( a ) = 0 if and only if a = 0, and (c) of(.) > 0 for all a # 0. We call any function f

f :R

70

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

with these properties an admissible nonlinearity. If the equilibrium xT = ( x , , x,, x 3 , x 4 , 0 ) = 0 is asymptotically stable in the large for all admissible nonlinearities f, we call system (2.8.2) absolutely stable (see Aizerman and Gantmacher [ l ] , Lefschetz [ l ] ) . System (2.8.2) may be viewed as a linear interconnection of isolated subsystems (Y1)and (Y,)described by i k

k

= -fkXk,

=

1,2,3,4,

(91)

and 6 = -rpa -f(o).

Systems (Y,)and (Y2) are interconnected to form system (2.8.2) by means of the matrices CT, = [ l , 1 , 1 , 1 ] and C , , = [fi1,fi2,p3,fi4],where the notation of Eq. (2.3.14) is used. Without loss of generality we assume in the following that p1 I p, I p 3 I p4. Let z I T = ( x , , x 2 , x 3 , x 4 ) and z2 = 0 . For (Y,) and (9, choose ) u,(z,) = c,z:zl and u2(z2)= c 2 z Z 2 where , c1 > 0 and c, > 0 are constants. Then DU2(Y*)(Z2)I -2rpcz lz212,

IVU,(Zz>I

2c2Iz2l

for all z,E R4 and z2 E R . The norms of C , , and C , , induced by the Euclidean norm are 11 C , , 11 = 2 and 1 C , , / = Hypothesis (i) of Corollary 2.4.28 is satisfied. The test matrix S of this corollary is specified by

(xf=

s,, =

-2c(,c,p1,

s,, =

s,1

Choosing a, = 1/(4c,) and form

= c(,

2c(,c,

=

s,,

+

=

(,.

-2a2rpc,, i= I

p,.)"'.

1 / [ 2 c 2 ( x i 2 ,p ~ ) ' ' 2 ] matrix , S assumes the

This matrix is negative definite if and only if

2 ti2 < 4

1 3

ti =

(2Pi>/(p,p r ) .

(2.8.3)

i= 1

I t follows from Corollary 2.4.28 that the equilibrium x = 0 of Eq. (2.8.2) is exponentially stable in the large for any admissible function f if inequality (2.8.3) is satisfied.

2.8

APPLICATIONS

71

We can also apply Corollary 2.5.21. The test matrix D of this corollary is given by

Matrix D has positive successive principal minors if and only if inequality (2.8.3) is satisfied. Thus, with the above choice of a, and a,, Corollaries 2.4.28 and 2.5.21 yield the same result. Utilizing an approach of the type discussed in Section 2.6, Piontkovskii and Rutkovskaya [11 obtain the stability condition

c 45: 4

i= 1

<1

(2.8.4)

using the above Lyapunov functions u1(2,) and u2 (2,) as components of a vector Lyapunov function. Condition (2.8.3) is clearly less conservative than condition (2.8.4). 2.8.5. Example. Let us reconsider system (2.8.2). For (9,) choose u l ( z l ) = lz,l and for (9, choose ) u2(z2)=1z21. Then D U ~ ~ ~I, - ~ p l ( Iz,] Z ~ and ) D Z I , ( ~ ~ ) (5 Z ,-rplal. ) Note that u, and u2 are globally Lipschitzian with L1 = L, = 1. Hypothesis (i) of Theorem 2.5.1 5 is clearly satisfied. Hypothesis (ii) of this theorem is also satisfied with a,, = u2, = 0, a 1 2= 2 and a,, = The test matrix D of this theorem assumes the form

(xf=

This matrix has positive successive principal minors if and only if the inequality 4

(2.8.6)

is satisfied. It follows from Theorem 2.5.15 that the equilibrium x = 0 of system (2.8.2) is uniformly asymptotically stable in the large if inequality (2.8.6) is true. Thus, with the preceding choices of Lyapunov functions and weighting vector, Corollaries 2.4.28 and 2.5.21 and Theorem 2.5.15 yield the same stability condition for system (2.8.2).

2.8.7. Example. Let us alter system (2.8.2) by assuming thatf((a) is given by f(0) = a((a2-a2),

a > 0,

(2.8.8)

72

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

which is not an admissible nonlinearity in the sense of Example 2.8.1. Specifically, in this case we have af(o) > 0 for all 101> a, lim,o,+mof(o) = 03, f ( 0 ) = O, f(a) = O,f( - a ) = 0, and ~ - f ( o<) 0 for all 0 < 101 < a. Utilizing either Theorem 2.4.34 or 2.5.26 and proceeding in an identical fashion as in Example 2.8.1, we conclude that all solutions of system (2.8.2) withf(a) specified by Eq. (2.8.8) are uniformly bounded, and in fact, uniformly ultimately bounded if inequality (2.8.3) is satisfied. 2.8.9. Example, (Indirect Control Problem.) An important class of problems arising in automatic control theory is the indirect control problem (see, e.g., Aizerman and Gantmacher [l], Lefschetz [l]) characterized by the set of equations

+ bf(0)

1

=

Ax

6

=

- p 0 - rf(0)

+ aTx

(2.8.10)

where x E R", (T E R, A is a stable n x n matrix (i.e., all eigenvalues of A have negative real parts), b E R", p > 0, r > 0, a E R", and f : R + R has the following properties: (a) it is continuous on R , (b) f ( a ) = 0 if and only if r~ = 0, and (c) 0 < of(a) < ka2 for all 0 # 0, where k > 0 is a constant. System (2.8.10) is called absolutely stable if its equilibrium (xT,a) = 0 is asymptotically stable in the large for any admissible nonlinearity f with the above properties. System (2.8.10) may be viewed as a nonlinear interconnection of isolated and (Y;)described by subsystems (9,)

1

and

6

=

=

AX

(91

-pa - rf(0).

)

(92)

Using the notation of Eq. (2.3.16), these subsystems are interconnected to form composite system (2.8.10) by means of the relations g , 2 ( 0 )= f ( o ) b and q z l(x) = a'x. In the subsequent analysis, as well as in later examples, we utilize the following well-known result (see, e.g., Hahn [2, p. 1171). 2.8.1 1. Theorem. Let y

E

R", let B be ann x n matrix, and consider the equation j,

=

(2.8.12)

By.

If all eigenvalues of B have negative real parts or if at least one eigenvalue has positive real part, then there exists a Lyapunov function of the form v ( y ) = yTPy,

PT = P

(2.8.13)

-Y'G

(2.8.14)

such that DU(2.&?.12)(Y) =

is definite, where

2.8

APPLICATIONS

73

-C

=

BTP i- PB.

(2.8.1 5)

Thus, if the conditions of Theorem 2.8.1 1 are satisfied, then it is possible to construct a Lyapunov function v( y) by assuming a definite matrix Cand solving the Lyapunov matrix equation (2.8.15). Returning to the problem on hand, since A of (9,) is a stable matrix it follows from Theorem 2.8. I 1 that there exists a function u1 : R" --t R and constants c l i > 0, i = 1,2,3,4, such that C111Xl2

-C13IXl2, D~,(,,,(X) I

I v1(x) 5 c121x12,

IVv,(x)I I c141xI

for all x E R". For (9,we ) choose v,(o) = a2/2. Then D Y , ( ~ ~ ) I ( O- p) and IVu,(o)l= lo1 for all o E R. Hypotheses (i)-(iii) of Corollary 2.4.25 are now satisfied with k,, = k Ibl and k,, = lal. Choosing ctl = l / ( k lbl) and tl, = c14/lal,matrix S of Corollary 2.4.25 assumes the form

This matrix is negative definite if and only if

<

(PCl3>/(bl

Ibl c14).

(2.8.16)

It follows from Corollary 2.4.25 that the equilibrium (xT,o)= 0 of composite system (2.8.10) is asymptotically stable in the large (i.e., system (2.8.10) is absolutely stable) if inequality (2.8.16) is true. We can apply Corollary 2.5.20 as well. The test matrix D of this corollary is given by

Matrix D has positive successive principal minors if and only if inequality (2.8.16) is satisfied. Therefore, with the above choice of CI, and a,, Corollaries 2.4.25 and 2.5.20 yield the same stability result. Using an approach of the type discussed in Section 2.6 involving Lyapunov functions u l ( x ) and v,(a) as components of a vector Lyapunov function, Piontkovskii and Rutkovskaya [1) obtain the stability condition k < C(Pcl3>/(lallbl C14)l ( ~ l l / ~ l z ) 1 ' 2 .

(2.8.17)

~ ~I~, it follows that condition (2.8.17) is in general more Since ( c , , / c , ~ )I conservative than condition (2.8.16).

74

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

To determine the constants ~ 1 and 3 c14 in (2.8.16), let v ( x ) = xTPx with P T = P, so that Dv,(su,) (x) = - xTCx. Given a positive definite matrix C, we can solve for P by solving the Lyapunov matrix equation - C = A T P + PA. In so doing we obtain &(P) 1x1, I u1 (x) I A,(P) [XI,, Dv,(,,,(x) I -A,(C)lx12, and I V v , ( x ) ) 1 2 A M ( P )1x1. Inequality (2.8.16) assumes now the form k < :[Am(c>/AM(P>l[P/(bl lbl>1. (2.8.18) To obtain the least conservative result, matrix C needs to be chosen in such a fashion as to maximize A,(C)/A,(P). 2.8.19. Example. Choosing an appropriate nonsingular linear transformation, we can represent the indirect control problem of Example 2.8.9 equivalently by the set of equations 1, = A , x l f 2

+ b,f(o) + b2f(o) - rf(o) + alTxl+ a2Tx2

(2.8.20)

= A 2 ~ 2

6 = -po

where x, E R"',x 2 E R"', A , is an n , x n , matrix, A , is an n, x n 2 matrix, b , E R ' ,6 , E R"', a , E R"', a , E R"', n , +n, = n, and the remaining symbols are as defined in Example 2.8.9. System (2.8.20) may be viewed as a nonlinear interconnection of three isolated subsystems (Y,),(Y,),(Y3) of the form 2, = A , x , f, = A 2 x Z

6 = - p o - rf(0).

Using the notation of Eq. (2.3.16), these subsystems are interconnected to form composite system (2.8.20) by means of the relations g12(x2) = Y21 ( X I > = 0, g13(a) = f ( a ) b l , g 2 3 ( o ) =f(a)b2> g31(xl) =alTxI, and g32b2)

= a2Tx2.

Assume that all eigenvalues of A , have positive real parts and that A , is stable. In accordance with Theorem 2.8.1 I , there exist 0 , : R"' + R , v,: R 2+ R,and positive constants cij,i = 1,2 and j = I , 2,3,4, such that -c11Ix1l2

I u1(x,) I -c121x12>

DOi,y,j(xi)

5

-Ci31xi12,

IVvi(Xi)l I C i 4 1 x i 1 ,

c211x212

5

u2(x2)

Dv2(Y2)(x2)

1vv2(x2)1

for all x, E R ' and for all x2 E R"'. For (Y3) choose u3(0) = 40,. Then Dv3,,3j(0) I 1 ~ for 1 all o~ R.

5 Czzlxz12,

I -c23

IX2l29

I Cz41x21,

and IVv3(o)l=

~ J O ) ~

2.8

75

APPLICATIONS

Using the notation of Theorems 2.4.31 and 2.5.24, we obtain a 1 3= c14k Ib, 1, = c2,klb21, a 3 L= lal[, a32 =la,/, and a12= a2*= a l l = a 2 2= a33= 0. Hypotheses (i) and (ii) of Theorem 2.4.31 are thus clearly satisfied. The test matrix D of Theorem 2.5.24 assumes the form

a23

0

-Ci4k\bil

0

c23

-c24klb1

-la11

-la21

3

D = [

1

.

P

This matrix has positive successive principal minors if and only if the inequality c13 c 2 3

k < c23 C i 4

la1 I h

P

h

3 C24

1%

(2.8.21)

lb21

is satisfied. It follows from Theorem 2.5.24 that the equilibrium xT = (xlT,x ~cr) = ~ 0, is unstable (for all admissible nonlinearities , f ) if inequality (2.8.21) holds. 2.8.22. Example. (System with Two Nonlinearities.) Consider the system described by the set of equations 5, 0 1

= AIZl

=

+ b,fl(fJl),

i 2 =

T c1 z2,

A,z,

+ b,f2(cr2),

(2.8.23)

6, = C2TZ1,

where for i = 1,2, zi E RnL,Ai is a stable ni x ni matrix, c1 E R"', c2 E R"', and f i : R + R has the following properties: (a) it is continuous on R, (b) fi(ai) = 0 if and only if cri = 0, and (c) 0 < crifi(cri) < kicri2 for all cri # 0, where k i > 0 is a constant. System (2.8.23) may be viewed as a nonlinear interconnection of two linear isolated subsystems (9,) and (Y2), (91)

i, = A , z , , i 2

=

A2~2,

(92)

which are interconnected to form composite system (2.8.23) by means of the relations g12(z2)= b , f l (crl) and gZ1(z1)= b 2 f 2 ( 0 2 )where , the notation of Eq. (2.3.16) is used. Since A, and A, are stable matrices there exist ui: Rni-+ R and constants cij > 0, i = 1 , 2 , j = 1,2,3,4, such that 2 cil /zi/

IUifzi)

s

ci2 IziI2,

2

Dui(yi)(zi)I -ci3 [zi)

3

for all zi E Rni.Thus, hypotheses (i) and (ii) of Corollary 2.4.25 are satisfied.

76

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Hypothesis (iii) is satisfied with k , , = k , JbllIc, 1 and k,, = k , Ib,J Jc,J. Choosing c ( ~= ~24/(k1[clJlb, 1) and c(, = ~,4/(k2Ic21 lb21), matrix S of Corollary 2.4.25 assumes the form

This matrix is negative definite if and only if k J k2

<

( C 1 3 C 2 3 ) / ( C 1 4 C 2 4 / b l / lb21 ('I/ 1'21).

(2.8.24)

It follows from Corollary 2.4.25 that the equilibrium xT= (xIT,x , ~ ) = 0 of system (2.8.23) is exponentially stable in the large if (2.8.24) holds. Following an approach of the type discussed in Section 2.6 and using the above Lyapunov functions vl(zl) and v 2 ( z 2 ) as components of a vector Lyapunov function, Piontkovskii and Rutkovskaya [I] obtain the inequality

as a sufficient condition for exponential stability. Since (cl, C , ~ ) / ( C ~ , c,,) 5 I , condition (2.8.25) is in general more conservative than condition (2.8.24). 2.8.26. Example. The purpose of the present simple example is to provide a case where the equilibrium is asymptotically stable but not necessarily exponentially stable and to present an example where the results of Section 2.5 involving M-matrices do not predict stability while those of Section 2.4 do predict stability, using identical Lyapunov functions for the subsystems. Specifically, consider

(2.8.27) where x , E R and x, E R . This system may be viewed as a nonlinear interconnection of isolated subsystems (Y,),(Y2),

f, x2

= -x13,

=

-x,

5

(91)

.

(92)

Choosing u l ( x , ) = x i 2 and v2(x2)= x22,we have D u l ~ y l ~ ( x=l -2x14 ) and D U , ~ , , , ( ~=, -) 2 ~ Using ~ ~ .the notation of Theorem 2.4.2 we make the identifications $, 1 ( ~ )= +h12(y) = r 2 , $ , 3 ( ~ )= r4, I ) ~ ~ ( Y ) = $ 2 2 ( ~ )= r 2 , and $ 2 3 (Y) = r'. The interconnecting structure of Eq. (2.8.27) is characterized by g1(x,,x2)= - 1 . 5 )x2I3 ~ ~ and g2(x1,x2)= x 1 2 x 2 , ,where the notation of

2.8

APPLICATIONS

77

Eq. (2.3.10) has been used. We now have v ~ , ( x l ) g l ( ~ l , x 2=) (2xl)(- 1.5X,)ix,1~= I X ~ I ~ ( - ~ ) I X ~ I ~ =

$1 3

(1x11)1’2

(- 3) $23 (lxzl)l/z

and ~ ’ ) iX1l2 v ~ z ( X 2 ) ~ 2 ( x , , x=2 )( ~ x ~ ) ( x ~ ~5x1x2I3(2) =

$23(1x21)1’2(2)$13(Ix11)1~2.

In the notation of Theorem 2.4.2 we now have a , , = a Z 2= 0, a,, = -3, uZ1= 2, 0, = - 2, and o2 = -2. Choosing C L ~= u2 = I , matrix S of Theorem 2.4.2 assumes the form

Since S is negative definite it follows from Theorem 2.4.2 that the equilibrium (xl, x , ) ~= x = 0 of Eq. (2.8.27) is asymptotically stable in the large. However, since the comparison functions $,, i = I , 2, j = 1,2,3, are not all of the same order of magnitude, we cannot conclude that the equilibrium is exponentially’ stable (see Theorem 2.4.20). Next, note that - S has positive off-diagonal terms and thus the results of Section 2.5 are not directly applicable. However, using the same Lyapunov functions as above, let us attempt to establish stability, using a test matrix D = [d,] for which d, < 0, i # j . In the notation of Theorem 2.5.1 1 we have the estimates VUl (Xd9, ( X l ? XZ)

$13(lx11)1~2(3)~23(Ixz1)1~2

vv, (xz) 9 2 (XI

$23

3

x2)

(1x2

(2) $ 13 (1x1

Hypotheses (i) and (ii) of Theorem 2.5.1 I are now satisfied. The test matrix of hypothesis (iii) assumes the form 2 .=[-2

-3 2

]

Since the successive principal minors of D are not both positive, Theorem 2.5.1 1 fails to predict asymptotic stability. Thus, using the same Lyapunov functions u1 (z,) and u 2 ( z z )we were able to predict stability for system (2.8.27) using Theorem 2.4.2, but failed to do so using Theorem 2.5.1 1. 2.8.28. Example. (The Hicks Conditions in Economics.) Let p i 2 0, i = I , ...,n, denote the prices of n interrelated commodities supplied from the same or related sources which are demanded by the same or related industries

78

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and let fi(pl, ...,pn) denote the excess demand function of commodity i. Such a multimarket system may be represented by the equation P =f(p)

(2.8.29)

where pT = ( p l , ...,p,) and where f: R n - t R" is assumed to be continuously differentiable with respect to all of its arguments. Assume that p > 0 is an isolated equilibrium price vector, so that f(p) = 0. Linearizing Eq. (2.8.29) about the equilibrium p , we obtain the set of equations

D~ = C a,(pj-pj), n

i = 1 ,..., n,

j= 1

(2.8.30)

where a,. = [dfi(p)/dpj],=a. Here we need to restrict ourselves to "small" initial deviations from the equilibrium. As well as providing mathematical simplicity, this restriction has the advantage of assuring us that in the presence of asymptotic stability, no price will become negative on its path to equilibrium, since it is never far from the equilibrium which we have already assumed to be positive. Here we are invoking theprinciple of stability in thefirst approximation (see Hahn [ 1, p. 1223) which asserts that we can deduce the stability properties of the equilibrium p of Eq. (2.8.29) from Eq. (2.8.30), provided that A = [aij] is not a critical matrix (i.e., A has only eigenvalues with negative real parts or at least one eigenvalue with positive real part). Letting xi = pi- pi, Eq. (2.8.30) assumes the form 1. = a..x. ,I I +

2

aijxj,

j=l,i#j

i = 1, ..., n.

(2.8.31)

Assuming that all commodities are gross substitutes for one another, we have the conditions ay 2 0, i # j . We also make the realistic assumptions a i i < O , i = l ) ...) n. Noting that the asymptotic stability results of Sections 2.4 and 2.5 can be modified in an obvious way to yield local conditions rather than global ones, we may view system (2.8.31) as a composite system of n isolated subsystems

(Z)>

x.L = a..x. 11 I )

(%)

which are interconnected by means of the relations gi(x) = Cl=l,i # j a i j ~ j , xT= ( x l ,..., x,,), where the notation of Eq. (2.3.10) is used. For each isolated subsystem we choose

(x) &)

=

(Xi(.

Note that vi is Lipschitzian with Li = 1. Along solutions of (q) we have Dui(yi)(xi) = aii IXil. We are now in a position to apply Theorem 2.5.15. The test matrix D

= [d,]

2.8

79

APPLICATIONS

of this theorem is specified by

4

laii[,

d.. =

-/ail,

i = j

i f j.

It follows from Theorem 2.5.15 that the equilibrium x = 0 of Eq. (2.8.31) (and hence, the equilibrium p of Eq. (2.8.29)) is asymptotically stable if the successive principal minors of matrix Da re all positive. To put it an equivalent way, the equilibrium price vector p of Eq. (2.8.29) is asymptotically stable if a11

a12

..'

alk

a21

a22

..'

a2k

.

...

.

> 0,

k = 1, ..., n.

(2.8.32)

Conditions (2.8.32) are called the Hicks conditions in economics (see, e.g., Quirk and Saposnik [ 11, Metzler [l], McKenzie [I]). Since D is an M-matrix, we can express these conditions equivalently by requiring the existence of real constants Ai > 0, i = 1, ...,I, such that the inequalities laiiJ-

C n

j = 1, i + j

(Aj//$) laji[> 0,

i = I , ...,n,

(2.8.33)

are true (see Corollary 2.5.3). 2.8.34. Example. (Linear Resistor-Time Varying Capacitor Circuits.) A large class of time varying capacitor-linear resistor networks (see Sandberg [ 101,

Mitra and So [I]) can be described by equations of the form

+ [ADl(t)+BD,(t)]x= b(t)

(2.8.35)

where x E R", A = [ a @ ]and , B = [bi] are constant n x n matrices with aii > 0 and bii > 0, i = 1, ...,n, D , ( t ) and D , ( t ) are diagonal matrices whose diagonal elements [ D l ( t ) l i i , [D2(r)liiare continuous nonnegative functions on J = [ t o , a),and b ( t ) is an n-vector bounded and continuous on J . Henceforth it is assumed that [ D 1 ( t ) l i j + [ D z ( t ) l i2i 6 > 0, i = I,...,n for all ~ E J . Presently we are interested in the asymptotic stability of the equilibrium x = 0. For this reason we let b ( t ) = 0 for all t E J and consider the free or unforced system f

+ [ A D , ( t ) + B D , ( t ) ] x = 0.

(2.8.36)

This system may be viewed as n isolated subsystems (Sq), f i = - { a i i [ D , ( t ) ] i i + b i i [ D 2 ( t ) ] i i }4 xir n i i ( t ) X i ,

(Sq)

11

80

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

which are interconnected by the relations g i ( x j , t ) (using the notation of Eq. (2.3.16)), g , j . ( x j , t ) = - ( a ~ [ ~ l ( t > I j j + b , j . C D z ( t ) I j j >4 x jm , ( t ) x j ,

i ~

j

,

where r n g ( t ) is defined in the obvious way. For (q) we choose

u i ( x i )= ailxil where Ai > 0 is a constant. Note that ui is Lipschitz continuous for all x i E R with Lipschitz constant Li = A i . Along solutions of (q) we have t ) = &mii(l)Ixil.

Dui(yi)(xi,

It now follows from Corollary 2.5.16 that the equilibrium x = 0 of Eq. (2.8.36) is uniformly asymptotically stable in the large if I1

~ , ( t ) Irnii(t)l -

C

i=l,i#j

(Ai/Aj) Imi(t)l 2 y > 0,

.j = I ,

...,n. (2.8.37)

Now assume that

,,

0.. -

C I1

(Ai/Aj)/ail 2

8 > 0,

bjj

C n

-

(AJAj)/bc/2 6 > 0,

i = 1, i # j

i = 1, i#j

j = 1,

...,Iz.

(2.8.38)

Applying the definition of m G ( t )and the properties of D ,( t ) , D , ( t ) to (2.8.37), we obtain n

ajj-

1

i=l,i#j

(&/Aj)laijI

2 a [ D l ( t ) + D 2 ( t ) l j j2

6 6 > 0.

Therefore, the equilibrium x = 0 of the free system (2.8.36) is uniformly asymptotically stable in the large if inequalities (2.8.38) are true. I t is interesting to note that stability conditions (2.8.38) were obtained by Mitra and So [ I ] (see also Sandberg [lo]) by methods which differ significantly from the present approach.

2.8.39. Example. Using Theorem 2.7.3, we now establish the following estimate for the trajectory behavior of composite system (2.8.35). Assume that for unforced system (2.8.36) there exist constants Ai > 0, i = 1 , ..., n, and 8 > 0, such that the inequalities (2.8.38) hold. There is no loss of generality 2;' = 1. Also assume that for forced system (2.8.35), in assuming that

xy=

2.8

81

APPLICATIONS

x:=

AiIbi(t)l I k for all t E J, where b i ( t ) denotes the ith component of vector b ( t ) . Let c = &'h,where 6 > 0 is defined in Example 2.8.34. If a > k/c and C;= lilxiol I a for the initial vector xo = ( x l 0 ,..., xn0)', then for t 2 to,

+

(2.8.40)

Ix(t; xo,to)l I [a- k / c ] e-c(r-ro) k / c .

To obtain this estimate, we choose again u ( x ) = C:= lilxil. Simple computations yield -cv(x>

oq2.8.35)(x)

+ k,

where the constants c and k are defined above. In Theorem 2.7.3 let G(u,t ) = - cu + k. Solving the equation ij = G ( P , t )

we obtain

p ( t ; p o , t o )= (po-kk/c)e-"(r-'o)+ k/c.

Choosing

i

So(?,) = x

and

=(xl,

..., X,)T

S ( t ) = B([a-k/c]

recalling that

x:= l i 2

=

x

1

i= 1

e-'('-'O)

+k/c),

1 by assumption, and noting that

inf u(x) = (a - k / c )e-'('-'O) E

S(')

+ k/c

it is clear that all hypotheses of Theorem 2.7.3 are satisfied. Thus, estimate (2.8.40) follows directly from the properties of the boundary of S(r), d S ( t ) . It is interesting to note that estimate (2.8.40) was obtained by Mitra and So [I] (refer also to Sandberg [lo]) by quite a different method. Observe that in Example 2.8.34, the Lyapunov function u ( x ) = lilxil and the derivative B z + ~ , ~ . can ~ ~ )be( xestimated ) by comparison functions I,!12,I,!13E K R which are of the same order of magnitude. In accordance with Theorem 2.2.26, we can actually deduce that the unforced system (2.8.36) is exponentially stable in the large. Indeed, letting b ( t ) = 0, we obtain from inequality (2.8.40) the estimate

xy=

Ix(t;xo,to)l I ae-'('-'O),

t 2 to,

n

C lilxiolI a.

i= 1

2.8.41. Example. (Nonlinear Transistor-Linear Resistor Networks.) Consider

nonlinear time-varying systems described by x

+ A f ( x , t )+ B g ( x , t ) = b ( t )

(2.8.42)

82

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where x E R", A = [ae], and B = [b,] are constant matrices with aii > 0 and bii > 0, where f:R" x J + R" and g : R" x J + R" are continuously differentiable in x and continuous in t , where f ( x , t ) = 0 and g(x,t ) = 0 for all t E J if and only if x = 0, wheref,(x, t ) = f i ( x i , t ) and g i ( x ,t ) = gi(xi, t ) , and where b ( t ) is a bounded continuous n-vector. Henceforth it is assumed that , 2 6 > 0 for all xi # 0, t E J, and that [ A ( x i ,? ) / x i ]2 6 > 0, [ . q i ( x i ?)/xi] [LJJ(xi, t)/d~~]l,,=~ 2 q > 0, [ d g i ( x i ,t)/d~~]l,,=~ 2 v] > 0 for all t E J . Presently, the asymptotic stability of the equilibrium x = 0 is of interest. For this reason we let b ( t ) = 0 for all t E J and consider the free system

1 + Af(x,t )

+ Bg(x, t ) = 0.

(2.8.43)

Equation (2.8.42) can be used t o model a great variety of physical systems. For example, a class of nonlinear transistor-linear resistor networks is described by Eq. (2.8.42), where the functions , J ( x i ,t ) = f i ( x i ) ,g i ( x i ,t ) E gi(xi) are assumed to be monotonically increasing in xi(see Sandberg [lo]). The free system (2.8.43) may be viewed as n isolated subsystems (Yi),

(8)

li = m i i ( x i , t ) x i

which are interconnected by the relations g i i ( x i ,t ) (using the notation of

XI

# 0,

(2.8.44)

XI =

0. (2.8.45)

I t follows from Corollary 2.5.16 that the equilibrium x = 0 of Eq. (2.8.43) is uniformly asymptotically stable in the large if

for all s E R" and t

E

J . Applying the definition of m e ( x j ,t ) and taking the

2.8

83

APPLICATIONS

properties of f i ( x i ,t ) and g i ( x i ,t ) into account, it follows similarly as in Example 2.8.34 that the equilibrium x = 0 of unforced system (2.8.43) is uniformly asymptotically stable in the large if there exist constants Ai > 0, i = 1, ...,n, such that a,.JJ

i= 1, i # j

i = 1, i # j

,j = 1 , ...,n.

(2.8.47)

Following a procedure similar to that of Example 2.8.39, it is also possible to obtain an estimate for I x ( t ; xo,t o ) [ , by invoking Theorem 2.7.3. Interestingly, the stability conditions (2.8.47) were obtained earlier by quite different methods (see Mitra and So [I], Sandberg [IC]). 2.8.48. Example. Consider the composite system described by the set of

equations

I

(2.8.49)

where t E J, zi E R"', and A i ( t ) and C g ( t )are real continuous n i x ni and nix n j matrices, respectively. Let Zf= ni = n, let xT= (zIT,..., zlT), and let g i ( x ,t ) = C:.= C , ( t ) z j , so that ii = Ai(t)Zi+ g i ( X , t ) ,

i

= 1,

...)I.

(2.8.50)

This system is clearly a special case of system (9") with decomposition (C;). Now assume that IICii(t)ll< 1, i , j = I, ..., I, for all r E J . Choose S i ( t )= @(pi) = {zi E Rn1: lzil < p i } , Soi(t)E Bi(ai), 0 < ai< pi,i = 1, ...,I, and 1

gi(x, t ) : l g i ( x ,t ) l <

1 pi

j= 1

for all x E

1

X

j = I

B'(Bj) and t E J

1

.

Since Igi(x,t)l=IC:.=lCg(t)zjIIC:.=lCCij(t)ll lzjl
xf=

xi=

(2.8.51) for all t , < t , , t , , t , E J .

84

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Direct application of Theorem 2.7.13 yields now the following result. Composite system (2.8.49) is uniformly stable with respect to

for all t E J and if inequality (2.8.51) holds for each i. It is possible to generalize the preceding result. Let ai: J + R', pi: J + R + be continuous functions and remove the restriction IICij(t)ll< 1. Following a similar procedure as before, we can show that composite system (2.8.49) is uniformly stable with respect to Bj(aj(t)), B j ( p j ( t ) ) } ,0 < 6 I ccj(t)< p j ( t ) , j = 1 , ...,I, t E J, if for each i the inequality

xi=

{Xi=l

1;

{AM

1

~wi(t>l + ( l / a i ( t > ) 11cij([)11 pj(t> j = 1

is satisfied for all t , < t,, t , , t ,

E

J.

dt < l n ~ ~ i ( t ) / a i i t > l

2.8.52. Example. (Multiple Feedback Systems: Direct Control Case.) We begin by considering I isolated subsystems described by the set of equations

(x)

i = I , ..., I, where zi E R"', A , is a stable n, x ni matrix, 6, E Rni,c, E R",,dii E R , ni E R and,fi: R -+ R is a continuous function such thatfi(a,) = 0 if and only if n, = 0 and 0 < aifi(ai)I k i a i 2 for all ni # 0, where k i > 0 is a constant.

I n this case we call f, an "admissible nonlinearity." System (q) is known as the direct control problem (see, e.g., Aizerman and Gantmacher [I], Lefschetz [ 11). Next we consider the composite system

ii

= A i z i - bifi(ai)

yi

z

CiTZ.

(2.8.53)

where yT = (yl, ..., y , ) E R' and d: = ( d i l ,..., dil)E R'. Let let xT = (zlT, ..., 2:) E R", and assume that x = 0 is the only equilibrium of system (2.8.53). It is important to note that in this example the interconnecting structure of composite system (2.8.53) does not enter additatively into the system description. Given as above, system (2.8.53) is not a special case of composite i = 1 , ..., I ,

zi=,n, = n,

(x)

2.8

APPLICATIONS

85

system ( Y )with decomposition (Xi). However, it is a special case of composite system (9") with decomposition (Z;). Let us first apply Theorem 2.5.22. We choose Ui(Zi)

=

Z?PiZi

where Pi = P? is a positive definite matrix which we will need to determine for some particular choice of a positive definite matrix Ci = C : via the Lyapunov equation

-ci

=

A?Pi

+ PiAi.

(2.8.54)

We have Arn(pi)/zi12 5 ui(zi) _< A,(pi) /zi12> and using the notation of Theorems 2.4.29 and 2.5.22, we have

Hypotheses (i) and (ii) of Theorem 2.4.29 are now clearly satisfied. The matrix D = [d,] of Theorem 2.5.22 is given by

From Theorem 2.5.22 it now follows that the equilibrium x = 0 of composite system (2.8.53) is exponentially stable in the large for every admissible nonlinearity (i.e., system (2.8.53) is absolutely stable) if all principal minors of matrix D are positive. In order to obtain the least conservative results, we need to choose matrix Ci so that the ratio A,(C,)/A,(P,) is maximized. Finally it should be noted that Eq. (2.8.53) can serve as a model of a large class of practical systems endowed with multiple nonlinearities. 2.8.55, Example. We have pointed out before that in a sense the method of

analysis advanced herein is more important than the individual results of the preceding sections. To demonstrate this, we reconsider system (2.8.53) in an attempt to obtain less conservative stability conditions. This time we choose for each a Lyapunov function of the Lur6 form (see, e.g., Aizerman and Gantmacher [11, Lefschetz [l]),

(x)

(2.8.56) where Pi= P? is a positive definite matrix and (3, > 0 is a constant. Clearly, ui(zi)is positive definite and radially unbounded.

II

86

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

For composite system (2.8.53) we choose a Lyapunov function of the form

c I

v(x)

=

i= 1

(2.8.57)

LYiDi(Zi)

where aT = (a1,... ,a,) > 0 is a weighting vector. This function is positive definite for all x E R" and is radially unbounded. Along solutions of Eq. (2.8.53) we have I

1

i= 1

i= 1

+ 1 aibi&(ai) c d,cTAjzj I

I

i= 1

j = 1

1

I

where Ci = -(A:Pi+

PiA i ) .Adding and subtracting the nonnegative quantity Ef=lai(ai-(fi(ai)/'i))fi(ai>to D v ( 2 . 8 . 5 3 ) ( ~we ) > obtain

where wT = (zIT,. .., z1T, fl (al),. ..,A(a,)), where

is an n x n matrix, where S s.. =

= [sJ

is an n x I matrix, where each sG,

+A:CidiiaiPi + &&ai

.

z =J

- PibiUi,

.

i#j is an n,-vector, and R r.. =

=

[rJ is an I x 1matrix, and where

- ai pidiiC:bi - (Ui/ki),

i=j

-3(aiBidijcj7bj+aiPjdjic:bi),

i

+j.

It now follows that system (2.8.53) is absolutely stable if the symmetric ( n + / ) x ( n + / ) matrix (2.8.58) is negative definite. I t is interesting to note that this result is somewhat similar to one obtained by Lefschetz [I], using quite different methods.

2.8

APPLICATIONS

87

Under suitable conditions, frequency domain techniques can be used to construct Lyapunov functions of the form (2.8.56) for subsystem (q)(see Bose [l], Bose and Michel [l, 21). If the pairs ( A i , b i )and (A?,ci) are completely controllable and completely observable, respectively, (see Kalman, Ho, and Narendra [l]) then the Popov criterion (see Popov [ l ] ) along with additional reasonable conditions guarantee the existence of a Lyapunov function for (Yi) of the Lurk type (by the Kalman-Yacubovich Lemma) whose derivative is negative definite (see Kalman [l], Yacubovich [l]). In fact, an algorithm based on the proof of the Kalman-Yacubovich Lemma can be implemented on the computer to construct the Lyapunov function (2.8.56). However, this approach is not particularly satisfactory, for it is not explicit. We will make extensive use of graphical methods (including Popov-type conditions) in Chapter VI, where we establish results which are much easier to use and which are applicable to a larger class of problems. 2.8.59. Example. (Multiple Feedback Systems: Indirect Control Case.) Consider systems described by the set of equations i i =

A i z i - bifi(oi)

ki = diTy =

c d..y .

j= 1

B

(2.8.60)

J’

i = 1, ...,I, where y i > 0 is a constant and all other symbols are defined similarly as those in Eq. (2.8.53). In addition, assume that lim,o,+mJ;A(q)dq = a. This system may be viewed as a nonlinear interconnection of isolated subsystems (Sq.), 2. = A 1. z1. - b1. f I. ( o1. ) ki = dii[~;zi-Yif,(oi)],

(.4”i>

i = 1, ... , I . For each (q) we choose a Lurt type Lyapunov function

v i ( z i ci) , = z?Pi zi

+ Pi l i f i ( q ) dq

(2.8.61)

where Pi = PT is a positive definite matrix and pi > 0 is a constant. For system (2.8.60) we choose a Lyapunov function of the form v(x,a) =

I

CliUi(Zi,fJi) i= 1

(2.8.62)

where oT= (ol,..., ol), xT= ( z l T ,. .., z L T )and , uT = ( a l , ..., a,)> 0. This function is clearly positive definite and radially unbounded. Along solutions of

88

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Eq. (2.8.60) we have

where w is defined as in Example 2.8.55, of (2.8.58), 3 = [S,] is given by s.. -v =

and

w

=

[TJ

c" is of the same form as submatrix C

[

+ c i d i i a i / j i- aiPibi,

i

12 c1. dJ .I . J./j. J'

i # .i,

=j

is specified by F . . = + ( g .I / j .I d .y .Y j + a J. / j j. dj.i .Y i ).

Noting that (xT,oT)= 0 is the only invariant subset of the set E = {(xT, )'a E R"" : h ( , , 8 , , 0 ) ( x , o) = 0}, it follows from Theorem 2.2.30 that the origin (x', )'a = 0 of system (2.8.60) is asymptotically stable in the large for all admissible nonlinearities if the matrix

is negative definite. This result is somewhat similar to one obtained by Lefschetz [l], using a different approach. It is also interesting to note that Pai and Narayan [ I ] have used a set of equations similar to Eq. (2.8.60) and a Lyapunov function of the form (2.8.62) to analyze multi-machine power systems.

2.8.63. Example. (Stabilization.) Consider systems described by equations of the form

ij = A,zi

+ C 1

C,zj,

i

j=l,i+j

=

I , ..., I,

(2.8.64)

where z , E Rnt,A i is an n, x n , matrix, and C, is an n , x n j matrix. Suppose this system, called an uncompensated system, is unstable or that its degree of stability is not acceptable. Associated with (2.8.64), consider the compensated system ii =

A;zi

+

I j = 1, i # j

C,zj

+ Biui,

i = I,

..., I,

where ui E Rm,and B; is an n i x mi matrix. Let Biui be of the form I

Biu; = Bi

1 E,zj

j = 1

I

4

C F,zj,

j= 1

(2.8.65)

2.9

NOTES AND REFERENCES

89

where E, is an m i x n j matrix. System (2.8.65) may be viewed as a linear interconnection of 1 isolated subsystems (q.),

i i = AiZi + BiEiiZi = AiZi

+ FiiZi,

(%)

i = 1, ..., 1. If (Ai,Bi) is controllable, we can choose the local feedback matrix Eii in such a fashion that (9J has a desired degree of exponential stability. To determine this property, choose Ui(Zi) = Z?PiZi,

Pi

=

P?

where Pi is a positive definite matrix which needs to be determined for some choice of a positive definite matrix Ci, via the Lyapunov equation - ci = (Ai+Fii)TPi

+ Pi(Ai+Fii).

This yields the estimates Am(P)(ziI22 ui(zi)I A,(Pi)l~i(2, /Vui(zi)/I 2,I,,,,(Pi)~zi~,and Dui(yi)(zi) I -A,(C,) Izi12. In accordance with Corollary 2.5.21, the compensated system (2.8.65) will be exponentially stable in the large if the principal minors of the test matrix D = [d,] are all positive, where

This is the case if for example

Thus, for given matrices Bi, i = 1, ..., 1, we need to choose feedback matrices EG,i, j = 1, ..., I, and matrices C i ,i = 1, ..., I, in such a fashion that the ratios Am(Ci)/AM(Pi),i = 1 , ..., I are maximized and such that IlC,+F,II i \\CG\l, i, j = 1, ..., I, i # j . Finally, we can invoke Corollary 2.5.23 to obtain an indication of the degree of stability of the compensated system, by computing p = mink Re[A,(D)], k = 1, ...,1. 2.9 Notes and References

For existence and uniqueness results of solutions for ordinary differential equations refer to Coddington and Levinson [1). Standard references on the Lyapunov theory include Hahn [2], Yoshizawa [J], Krasovskii [l], and LaSalle and Lefschetz [l]. Our exposition is heavily influenced by Hahn [2], where an extensive discussion of comparison functions of class K is given (Hahn [2, pp. 95-97]). For a good overview of comparison theorems and the comparison principle, refer to Lakshmikantham and Leela [l, 21, Szarski [1],

90

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and Walter [l]. The original results in this area include Miiller [l] and Kamke [l]. For subsequent work, refer to Wazewski [I], Grimmer [l] and LaSalle [I]. Standard references on absolute stability are the monographs by Aizerman and Gantmacher [I] and Lefschetz [l]. Nice sources on M-matrices include the papers by Ostrowski [1] and Fiedler and Ptak [l] and the books by Bellman [3] and Gantmacher [ l , 21. See also the report by Araki [2]. Sections 2.3 and 2.4 are based on references by Porter and Michel [l, 23, Michel and Porter 131, Michel [S, 71, Bose and Michel [ l , 21, and Bose [l]. For related results refer to Thompson [I, 21 and Thompson and Koenig [l]. Section 2.5 is an adaptation and expansion of results reported by Michel and Porter [3], Araki and Kondo [l], Michel (75-7, 91, Araki [l, 41, Rasmussen and Michel [2,4], and Rasmussen [l]. The introduction of vector Lyapunov functions by Bellman [2] gave rise to the qualitative analysis of large scale systems, using the approach advanced herein. Bailey [ l , 21 was the first to apply the comparison principle to vector Lyapunov functions in analyzing nonlinear composite systems with linear interconnecting structure. Subsequent extensions include the work of Matrosov [ 1, 21 whose approach we follow in Section 2.6. Refer also to Piontkovskii and Rutkovskaya [I]. Additional related references on Lyapunov stability of composite systems include Weissenberger [I], Matzer [ 11, Grujii. and Siljak [ 2 ] , Tokumaru, Adachi, and Amemiya [2], and Athans, Sandell, and Varaiya [I]. Section 2.7 is based on results by Michel [2,4, 91. Related work is contained i n Michel [ I , 31, Michel and Heinen [I-41, and Michel and Porter [ I , 21. For results on practical stability and finite time stability, refer to LaSalle and Lefschetz [I], Weiss and lnfante [I], and Michel and Porter [4]. The source of Examples 2.8.1, 2.8.9, and 2.8.22 is Piontkovskii and Rutkovskaya [I]. Our treatment of these examples follows Michel [ S , 71. For a good qualitative treatment of economic systems refer to Quirk and Saposnik [I]. The sources of Examples 2.8.34, 2.8.39, and 2.8.41 are Sandberg [lo], Mitra and So [I], and the related work by Rosenbrock [ I , 21. In our approach to these examples we follow Michel [9]. Multiple feedback systems similar to those of Examples 2.8.52 and 2.8.59 are treated by Lefschetz [l] .The present results, which are not entirely identical to those of Lefschetz [l], are based on references by Bose [I] and Bose and Michel [1,2]. Sources concerned with systems containing many nonlinearities are numerous. For some of these refer to Narendra and Neuman [l], Narendra and Taylor [l], McClamroch and Janculescu [I], and Blight and McClamroch [l].

C H A P T E R I11

Discrete Time Systems and Sampled Data $stems

In the present chapter we continue the stability analysis of large scale systems described on finite-dimensional spaces. In particular, we show how the methods of Chapter I1 need to be modified for the case of discrete time systems described by ordinary difference equations and also for the case of sampled data systems. Although we develop only selected results in this chapter, it will become apparent that most of the results and comments of Chapter I1 can in principle be modified to accommodate corresponding systems described by ordinary difference equations. However, it will also become apparent that these modifications are not'completely obvious in every case. Difference equations are important in their own right in applications. They are also frequently used to represent sampled data systems at discrete points in time. However, a complete description of sampled data systems, at all points in time, provides us with the first important class of hybrid systems. Later, in Chapter V, we will be able to analyze hybrid systems (systems described by a mixture of equations) defined on infinite-dimensional spaces. This chapter consists of five parts. In the first section we briefly introduce 91

111

92

DISCRETE TIME AND SAMPLED DATA SYSTEMS

the types of difference equations considered, while in the second section we characterize the classes of composite systems which we will treat. In the third section we state and prove selected results for stability, asymptotic stability, exponential stability, and instability of interconnected systems described by ordinary difference equations. In the fourth section we apply these results to some specific examples. In the fifth section we consider the stability analysis of a class of composite sampled data systems, valid for all points in time. We conclude our presentation with a brief discussion of pertinent literature in the last section. As in Chapter 11, the objective in this chapter will be to analyze large scale systems in terms of lower order subsystems and in terms of interconnecting structure. 3.1

Systems Described by Difference Equations

We consider discrete time systems described by difference equations of the form x ( z + 1) = SCx(T),~l (3.1.1) where E I g {t,+k}, to 2 0, k = 0, 1,2,..., X E R",a n d g : R " x I + R".Note that for every x, E R" and for every to 2 0, Eq. (3.1.1) has a unique solution x(z; x,, r,) which is defined for all z E I , with x(t,; x, t o ) = x,. We assume that g ( x , z) = x for all z E I if and only if x = 0. Thus, Eq. (3. I . 1) admits the trivial solution x = 0 which is in fact the only equilibrium point. We can characterize the equilibrium of Eq. (3.1.1) as being uniformly stable, uniformly asymptotically stable, exponentially stable, unstable, completely unstable, etc., and we can characterize the solutions of Eq. (3.1.1) as being uniformly bounded, uniformly ultimately bounded, etc., by rephrasing Definitions 2.2.1-2.2. I 1 verbatim, save that t E J is replaced by z E I. The Lyapunov results for system (3.1.1) involve the existence of mappings u : R" x I + R and the first difference D u ( x , z ) along solutions of Eq. (3.1.1) expressed by D q 3 . l . l , ( x , r )=

uCg(x,9, z +

11 - u ( x , z ) .

Such functions are characterized as being positive definite, negative definite, radially unbounded, decrescent, positive semidefinite, and negative semidefinite by modifying Definitions 2.2.16-2.2.20, respectively, in an obvious way, replacing t E J by z E I . Lyapunov theorems for stability, uniform stability, uniform asymptotic stability, exponential stability, uniform asymptotic stability in the large, exponential stability in the large, instability and complete instability of the

3.2

93

LARGE SCALE SYSTEMS

equilibrium of Eq. (3.1.1) as well as Lyapunov-type theorems for uniform boundedness and uniform ultimate boundedness of the solutions of Eq. (3.1.1) have the form of Theorems 2.2.21-2.2.28, with t E J replaced by z E I.

3.2 Large Scale Systems We consider discrete time systems described by equations of the form

zi(z+1) =fiCzi(z>,zl

(xi)

+ gi[zl(z>, ...,z[(z),zl,

(3.2.1)

i = 1, ...,I, where z E I, zi E Rni,fi: Rnix I-+Rni,and gi: R " ' x ... x R"'x I - + R"'. Letting n, = n, xT= (z,', ..., z?),f(x, z)' = [fl(zl, z)', ...,h(zl, z)'], and

ct=

g ( X , T)'

=

b i (21, ..

., ZI

T),'

...,Sr(Zi, ..,, Zi > 711' A [gi (X, T I T , ...,91(X, z)T1,

we can represent Eq. (3.2.1) equivalently as

(9)

x ( z + l ) =f[x(z),zl

+S[X(T),Zl

2 h[x(z),z].

(3.2.2)

We call system (Y ),which is of the same form as Eq. (3.1.1), a composite system or an interconnected system with decomposition ( X i ) . System (9') may be viewed as a nonlinear and time varying interconnection of 1 isolated subsystems (q) described by equations of the form

($1

Zi(T+l)

=fi[z,(z>,71.

(3.2.3)

We denote the unique solutions of (9') and (Yi) b y x(z; xo,to) and zi(z; zio, to), respectively. Furthermore, we assume that x = 0 and zi = 0 are the only equilibrium points of (9'and ) respectively. The interconnecting structure enters additatively into the description of system (9')As . such, this system is a special case of composite systems described by equations of the form

(x),

(W

Zi(z

1) = fi [Zi(z), Si(Z1 (z), ...>ZI (TI, = fi[zi(z),gi(x(z),z),z]

cf=

T)> z]

4 hi[x(z>,~I,

(3.2.4)

i = I , ...,I, where zi E Rni, z E I , ni = n, xT = (zlT, ..., z;) E R", gi: R" x I - + Rri,fi: R"i x Pi x I + Rni,and hi: R" x I ---f Rni. Letting h(x, z)' = [h (x,,')T ...,h, (x, z)'], Eq. (3.2.4) can be rewritten as

(9'")

x(z+l)

=

h[x(z),z].

(3.2.5)

Given to 2 0 and xo E R", we let x(z;x,,t,) denote the unique solution of Eq. (3.2.5) for T E: I such that x(to; xo,to) = xo. Furthermore, we assume that Eq. (3.2.5) possesses only one equilibrium, x = 0.

94

111

DISCRETE TIME AND SAMPLED DATA SYSTEMS

3.3 Stability and Instability of Large Scale Systems

We now state and prove selected qualitative results for composite systems

(9’)and (9’”).

3.3.1. Definition. Isolated subsystem (q) possesses Property A if there exists a function vi: Rni x I - , R , $ i 2 E K R , +hi3 E K , and a constant oi E R such that $il(lZil) 5 ui(zi,z) for all zi E Rni and

tE

$i2(Izil),

I.

Dui(v,,(zi,T) 5

0i$i3(Izil)

(z)

If 0,< 0, then the equilibrium of is uniformly asymptotically stable in the large. If ci = 0, the equilibrium is uniformly stable and if 0,> 0, the equilibrium of (q) may be unstable. 3.3.2. Theorem. The equilibrium x = 0 of composite system (9’with ) decomposition (Xi) is uniformly asymptotically stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (q) possesses Property A ; (ii) given ui of hypothesis (i), there is a constant L, > 0 such that

I vi (z,’, z) - v,(ZY, 2 ) I 5 Li lZi’ - z; I for all z,’, zy E Rni and T E I ; (iii) given t,ki3 of hypothesis (i), there exist constants ay 2 0, i, j such that

= 1, ..., I,

1

Igi(Z1,...,~1,t)l5

C aij$j3(lzjI),

i

j =1

=

for all zi E Rni,i = 1, ..., I, and t E I ; and (iv) given a,of hypothesis (i), the successive principal minors of the test matrix D = [Idy] are all positive, where

d..

+

- (ai Li aii),

i

=j

i # j.

Proof. Given vi of hypothesis (i), let txT = (a1, ...,a,) > 0 be an arbitrary constant vector and choose as a Lyapunov function for ( Y ) ,

c I

u(x,z) =

i= 1

aivi(zi,2).

(3.3.3)

In view of hypothesis (i), v is positive definite, decrescent, and radially unbounded. Along solutions of (9’we ) have, taking hypotheses (i)-(iv) into

3.3

STABILITY AND INSTABILITY OF LARGE SCALE SYSTEMS

95

account

=

-g'Dw

A

-YTW

where wT = [$13(lzl/),...,$13(1z11)], aTD = yT, and matrix D is defined in hypothesis (iv). Since D is an M-matrix, it follows from Theorem 2.5.2 that D-' 2 0 exists. Thus, a = ( D - ' ) T y . Using a similar argument as in the proof of Theorem 2.5.15, we can choose y > 0 so that a > 0. Thus, DV(,)(X,

z) I -yTw

< 0,

x # 0,

i.e., Du(,,(x, z) is negative definite for all x E R" and z E I. Therefore, the equilibrium of (9') is uniformly asymptotically stable in the large. Next, we prove an instability result. Recall the notation Bi(ri) = {zi E Rni : lzil < r i } .

(x.)

3.3.4. Definition. For let there exist a function ui: Bi(ri)x I + R , $ i l , $i2, $i3 E K , and constants b,,, biz,ai E R such that ail

$il(lzil> 5 ui(zi,z) 5 ~ i ~ $ i z ( l z i l > ~Dui(yi)(zi,z) 5 oi$i3(I~il)

for all zi E Bi(ri)and z E I. If hi, = hi, = - 1 we say that C. If ail = di2 = 1 we say that (q) has Property A".

(x)possesses Property

If (q.) has Property C with D~ < 0, then the equilibrium zi = 0 is completely unstable. If (Yi) has Property A" and ci < 0, the equilibrium is uniformly asymptotically stable. 3.3.5. Theorem. Let N # 0, N c L = { 1, ..., Z>. Assume that for composite system (9) with decomposition (Xi)the following conditions are true. has Property C and if i $ N , (i) If i E N , then isolated subsystem i E L, then (q) has Property A";

(x)

111

96

DISCRETE TIME AND SAMPLED DATA SYSTEMS

(ii) given uiof hypothesis (i), there is a constant Li> 0 such that

1 Ui(Zi’,

1

z) - Di(ZY, z) 5 Li IZi’

-.;I

for all zi’, z; E Bi(ri)and t E I ; (iii) given t,bi3 of hypothesis (i), there exist constants a! 2 0, i , j E L, such that 1

Igi(z1, ...,zltz)I I

C1 ac$j3(lzjl),

j=

i = 1 7 ...,I,

for all zi E Bi(ri), i E L,and 7 E Z;and (iv) given ai of hypothesis (i), the successive principal minors of the test matrix D = [dJ are all positive, where

- (ai+ Li aii),

i =j

-Liag,

i # j.

If N # L, the equilibrium x = 0 of system (9) is unstable. If N = L, the equilibrium of (9) is completely unstable. Proof. For system (9) we choose the Lyapunov function given in Eq. (3.3.3). Invoking hypotheses (i)-(iv), it is clear that Du(,)(x, z) is negative definite for all zi E Bi(ri), i E L, and z E I. Now consider the set

D

=

((~,z)~R”xI:z~fB r <~m( irn)r ,i , i E N , and zi=O, i $ N , and Z E Z } .

For ( x , ~ E) D we have - - x i E N ~ i $ i l ( \ z i 1 ) I u ( x , ~I ) -ZisN@i$i2(IZil). It follows that in every neighborhood of the origin x = 0 there is at least one point x’# 0 for which u(x’, 7) < 0 for all z E f. Also, u(x,z) is bounded from below on D. Thus, if N # L, it follows from Chetaev’s theorem (Theorem 2.2.28), modified for systems described by difference equations, that the equilibrium of (9’is )unstable. If N = L, then u is negative definite and the equilibrium is completely unstable. Because of hypothesis (ii), Theorem 3.3.2 can be satisfied only for Lyapunov functions of the norm type. It is possible to obtain results using higher order Lyapunov functions. However, as will be seen, in this case the results are not quite as simple as corresponding ones for systems described by ordinary differential equations. 3.3.6. Theorem. The equilibrium x = 0 of composite system (Y) with decomposition (Ci) is uniformly asymptotically stable in the large if the following conditions are satisfied. possesses Property A ; (i) Each isolated subsystem (ii) given vi and $ i 3 of hypothesis (i), there exist constants Si,p, E R ,

(x)

3.3 i, j = 1,

97

STABILITY A N D INSTABILITY OF LARGE SCALE SYSTEMS

..., I, such that

for all zi E Rni,zj E R”’, i,j = 1 , ..., I, and z E I ; ...,a,) (iii) given oiof hypothesis (i), there exists a vector aT = (al, >0 such that the matrix B = [be] is positive definite, where

(3.3.7) and

(3.3.8)

and

di + Pii,

i

=j

(3.3.9)

i # j.

Proof. For (9) we choose the Lyapunov function (3.3.3) which is clearly positive definite, radially unbounded, and decrescent. Along solutions of ( Y ) we have, using hypotheses (i)-(iii), 1

D U ( , , ( X , ~= )

C a i { u i C J ; . ( z i , 7 ) + ~ i ( x , z )7, + l I i= 1

c 1

=

+

i= 1

ai{uiCf,(zi,7),

-ui(zi,z))

z + 1 1 - Ui(Zi,T)}

1

C ai {ui [IJ;.(zi z>+gi(x, z), 7 + 11 - IJ;:(zi

i= 1

3

ui

9 7)2

z

+ 111

Since B = [bi] is positive definite, it follows that Dv(,,(x,7) is negative definite for all x E R” and 7 E I. This completes the proof. W

98

111

DISCRETE TIME AND SAMPLED DATA SYSTEMS

Jn order to satisfy hypothesis (iii) of Theorem 2.3.6, we need --oi+diZ > 0 and ai ci2akd$ > 0. Note that if in this theorem all I,+ii E K R and are of the same order of magnitude, then we can conclude that the equilibrium x = 0 of (9’)is exponentially stable in the large. Also, if matrix B is only positive semidefinite, then the equilibrium is uniformly stable. Now let C = diagrc,, ..., cr], A = diagra,, ..., at], D = [d,l, and B = [bii], with elements as specified in (3.3.7)--(3.3.9). Then clearly B = C A C - DTAD.

(3.3.10)

We now prove the following simpler result involving M-matrices. 3.3.11. Corollary. Assume that hypotheses (i) and (ii) of Theorem 3.3.6 are true with -oi+di2 > 0, di+Bii 2 0, i = 1, ..., I, and fiq.2 0, i, j = 1, ..., 1, i # j . The equilibrium x = 0 of system (9’)is uniformly asymptotically stable in the large if all successive principal minors of the test matrix C - D are positive.

PvooJ By the above assumptions, D 2 0, C > 0, and C - D is an M-matrix. From Corollary 2.5.8 it follows that there exists A = diag[a,, ...,a,] > 0 such that the matrix (3.3.10) is positive definite. Therefore all assumptions of Theorem 3.3.6 are satisfied, which completes the proof. Combining the ideas in the proofs of Theorems 3.3.5 and 3.3.6 and Corollary 3.3.11, we can prove the following instability results. 3.3.12. Theorem. Assume that hypothesis (i) ofTheorem 3.3.5 is true. Assume that hypotheses (ii) and (iii) of Theorem 3.3.6 are true for ziE Bi(ri),i = 1, .. .,I, and z E 1. If N # L, the equilibrium of (9’is)unstable and if N = L , the equilibrium of (9) is completely unstable. 3.3.13. Corollary. Assume that hypothesis (i) ofTheorem 3.3.5 is true. Assume that hypotheses (ii) and (iii) ofTheorem 3.3.6 are true for zi E Bi(ri),i = I, ..., I, and T E I, with - o i + d i 2 > 0, di+fiii 2 0, i = 1, ..., 1, and fig 2 0, i, j = 1, ..., I, i # j . Assume that all successive principal minors of the test matrix C - D are positive. If N # L , the equilibrium of (9’is )unstable and if N = L, the equilibrium of (9) is completely unstable.

In the remaining results we consider composite system (9”’). These results are of course also applicable to system (9’). 3.3.14. Theorem. The equilibrium x = 0 of composite system (9”’) with decomposition (C:) is uniformly asymptotically stable in the large if the following conditions are satisfied. (i) There exist functions vi: R“ x I + R , i = 1, ..., I, and I,+il, I,+i2 E KR, i = 1, ..., I, such that ~

i (Izil) l

5 ui(x, z) 5

+i*(IXI)

3.3

99

STABILITY AND INSTABILITY OF LARGE SCALE SYSTEMS

for all zi E Rni,x E R", and z E I ; (ii) there exist $ i 4 E K , i = 1, ...,I , constants b, E R , i, j constants uii E R, i, j = I , ... 7 I, i # j , such that

=

1, ..., I, and

1

for all zi E Rni7x E R",z E I, i = 1, ..., I ; and (iii) there exists a vector aT = ( a l , ...,al) > 0 such that the test matrix S = [s,] is negative definite, where

Proof. Given ui in hypothesis (i) and a in hypothesis (iii) we choose

u(x,z) =

I

1 CliUi(X,Z).

(3.3.15)

i= 1

This function is clearly positive definite, radially unbounded, and decrescent. In view of hypotheses (i)-(iii) we have along solutions of (Y"),

c ai{ui[hi(x,z), I

DU(,..,(X,Z) =

=

z + 1) - U i ( X , Z ) }

i= 1

uTRu = u T ( ( R + R T ) / 2 ) u= U ~ S IU I,(S)~U~~

where uT = [ $ 1 4 ( ~ z 1 ~...,$14(1zf1)1/2], )1~2, matrix R

and S

= [sii]

= [r,]

is specified by

is given in hypothesis (iii). Since S is negative definite we have

I, ( S ) < 0. Therefore Du(,..)(x,z) is negative definite for all x E R" and z E I , which completes the proof.

W

Note that if $ii E K R and are of the same order of magnitude then the equilibrium x = 0 of system (Y")is exponentially stable in the large. Also, if

IOU

111

DISCRETE TIME A N D SAMPLED DATA SYSTEMS

in the above theorem matrix S is only negative semidefinite, then the equilibrium of (9”’) is uniformly stable. 3.3.16. Theorem. Let L = ( I , ..., I} =) N # 0. Assume that for composite system (9”‘) with decomposition (C;) the following conditions hold. (i) There exist functions u,: Bi(ri) x I - + R , for some ri > 0, i = 1 , ..., I, and $ i , , Gi2 E K such that 5

-*il(lxl) $il(lzil)

I

Ui(X,Z)

5 ui(x,z) 5

iE N

-$i2(14),

i

Gi2(lxl)y

B

N, i E L

for all ziE Bi(ri), i E L, x E Xi= Bi(ri),and z E I ; (ii) there exist t+hi4E K , i E L, constants b, E R, i, j E L, and constants aGE R, i, j E L , i # j , such that

for all ziE Bi(ri), i E L, x E Xi= Bi(ri),and 7 E I ; ...,a,)> 0 such that the matrix S (iii) there exists a vector aT = (al, [xu] is negative definite, where 3.. rJ =

{ C‘

akbki,

.

l = J

=

.

k=l

(ai a,

+a j a j i ) / 2 ,

i # j.

If N # L, the equilibrium x = 0 of system (9”) is unstable. If N equilibrium of (9”’) is completely unstable.

= L,

the

Proof. For (9”) we choose the Lyapunov function (3.3.15). In view of (ii) and (iii), Duo..,(x, T ) is negative definite for all x E R” and z E I. From (i) it follows that in every neighborhood of x = 0 there exists a point x’ # 0 such that u ( x ’ , z) < 0 for all T E I . Also, u is bounded on B ( r ) x I where r < minri. The conclusion of the theorem follows from Theorem 2.2.28 (modified for difference equations). We conclude this section by noting that it is also possible to establish results for boundedness and ultimate boundedness of solutions for systems (Y) and (9”’) described by Eqs. (3.2.2) and (3.2.5), respectively. In addition, it is also possible to formulate results for discrete time systems which are analagous to those of Sections 2.6 and 2.7.

3.4

EXAMPLES

101

3.4 Examples

We now apply the preceding results to some specific examples. 3.4.1. Example. Consider systems described by the set of equations =~,z,(~)+~,f1('11~~))

Z,(Z+l)

(3.4.2) '12w

= CZTZ1(4

where 7 E I , zi E P i ,A i is an ni x ni matrix such that llAiil = (AM(A:Ai))1'2 < 1, i = 1,2, c1 E Rn2, c2 E R"', fi: R + R , f i ( q i ) = 0 if and only if qi = 0, 0 < qifi(yli) < kiqi2for all qr # 0, and k i > 0 is a constant, i = I , 2. System (3.4.2) may be viewed as a nonlinear time-invariant interconnection of two isolated subsystems (Y;), (Y2), Zi(Z+

1) = A,z,(r),

(a

i = 1,2, interconnected by the relations g1(x)= b, f , (ql) and g 2 (x) = b, f 2 (q2).System (3.4.2) is clearly a special case of system (3.2.1). For (Yi) choose ui(zi)= IziJ which is Lipschitz continuous with Li = I , i = I , 2. Since Dui(yi)(zi)5 (liAiil - 1) Izil, hypotheses (i) and (ii) of Theorem 3.3.2 are satisfied. Hypothesis (iii) is also satisfied with a,, = a22= 0, a12= k , Ibll I c l [ , and aZ1= k , lb21 Ic21. Matrix D i n hypothesis (iv) is given by

It follows from Theorem 3.3.2 that the equilibrium xT= (zlT,zZT)= 0 of system (3.4.2) is uniformly asymptotically stable in the large if all successive principal minors of D are positive, i.e., if

(3.4.3) 3.4.4. Example. We reconsider system (3.4.2). This time we assume that

there exists a constant a, > 1 such that IA,zlI 2 a, Izl I for all z1 E R"' and that llA211 < 1. For (Yl) choose ul(zl) = - /zl/ and for (97;choose ) uz(z2) = 1z21. Then ui is Lipschitz continuous with Li = 1, i = 1,2. Since Dul(yl)(zl)I ( l - - a 1 ) ~ z l and ~ I( ~ ~ -Al)lz21, z ~ ~ hypotheses (i) and (ii) of Theorem 3.3.5 are satisfied. Hypothesis (iii) of this theorem is also true with a,, = a 2 2 = 0 , a , 2 = k , ~ 6 , ~ ~ c , ~ , a =k21b211c21.MatrixDofhypothesis nda2,

102

111

DISCRETE TIME AND SAMPLED DATA SYSTEMS

(iv) is given by

It follows from Theorem 3.3.5 that the equilibrium of system (3.4.2) is unstable if

3.4.5. Example. Let us reconsider Example 3.4.1, where we assume liAill< 1, i = 1,2. For (q)we choose u , ( z , ) = [ z ~ ( ~i = , 1,2. In this hypotheses (i) and (ii) of Theorem 3.3.6 are satisfied for i = 1,2, oi=IIAiI12-l~ 6i=IIAiII, Bii=O, P12=klIb1lI~113 P 2 1 = k 2 I b z ( ( c 2 ( , t,hj3(1zi1) = I Z , ~ ~ .Using the notation of Eqs. (3.3.7)-(3.3.10), we have

that case with and

so that

It follows from Corollary 3.3.11 that the equilibrium of system (3.4.2) is uniformly asymptotically stable in the large if all principal minors of C- D are positive. This is the case if and only if inequality (3.4.3) is true. Thus, given our choices of Lyapunov functions, Theorem 3.3.2 and Corollary 3.3.11 yield the same stability condition for system (3.4.2).

3.4.6. Example. We consider two subsystems (q), i = 1,2, described by Zi(7-k

1)

=

AiZi(z)

o,(z) = c;zi(T)

+ bi.f;.(Gi(T))

(q)

where z,E R",, A , is an n, x n, matrix, b, E R'", cii E R",, ,h:R.-+ R , fi(oi) = 0 if and only if oi= 0,O < oi.fi(oi)5 kioi2 for all oi # 0, and k , > 0 is a constant for i = 1,2. We interconnect (Y1)and (.U;) to form a system described by

(3.4.7)

3.4

103

EXAMPLES

where c12 E Rn2and c21E R"'. Note that in this case the interconnecting structure does not enter additatively into the system description. Thus, with (Y1) and (Y;) specified as above, system (3.4.7) may be viewed as a special case of system (9") with decomposition (Xy). If we choose ui(zi)= IziI2, then hypothesis (i) of Theorem 3.3.14 is satisfied. Now = I A l z l + b l f l ( ~ l ) 1 2- 1 Z 1 l 2

DUl(Z1)

(11.11

112-1 +2kl l l A l II . Pll. Ic1 I I + k12lbl12 +1112>

+ {2kl IIA 1II . Ibl I . l C l 2 l + 2kl + {k12l b I l 2 . l C 1 2 l 2 1 k 2 I 2

P I

1Z1l2

l2 . Ic11 I . I c 1 2 I1 1z1 I . I Z Z I

with a similar inequality being also true for Du2(z2). Let tjr4(Izil) i = 1,2. Then hypothesis (ii) of Theorem 3.3.14 is satisfied with 1%

bll

=

~ / 1 ~ 1 1 1 + ~ 1 1 ~ 1 1 ~ 1~ 1 1 1 ~ 2b 1 2

b22

=

~ l l ~ z l l + ~ 2 1 ~ 2 1 . 1 ~ 22 1 > 2

a12

=

2kl

021

=

2 k 2 l l ~ 2 l l ~ I b 2 l . l c 2 l l 2kz2 +

1,

ll A 1 /I. 141.Ic121+

Choosing a1 = a2 = 1, matrix S s11 =

b21

2k12l b 1 l 2 .

= Isg]

= IziI2,

=

k121~112-IC1212

=

~ Z 2 1 ~ 2 l 2 . I ~ 2 1 I 2

IClll. ICIZI

1bZl2-Ic221-Icz1I.

of Theorem 3.3.14 is specified by 1

~ 1 1 ~ 1 / 1 + ~ 1 1 ~ 1 1 ~ 1 -~ 1+1k 12 2~12b 2 1 2 ~ I C z 1 1 z

s22

= ~ l l ~ 2 1 1 + ~ 2 1 ~ 2 1 . 1 ~-z 1z +1k~1 221 b 1 1 2 + 1 2 1 2

s12

=

$21

=

kl

II.1~II.1~lI.l~121+~121~112.1~111.1~121

+ k 2 11.12 /I . lb2l . k 2 l I + k 2 2 P 2 I 2

'

.

Ic221 l C 2 l

I.

It follows from Theorem 3.3.14 that the equilibrium xT= ( z ~ ~ , = z 0~ of ~ ) system (3.4.7) is uniformly asymptotically stable in the large if matrix S is negative definite, i.e., if I I I . 1 1 1 1 + ~ l I ~ , I . l ~ l 1 1 3 2+ k 2 Z l b 2 1 2 . 1 C 2 1 1 2

+

~ I I . 1 2 1 1 + k 2 1 ~ 2 1 ~ / ~ 2 z 1 k121bl12.1clz12 ~2

and

<1 <1

111

104

DISCRETE TIME AND SAMPLED DATA SYSTEMS

3.5

Sampled Data Systems

We consider systems described by equations of the form

i = 1, ..., 1, where zi E R"',fi: Rni x [0, m) -+ R"', f i ( z i ,t ) = 0 for all t E [0, 00) if and only if zi = 0, C, is an n i x n j matrix, T > 0 is a constant, called the sampling time, and i i ( t )denotes the right-hand derivative of z ( t ) with respect to t . This system may be viewed as a linear interconnection of 1 isolated subsystems (q), described by

(a

(3.5.2)

ii = f i ( q , t ) .

Letting xT= ( z l T ,. and

,

c,2

...

0

... ..

c,2

...

system (3.5.1) can be represented by the equation i ( t ) = f [ x ( t ) ,t

(9)

]

+ Cx(kT)

F [ x ( t ) , x ( k T ) ,t ] ,

kT 5 f < ( k + 1)T,

k = 0,1,2, ...,

(3.5.3)

where x E R",f': R" x [0, a)+ R", C is an n x n matrix, andf(x, t ) = 0 for all t E [0, m) if and only if x = 0. Henceforth we assume that Eq. (3.5.2) possesses for every z i ( 0 ) = zio a unique solution zi(t;zi(0),O)for all t 2 0 and that Eq. (3.5.3) has for every xo = x(0) a unique solution x(t;x(O),O) for all t 2 0. It should be noted that the terms Ui(t) =

Ui(kT)A

c 1

j=l,i#j

C,Zj(kT)

(3.5.4)

are forcing functions which assume constant values over each interval [kT, ( k 1) T ) , k = 0, 1 ,2, ..., and which change their values for each such interval according to zj(t),.j = 1 , ..., I, sampled at the discrete points in time k T , k = 0 , 1 , 2 ,....

+

3.5 SAMPLED

105

DATA SYSTEMS

3.5.5. Lemma. Assume that for each isolated subsystem (q)described by Eq. (3.5.2) there exists a continuous function ui: R"' x [0,co) -+ R and three positive constants cil, ci2,Li such that (i)

lzil 5 vi(zi,t)Icil lzil

(ii) D, qSp,) (zi, t )

=

lim inf,,,,

+

( l / h ){ui[zi+h .fi(zi,t ), t

+ h] - vi(zirt )>

2 -ci2 IZi(

(iii) Iui(zi',t)-ui(zf',t)l I LiIzi'-zf'I for all z i , zi', zf' E Rni and t E [0, co). Then the solution z i ( t ;z i ( k T ) ,k T ) (with z i ( k T ; z i ( k T ) , k T )= zi(kT)) of the forced system

+

i i ( t ) = fi(zi(t),t)

satisfies for all t

E

c 1

j = 1 , i#j

Cizj(kT),

k T 5 t < ( k + 1)T, (3.5.6)

[kT,(k+ 1) T ) the estimate

Izi ( t ;zi ( kT) ,k T )I 2 (I/cil) [eci2(kT)] [e-c121]Izi(kT)l - { ( I / c ~ , ) ( L ~ / c , ~ ) [I - e c ~ 2 ( k T ) e - c ~ z t l >

Proof. Let I , t + h E [ k T , (k

+ 1) T ) . Along solutions of Eq. (3.5.6) we have

ui[z,(t+h;Zi,t),t + h ] - Ui(Zi,t)

c I

Zi+h.f;.(Zi,f)+ h .

j=l,i#j

1

c i z j ( k T ) + o ( h ) , t+h - u i ( z i , t )

111

106

DISCRETE TIME AND SAMPLED DATA SYSTEMS

where use of hypothesis (iii) has been made. Invoking hypothesis (ii) we have

for all r

E

[kT,( k

+ 1) T ) . Let

and let u i ( t ) = u i [ z i ( t ; z i ( k T ) , k T )t ,] . Then D+u2 ~

-ci2ui

(3.5.7)

- ki,

and along solutions of (3.5.7) we have for all t E [kT,(k+ l)T),

Using hypothesis (i) we have

for all t

E

[kT, (k

+ 1) T ) , which concludes the proof.

3.5.8. Definition. Subsystem (8) described by Eq. (3.5.2) is said to possess Property k if there exists a continuous function u i : Rnix [O, co) R which satisfies hypotheses (i)-(iii) of Lemma 3.5.5. --f

We are now in a position to prove the following result.

3.5

107

SAMPLED DATA SYSTEMS

3.5.9. Theorem. The equilibrium x = 0 of composite system (9) with de-

composition (3.5.1) is uniformly asymptotically stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (q) possesses Property (ii) given ui of hypothesis (i), there exists a constant ci3 > 0 such that

z;

D u ~ ( , ~ ) ( z= ~ , ~lim ) sup(l/h){ui[zi((t+h);zi,t), t+h] - ui(zi,t)} h+O+

<

-ci3

lzil

for all zi E Rni and t E [0, 00); and (iii) all successive principal minors of the test matrix S where ((ci3/cil)ePCizT,

=

[sJ are positive,

.

.

Proof. Given ui of hypothesis (i), let aT = ( a l , ...,a,) > 0 be an arbitrary vector and choose as a Lyapunov function for (Y),

u(x,t) =

c LXiUi(Zi,t).

i= 1

Using hypothesis (i) we obtain

for all zi E Rni,i = 1 , ..., I, x E R" and t E [0, 00). Thus, u is positive definite, decrescent, and radially unbounded. Let t , t+h E [ k T , ( k + 1)T). In view of hypotheses (i) and (ii) we have D u ( , ) ( x , t ) = lim sup(l/h) h+O+

i= 1

aiui[zi(t+h;zi,t),t + h ] - a i u i ( z i , t )

108

111

DISCRETE TIME A N D SAMPLED DATA SYSTEMS

where w ( ~ T=)(12, ~ (kT)I,..., Izl(kT)I)and S i s the matrix defined in hypothesis (iii). Noting that w ( k T ) = 0 if and only if x = 0, and noting that S is an M-matrix, we conclude that Du(,,(x, t ) is negative definite for all x E R" and t E [0, a), using the same argument as in the proof of Theorem 2.5.15. Therefore, the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. It is possible to establish stability results for other classes of sampled data systems by the present approach. Using appropriate modifications, one can also obtain instability and boundedness conditions for sampled data systems by the present method. 3.6 Notes and References

For a treatment of the Lyapunov theory of systems described by ordinary difference equations, refer to Hahn [ l , 21 and Kalman and Bertram [2]. Sampled data systems are tfeated in numerous engineering texts (see, e.g., Kuo [I]). Theorems 3.3.2 and 3.3.5 are based on results reported by Michel [5, 71 while Theorems 3.3.14 and 3.3.16 are new. Theorems 3.3.6 and 3.3.12 and Corollaries 3.3. I 1 and 3.3.1 3 constitute an adaptation and expansion of results by Araki 141. Some of the examples of Section 3.4 are given in Michel [5, 71. Stability results for composite sampled data systems, such as those given in Section 3.5, are from Michel [IS, 71. For related stability results dealing with discrete time interconnected systems, refer to GrujiC and Siljak [l, 31.

CHAPTER IV

$stems Described Ly Stochustic Dzfferential Eqzlutions

Disturbances are important considerations in the qualitative analysis of large scale systems. For this reason, we now turn our attention to systems described by a large class of stochastic differential equations. Although our primary objective in this chapter is to establish conditions for asymptotic stability with probability one, exponential stability with probability one, and exponential stability in the quadratic mean of systems described by Tto differential equations, the present results can readily be modified to be applicable to other types of stochastic differential equations (and stochastic difference equations) and also, to stochastic stability types not explicitly considered herein. As in the preceding chapters, our objective is to analyze large scale systems in terms of lower order (and simpler) subsystems and in terms of the system interconnecting structure. This chapter consists of six parts. In the first section necessary nomenclature is established. In the second section, which consists of background material, the stochastic differential equations of interest are introduced and selected stability concepts and results are presented. Composite systems described by stochastic differential equations are formulated in the third section. 109

110

Iv

STOCHASTIC DIFFERENTIAL EQUATIONS

Scalar and vector Lyapunov functions are used in the fourth and fifth section, respectively, to establish stability results for the interconnected systems considered. These results are applied to specific examples in the sixth section. The chapter is concluded with a brief discussion of the pertinent literature in the last section. 4.1

Notation

In the present chapter we find it convenient to use the following additional nomenclature. Let V,, = a/au and V,, = a2/(duau), where u and u may be scalars or vectors. For example, if x E R" and u : R" + R, then V , u denotes the gradient vector of Y and V,, u represents the matrix defined by the elements d Z ~ / ( d xaxj), i i, j = I , .. .,n. The trace of a square matrix A is denoted by tr A . As before, we define for an arbitrary matrix D the norm IJDll by llDll= [AM(DTD)]"2and we define the norm llDiIrnby llDllnt= [tr(DTD)]'/'. Let T = R+ = [ O , c o ) and let (R,&,P) denote a probability space with probability measure P defined on the 0-algebra & of o-sets (oE R) in the sample space R. Any &-measurable function on R is called a random variable. A sequence of random variables indexed by t E T, {x,(w) E R", t E T } , is called a (continuous time parameter) stochastic process. Henceforth the w dependence is suppressed and we always assume that xo(w) x is known. Throughout this chapter we consider only Markov processes and we sometimes simply write x , in place of {x,, t E T } for such processes. For A E &, P ( A ) denotes the probability of event A and P ( A I B ) denotes the conditional probability of A under the condition B E d.Let E denote the expectation operator and let {x,,t E T } be a Markov process. Then EX,,x, denotes the expected value of x, at t E T if it is known that x, = x. If s = 0, the notation Ex x, is used. Subsequently, we will have occasion to use the quasimonoticity condition introduced earlier in Chapter 11. We will find it convenient to use the following convention. 4.1.1. Definition. A function b : R' x T + R is said to belong to class K' (i.e., u, < uj for all i # j and such b E K ' ) if for two vectors u, u E R', such that 0 I that 0 I ui= u i ,we have b(u, t ) < b(v,t ) for all t E T. 4.2 Systems Described by Stochastic Differential Equations

We consider systems which may be represented by equations of the form

(4.2.1) dx,= f(xt t ) dt + t ) dt, where t E T , x , E R " ,f:R " x T + R " , ( T : R X T - , R " ' ~ and , { l , , t ~ T is} an 2

5(xr3

4.2

STOCHASTIC DIFFERENTIAL EQUATIONS

111

independent increment Markov process. Subsequently we consider two specific types of stochastic processes for Eq. (4.2.1): (a) normalized mdimensional Wiener processes with independent components, denoted by {t,,t E T } A { z , , t E T } ,and (b) normalized m-dimensional Poisson step processes with independent components, denoted by {(,, t E T } $ {q,, t E T } . For an appropriate interpretation of Eq. (4.2.1) as an integral equation, refer to Arnold [I], Wong [l], and Kushner [l]. Henceforth we assume that the initial condition xo = x is known and that the trivial solution {x, = 0, t E T } is the only equilibrium of Eq. (4.2.1). The following is an existence and uniqueness result for Eq. (4.2.1). 4.2.2. Theorem. In Eq. (4.2.1) letf(.) and a(.) be measurable functions in ( x , t ) and letf( .) and a ( .) satisfy the conditions

Ifk t)12 I k ( l + lxIZ) lla(x, t>IG I k ( l + 1x1')

If(x+u,t)-f(x,t)l 5 klul //a(x+u,t)--ox,t)llm

5 klul

for all x E R", u E R", t E T, and some constant k > 0. Then there exists a separable and measurable Markov process {x,, t E T } satisfying Eq. (4.2.1) which is unique and sample continuous with probability one. In the following definitions, the qualitative behavior of the trivial solution of Eq. (4.2. I ) is characterized. 4.2.3. Definition. The trivial solution of Eq. (4.2. I ) is stable with probability one (stable w.p.1) if for any p > 0 and any d > 0 there is a 6 = 6(p,d) > 0

such that, if lxol< 6, then

I

P s u p ~ x , ~ 2 d ~ x oI = xp. {rsT

If the trivial solution is not stable w.p.1, then it is said to be unstable w.p.1. The trivial solution of Eq. (4.2.1) is asymptotically stable with probability one (AS w.p.1) if it is stable w.p.1 and if limr-,mx,= 0 with probability one, for all xo in some neighborhood U of the origin. If U = R", the trivial solution is said to be asymptotically stable in the large w.p.1 (ASL w.p.1). 4.2.4. Definition. The trivial solution of Eq. (4.2. I ) is exponentially stable in the large with probability one (ESL w.p.1) if the trivial solution is stable w.p.1 and for all Y > 0 and for all z E T and 8 > 0, there exist positive constants M = M ( r ) and such that if lxol < Y, then

112

Iv

STOCHASTIC DIFFERENTIAL EQUATIONS

The above definitions pertain to the sample functions x, on T, which are of particular interest in applications, because the observable behavior of a system is generally exhibited via sample functions. However, moments are also frequently of interest. 4.2.5. Definition. The trivial solution of Eq. (4.2. I ) is exponentially stable in

the large in the quadratic mean (ESL q.m.) if, given xo = x (i) for every B > 0 there is a 6 = 6(&) > 0 such that for any x E R", 1x1 < 6 implies

supEx1x,12 I 8, tsT

and (ii) given r > 0, there exist positive constants M ( r ) and 1x1< Y, then E X ) x , I 2I M(r)e-P1,

p such that when

t E T.

Stability results for Eq. (4.2.1) involve the existence of Lyapunov type functions u : R" x T + R. Henceforth we assume that u(x,t) possesses continuous first and second order partial derivatives in x and continuous first partial derivatives in t. Stability results for Eq. (4.2.1) also require that u(x,t ) be in the domain of the weak infinitesimal generator A for Eq. (4.2. l), given by Au(x, t )

=

lim [E,,,u(x,+,, t+6) - u ( x , t ) ] / 6 .

,-to+

(4.2.6)

Note that A is a linear operator. If in Eq. (4.2.1) ( 4 , = z , , t E T } is a normalized Wiener process with independent components, then Au(x, f) LL dpu(x, t )

=

v,u(x, t ) + [ V , u ( x , t>lTf(x,t )

+ ~tr[o(x,t)TV,,u(x,t)a(x,t)].

(4.2.7)

I f i n Eq. (4.2.1) {& = q r , t E T } is a normalized Poisson step process with independent components q , (which experience a jump in any interval of length A f with probability piAt + o ( A t ) ) and with zero mean jump distribution P i ( . ) ,then Au(x, 1 ) 2 9 u ( x , t ) = V , u ( x ,f ) 4- [V,V(X, t ) ] T f ( x , t ) m

r

(4.2.8) where a i ( x ,t ) denotes the ith column of a ( x , t ) , i.e., a(x,t>= Ca,(x,t),...,a,(x,t)l.

4.3

113

COMPOSITE SYSTEMS

If for the autonomous version of Eq. (4.2.1) we consider u(x) = xTPx, where P is a positive definite matrix, then straightforward computation yields 9 u ( x ) = 9 u ( x ) = 2f(~)~Px+tr[a(x)~Pa(x)]. I n the following results, A u ( x , t ) denotes eithergu(x, t ) or 9 u ( x , t ) .

4.2.9. Theorem. Suppose there exists a Lyapunov function u : R" x T-r R in the domain of the infinitesimal generator A and that there exist t+bl, t,b2 E K R and 1+9~E K such that

(i) *, 5 u(x,l) 5 (ii) Au(x,t) 5 -$j(IxI)

*2(1xI)

and

for all x E R" and t E T. Then the trivial solution of Eq. (4.2.1) is ASL w.p.1.

4.2.10. Theorem. If in Theorem 4.2.9 ( r ) = c1 r z , t+k2 ( r ) = c2 r 2 , and $ 3 ( r ) = c3r2 where cl, c2, and c3 are positive constants, then the trivial solution of Eq. (4.2.1) is ESL w.p.1 and ESL q.m. 4.3 Composite Systems

We consider composite or interconnected systems which may be described by equations of the form

Pi)

dw,' = fi(w,', t ) dt

I

+ g i( w , l, ..., wti, t ) dr + C aij(w,j,t) d(,j, j= 1

i = 1 , ..., I,

(4.3.1)

where wIi E . R " h: ~ , Rnix T + R"', aij;R"jx T-+ R " I ' ~gi: ~ R"' , x ... x R"'x T + R"i, (,' E Rmi,and {tt',t E T } is an independent increment Markov process. Letting nj = n, mj = m, xT = [ ( w ' ) ~ ..., , (w')'] E R", f ( x , t)' = [j,(wi,t)T,...,.f i ( w f , t ) T ] , g ( x , r ) T =[gl(wl,..., ~ ' , t,... ) ~,gf(wl, ..., ~ ' , t ) ~ ] ...,(tf)T1, we [gl(x,t)T, . . . , g f ( x , t ) T ] ,o ( x , t ) = [ay(wJ,t ) ] , and tT= can represent Eq. (4.3.1) equivalently as

x>=l

(9)

c:=,

dx, = f ( x , , t ) dt

A

~ ( x ,r ,) dt

+ o ( x t , t ) d t , + g ( x , , t ) dt

+ o(x,, t ) d5,

(4.3.2)

w h e r e 8 R" x T -+ R", a: R" x T + R"'", g: R" x T - t R", and tiE R". We call system (4.3.2), which is of the same form as Eq. (4.2.1), a composite or interconnected system (9')with decomposition (Xi).It may be viewed as a nonlinear and time varying interconnection (under disturbances) of 1 isolated subsystems ( g )described by equations of the form dw,'

= fr(wti, t ) dt

+ aii(wti, t ) dt,'.

(4.3.3)

Iv

114

STOCHASTIC DIFFERENTIAL EQUATIONS

We assume that the Markov process {&, t E T } in Eq. (4.3.2) has independent components. Furthermore, we assume once and for all that systems (9’)and (y;’) satisfy conditions of the type given in Theorem 4.2.2. We denote the unique solutions of (9’)and (8) by {x,, t E T ) (or simply x,) and ( w t i , t E T } (or simply w,’), respectively, with xo x and woi A wi known. Also, we assume that the origin is the only equilibrium position of (9’and )

(y;’).

In the following definition we let A i denote the infinitesimal generator (defined by Eq. (4.2.6)) for subsystem (8). 4.3.4. Definition. Isolated subsystem (y;’) is said to possess Property A if there exists a Lyapunov function ui in the domain of the infinitesimal generator Ai and if there exist $ i l , t,bi2 E K R , $i3 E K , and a constant c iE R , such that

(i) $i,(lwil) I ui(wi,t) I $ i 2 ( J w ~ i l ) and (ii) ~ ~ v , ( w ’ ,I t) 0

for all

~

$

~

~

and for all t

1.1,‘ E

(

1

E

~

j

~

l

)

T.

4.3.5. Definition. I f in Definition 4.3.4 + h i , ( r ) = c i l r 2 , $ i 2 ( ~ )= ciZr2,and t,bi3 (Y) = r 2 , where c;, and ci2are positive constants, then isolated subsystem (y;’) is said to possess Property B.

If oi < 0, then the conditions in Definitions 4.3.4 and 4.3.5 correspond to the hypotheses of stability Theorems 4.2.9 and 4.2.10, respectively. As such, CT~plays a similar role i n the present case as it did i n the preceding chapters, i.e., it is a measure of the degree of stability for (y;’). Subsequently, we require certain identities involving the infinitesimal generator of Lyapunov functions ui for system (Ci) described by Eq. (4.3.1), or equivalently, for system (9’) described by Eq. (4.3.2). I n the following remarks, A i = Tiidenotes the operation 2 specified by Eq. (4.2.7) for the ith isolated subsystem (y;’). Similarly, A i = g i denotes the operation 9 specified by Eq. (4.2.8) for (y;’). 4.3.6. Remark. For system ( X i ) , or equivalently, for system (9’),let {tr’= z,’,I E T ) , i = I , ..., I, be normalized Wiener processes and consider a Lyapunov function ui : Rnix T + R . Let 6, denote the Kronecker delta symbol. Then 2 L l i

(x,r ) = ~ ( xtlT , V, ui(wi, t )

+ + tr [a(x, t ) T ~ xuxi ( w i ,t ) o ( x , t ) ] + V, ui(wi, t )

I

=

C

j= 1

[f,(~~~,t)+y~(x,t)]~V,,~~(~~‘,t)

+4 C 1

j . h , ni = 1

+

t r [ a ~ j ( i ~ ~ j , r ) T V ~ , ~ , , , , , , v ~ ( ~ ~ ~t )’ ], r ) aY, nv ji (O~v~i~, t’),

4.4

+

1

=

1

=

t)+gj(x, t)]Td,~wiui(wi, t ) vtui(wi,t)

[fj(wj,

j =1

C

+

115

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

1

j, k , m= 1

tr[okj(wj7 t)T6ki8miVwiwiui(~i,r)~mj(~j, t)]

+

[fi (w' , t)+gi(x, t)ITVwi~i(wi7 t ) Y u i ( w i 7t )

+ 4 C tr[od(wj,t)TVwiwiui(wi,t)od(wj,t)] I

j =1

= p iui(wi,t )

+3

+ gi(x, t)T vWiui(wi,t )

1

tr[od(wj,t)TVwiwiui(wi,r)od(wj,t)l, j=l,i#j

i.e., we have 9 u i ( x ,t ) = g iui(wi,t )

C 1

-I-4

+ gi(x, t)TVwlui(wi, t ) tr[od(wj,t)TVwiw, ui(wi,t)oG(wj,t)].

j=l,i#j

(4.3.7)

If in particular, ol(wJ7t ) = 0 for all i # j , then Eq. (4.3.7) reduces to 9 u i ( x ,t )

=y

i u i ( d , t ) + gi(x,t)'v,, vi(wi, t ) .

(4.3.8)

4.3.9. Remark. For system (Xi), or equivalently, for system (Y), let {tti= qti, t E T } , i = 1, ..., I, be Poisson processes, assume that oi,(wj, t ) = 0 for all i # j , and consider a Lyapunov function u i : R"' x T + R. Straightforward computation yields 9Ui(X, r ) = g i U i ( w i ,t )

+ gi(x, f)TVw' ui(u';, t ) .

(4.3.10)

4.3.11. Remark. From Eqs. (4.3.8) and (4.3.10) it now follows that if 5,' or 5,' = qti, and if od(wi,t) = 0 for all i # j , then Aui(x, t ) when with Ai = 9 i

=

~ , u , ( w 't,)

5,' = zti and

Ai

+ gi(x, t)'Vw, ui(wi, t )

= g iwhen

= zti

(4.3.12)

lti= qti.

4.4 Analysis by Scalar Lyapunov Functions

We are now in a position to state and prove several stability results for with decomposition ( X i ) . In the present section we composite system (9) employ scalar Lyapunov functions consisting of a weighted sum of Lyapunov

IV

116

STOCHASTIC DIFFERENTIAL EQUATIONS

functions for the isolated subsystems (q). In our first result we consider systems with disturbances confined to the subsystem structure.

4.4.1. Theorem. Assume that composite system (9')satisfies the following conditions. (i) {&, t E T) is either a Wiener process or a Poisson process and there are no stochastic disturbances in the interconnecting structure, i.e., a,(wj, t) = 0, i # j ; (ii) each isolated subsystem possesses Property A ; (iii) given the Lyapunov functions vi and comparison functions $i3, i = 1, ..., 1, of hypothesis (ii), there exist real constants bij,i, j = 1, ...,I, such that

(x)

for all x E Rn and t E T; and (iv) there exist positive constants ai, i = 1, ..., I, such that the test matrix S = [s..]8 ' i, j = 1, ..., I, defined by

is negative definite, where oiis given in hypothesis (ii). Then the trivial solution of (9) is ASL w.p.1. Proof. Given the Lyapunov functions vi of hypothesis (ii) and the vector aT = (a,, ...,a,) > 0 of hypothesis (iv), choose for system (9) the Lyapunov

function

Since each subsystem (q) possesses Property A, we have

for all x E R"md t E T. Thus, u ( x , t) is positive definite, decrescent, and radially unbounded, and there exist *,,$, E KR such that for all x E Rn and t E T. From hypothesis (i) we have 4.3. 1 1 we have

a G ( w j ,t) r

Avi(x, t ) = A, vi(wi,t)

,

0 for all i # j, and from Remark

+ g i ( x , t)TVw,vi(wi,t ) .

4.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

117

From the linearity of A and from hypotheses (ii) and (iii) it now follows that

I

=

C N ~ [ A ~ v ~t)+gi(x, ( w ~ , t ) T V w i ~ i (t)] ~i7

i= 1

..., [$,3((~11)]112),and S is the test matrix given where uT = ([$13(Jw1 in hypothesis (iv). Since S is by assumption negative definite, it follows that A, (S) c 0 and I

Av(x, t )

AM(S)

11Cli3(IwiI)

i= 1

.

for all x E Rn and t E T. Therefore, Au(x, t ) is negative definite and there exists a function $, E K such that for all x E Rn and t E T. Thus, all conditions of Theorem 4.2.9 are satisfied for system (Y), which proves the theorem. Note that Theorem 4.4.1 is quite similar in form to Theorem 2.4.2. In fact, the test matrices of these two results are identical. Therefore, the observations made in Chapter 11 for results such as Theorem 2.4.2 are also presently applicable. Furthermore, if sij 2 0 for all i # j, then we can use the theory of M-matrices to eliminate the weighting vector a and obtain a result which is similar in form to Theorem 2.5.11. Next, we consider systems with disturbances in the interconnecting structure and in the subsystem structure. 4.4.2. Theorem. Assume that composite system ( 9 ) satisfies the following conditions. (i) (13,t E T) is a Wiener process and in general the interconnecting structure disturbances are nonzero, i.e., oij(wi, t ) 0,i, j = 1, ..., I ; (ii) each isolated subsystem possesses Property A ; (iii) given the Lyapunov functions ui and comparison functions Gi3, i = 1, ..., I, of hypothesis (ii), there exist real constants bij, i, j = 1, ..., I, such that

+

(x)

I

1

gi(x, t)Tvwivi(wi, t ) I [ I C / i 3 ( ( ~ ~ 1 ) ] ~ buC$j3(Iw'1)1112 /~ for all x E Rn and t E T;

j= 1

Iv

118

(iv) for each v i r i

STOCHASTIC DIFFERENTIAL EQUATIONS

= 1, ...,1, there

is a positive constant ei such that

(ui)TVwiwiUi(Wi,t ) u i

I eilui12

holds uniformly in w i and t E T for all ui E R"' ; (v) for each oC(ivj, t), i, j = 1, ..., I, i # . j , there exists a constant dg 2 0 such that IIay(wj,t)IIi dij$j3(IwiI) for all w Y ' E R " ~ , j =..., l , /;and (vi) there exist positive constants ai, i = 1, .. .,I, such that the test matrix S = [ S J , i, j = 1, ..., I, defined by

4

s..=

I

+3 1

ui(oi+bii)

k=l,k#i

ake,dki,

f[aibij+ajbji],

i =j

i#j

is negative definite, where ci is given in hypothesis (ii). Then the trivial solution of composite system (9) is ASL w.p.1. ProoJ: As in the proof of Theorem 4.4.1 we choose for system (Y) the Lyapunov function I

v(x,t) =

1 aiui(zi,t). i= 1

Since each subsystem (q) possesses Property A, it follows again that u(x, t ) is positive definite, decrescent, and radially unbounded for all X E R" and t E T. Thus, there exist $,, $ 2 E K R such that $l(lXl)

5

U(X,t)

5 $z(lxl)

for all x E R" and t E T. Using the linearity of the operator 9, using hypotheses (ii)-(vi), and invoking Eq. (4.3.7), we obtain for system (Y), 1

2U(X,t)

=

1a i 9 z ~ i ( x , t )

i= 1

4.4

119

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

+ C

i, j = I: i # j

tccieid,,.{[$j3(1~j1)]1/2}2 = uTsu

where uT = ([I+b13(\w11)]1’z, ...,[$,3(1w11)]1/2) and S is the test matrix given in hypothesis (vi). Since S is negative definite, it follows that l,(S) < 0 and ~ Z I ( Xt ), I uTSu is negative definite. Thus, there exists a function t+h3 E K such that yu(X,t) I

for all x

E

-IC/3(IXI)

R” and t E T. This completes the proof.

4.4.3. Theorem. If in Theorems 4.4.1 and 4.4.2 each isolated subsystem (q) possesses Property B, then the trivial solution of composite system (9’) is ESL w.p.1 and ESL q.m.

(z)

Proof. Since each subsystem possesses Property B it is easily shown that the comparison functions $1, t,h2, and I+b3 in the proofs of Theorems 4.4.1 and 4.4.2 can be chosen as =

min(ccicil)r2, 1

~+k~(r> = max(orici2)r2,

I+b3(r) =

i

1,(S)u2.

The conclusion of the theorem follows now from Theorem 4.2.10.

In our next result we consider the autonomous version of composite system

(9) with decomposition (Xi). We refer to the autonomous versions of (Y),

(x)

(x’),

(Xi), and as (Y’),(Xi’), and respectively. For each isolated subsystem (Yi’)we choose a quadratic Lyapunov function Vi(Wi) = (W’)’PiWi

(4.4.4)

where wi E Rni and Pi= P: is a positive definite (nix ni) matrix. 4.4.5. Theorem. Assume that for composite system (9”)the following conditions hold.

(i) {&, t E T } is either a Wiener process or a Poisson process and in general the interconnecting structure disturbances are nonzero, i.e., o,,.(wj) $ 0 , i, j = 1, ...,I ; possesses Property B with vi(wi)specified (ii) each isolated subsystem (q’) by Eq. (4.4.4);

120

Iv

STOCHASTIC DIFFERENTIAL EQUATIONS

(iii) given the matrices Pi of hypothesis (ii), there exist real constants b,,

i, j = 1, ...,1, such that

gi(X)TPiw i 5 for all x E R”; (iv) for each a,(wj), such that

31w iI

1

j = 1

b,J wil

i, j = 1, ..., I, i # j , there exists a constant d, 2 0

5

I.,(w’)Il:

d,lWjlZ

for all w j R”j, ~ j = 1, ..., I ; and (v) there exist positive constants cli, i = 1, ..., I, such that the test matrix S = [s,], i , j = 1, ..., I , defined by

is negative definite.

[+(ai 6 , + cc

hj i ) ,

i # j

Then the trivial solution o f composite system (9”)is ESL w.p.1 and ESL q.m. Proof. The proof follows directly from Theorems 4.4.2 and 4.4.3 and from the fact that Y = 9, given the quadratic functions u i . Note that in Theorems 4.4.2, 4.4.3, and 4.4.5, the interconnection disturbance terms dki, k , i = 1 , . . ., I, k # i, which express the magnitude of the disturbances, occur in the diagonal of the test matrix S . Their effect on the negative definiteness o f S is to make more restrictive the conditions on the remaining parameters of this matrix. Thus, the disturbance terms have in general a degrading influence on the stability properties of composite system (9). It is also interesting to note that the test matrices of Theorems 4.4.2, 4.4.3, and 4.4.5 have a similar structure as the test matrix of Theorem 3.3.14. Note that the stabilization procedure indicated in Chapter I1 is also applicable in the present case. The presence of additional terms in the diagonal of the test matrices S in Theorems 4.4.2, 4.4.3, and 4.4.5 prevents us from eliminating the weighting vector cc by the methods of Chapter 11. However, as can be seen from our next result, it is still possible to use M-matrix results to establish stability conditions for system (9’)which involve nonsymmetric test matrices without the presence of weighting factors. 4.4.6. Theorem. Assume that for composite system (9’)the following condi tions hold.

4.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

121

(i) {t,,t E T } is a Wiener process and in general the interconnecting disturbances are nonzero, i.e., o U ( w j , t ) f 0, i, j = 1, ..., I ; (ii) each isolated subsystem (8) possesses Property A ; (iii) given the Lyapunov functions ui and the comparison functions i+bi3, i = I , ..., /, of hypothesis (ii), there exist real constants aij,i, j = 1, ..., /, such that (a) a,j 2 0 for all i # j and (b) Vwiui(wi,t)Tgi(~, t)I Ci=,aiji+bj3(lwil)for all x E R" and t E T ; (iv) for each ui, i = 1, ..., I, there is a positive constant ei such that

(ui)TVWiW, ui(wi,t ) u ' Ieilui12 holds uniformly in wi and t E T for all ui E R"'; (v) for each o i j ( w i ,t), i, j = 1, ..., I , i # . j , there exists a constant dij2 0 such that

II 0" (

~ j1 ,

II i I du i+b j 3 ( I ~j I

for all w j E R"J,j = 1, ...,I, and t E T ; and (vi) the test matrix S = [sij]defined by

4

s..=

- (ai + aid,

-a,.v - 12,i d

i = j ~3

i # j

has positive successive principal minors. Then the trivial solution of composite system (9') is ASL w.p.1. Proof. Given the Lyapunov functions ui of hypothesis (ii), let cxT = ...,c(,) > 0 be an arbitrary constant vector and choose for system (9) the Lyapunov function (a1,

u(x,t) =

1

1 aiui(wi,t). i= 1

Since each subsystem (8) possesses Property A it follows that u ( x , t ) is positive definite, decrescent, and radially unbounded for all x E R" and t E T. As in the proof of Theorem 4.4.2, we have

i= 1

Yiui(wi,r)+gi(x,t)TVwiui(~i,t)

+-12 1'

j=i,i+j

tr[aU(w~,t)TVwiwlui(~i,t)aij(~~,t

122

Iv

STOCHASTIC DIFFERENTIAL EQUATIONS

Usage of hypotheses (iii)-(v) yields

Letting uT = ($, ( 1 w 1I), . . ., I / J ~(1 ~w'l)), defining S = [sJ as in hypothesis (vi), definingj.T = mTS,and using the identical argument as in the proof of Theorem 2.5.15, we can readily show that Y u ( . x , t ) 5 -yTU

< 0,

x # 0,

i.e., Y L > (t X ) is, negative definite for all x E R" and t E T. This completes the proof. There are several differences between the test matrices of Theorems 4.4.2, 4.4.3, 4.4.5, and the test matrix of Theorem 4.4.6. I n the case of the former, S is symmetric, there are no sign restrictions on the off-diagonal terms, S involves the weighting vector a,and the disturbance terms dg enter into the diagonal terms sii.In the case of the latter, S is in general not symmetric, the off-diagonal terms have to be nonpositive, S does not involve any weighting factors, and the disturbance terms enter into the off-diagonal terms sg, i Z , j . On the other hand, in d l results of this section, the disturbances for the isolated subsystems (8) are reflected in the diagonal elements of the test matrix S . Finally, it is important to note that if og(wj,t)= 0 for all i , j = I , ..., I, then (,Yp)becomes an ordinary differential equation. In that case, all results of the present section reduce t o corresponding results established for deterministic systems described by ordinary differential equations which we treated in Chapter I I . 4.5 Analysis by Vector Lyapunov Functions

In the present section we concern ourselves once more with composite with decomposition (Z,). T o simplify matters, we consider only system (9') the case when {<, ,t E T ) is a Wiener process. In contrast t o the approach of

4.5

ANALYSIS BY VECTOR LYAPUNOV FUNCTIONS

123

Section 4.4, where we utilized scalar Lyapunov functions, the results of the present section depend on the existence of vector Lyapunov functions constructed from scalar Lyapunov functions for the isolated subsystems (8). Henceforth we assume that all such functions are in the domain of their respective infinitesimal generator Yi. 4.5.1. Definition. Isolated subsystem (8) is said to possess Property E if there exists a scalar function ui: Rnix T - R in the domain of gi, a function $iE KR, and a continuous scalar function ai : R x T + R such that and

ui(wi,t) 2 $i(lwil),

ui(O,t)

= 0,

Yi ui(wi,t ) 5 ai(ui(wi,r ) , r ) for all w iE R"' and t E T and if luil is decrescent. If isolated subsystem (Yi) possesses Property E and if there exists a function K such that ai(ui(w',t ) , t ) I -cpi(lwiI) for all w' E RnLand t E T, then the trivial solution of (4)is ASL w.p. 1. If possesses Property E and if for some ci> 0, ai(ui,t ) I -ciui, then the trivial solution of is ESL w.p.1 (see Kushner [I] and Arnold [l]). In our first result, which is a comparison theorem, we use functions of class K'defined in Section 4.1. This result involves vector Lyapunov functions of the form u(x, t)' = ( u l (w', t ) , ..., u,(w', t ) ) . 'piE

(x)

(x)

4.5.2. Theorem. Assume that for composite system (9'the ) following conditions hold.

possesses Property E; (i) Each isolated subsystem (3) for each i = I , ..., I, there exists a function b i : R' x T + R, bi E K', (ii) such that

< bi(u(x, t ) ,t )

(4.5.3)

where u ( x , t)T = ( u l (d, r ) , ..., t)); let a ( . ) ' = ( a , ( . ) , ..., a , ( . ) ) and b(.)' = ( b , ( . ) , ..., b,(.)) where a i ( . ) (iii) and b i ( . ) , i = 1, ..., I , are defined in hypotheses (i) and (ii), and assume that the vector function [a( .) + b ( .)I satisfies the inequalities u,(wl',

la(u,r)+b(u,t)12

I k ( l + lu12)

and la(u,t)+b(u,t)-a(w,t)-b(w,t)l

I k Iu-wI

for all u, u' E R', u 2 0, w 2 0, for all r E T, and some k > 0;

124 (jv)

lv

STOCHASTIC DIFFERENTIAL EQUATIONS

there exists a function p : R' x T + R such that

p ( y ,t ) 2 and

c IVWi I

sup

Ui(Wi, t ) T o i j ( w j ,

v(x,r)
t)y

P(Y, t ) 5 4 1 + lvl".

(4.5.4)

(4.5.5)

Then v(x,, t ) I y , with probability one for all t E T, where yt is a solution of the vector stochastic differential equation

4, = Cdv,, t ) + b ( ~ , , t ) ldt + Y, dz, with y o 2 u(x,, 0) and with

(4.5.6)

a matrix-valued random process such that

Ilrtll:

(4.5.7)

5 P(A9 t).

Proof. C h o o s e v ( ~ , t = ) ~( u , ( w ' , t ) , ..., v,(w',t))asavector Lyapunovfuncwith vi(wi,t ) , i = 1, ...,I, given in hypothesis tion for composite system (9) (i). By Ito's lemma (see Wong [I]) we have dv(x,, t)T

where

= (du, ( w l , t ) , ...,du,(w',t))

d ~ i ( w , ' t, ) = ~'u;(w,~, t ) dt =

+ V,V~(W,',

t)'g(X,,

{V, v i ( w t i , t ) + V, ui(wvti>tIT [ A x , , t ) +g(xt t)I

++tr[a(x,,t)TV,,ui(w,i,

+ V,ui(w:,t)Ta(x,,t) =

t ) dz, 9

t>o(x,,t)]} dt

dz,

{V, vi(w:, t )+ vwivi(w:, tIT [.A(w,i, t ) + S i ( X , , t ) l

+-21 C' tr[og(wi, t)TVwiw
1

+ Vwiui(wti,t)T C1 ai(w;,

t ) dz,'

j=

+-21 C'

j=l,i+j

where

tr[aij(w,',t)TVwiwiui(W;,t)ag(wtj, i)]} dt

4.5

ANALYSIS BY VECTOR LYAPUNOV FUNCTIONS

Letting a( .)' = (al (.), ..., a'( .)), b( .)' the above inequality assumes the form du(xr,t)

= ( b , (.), ...,b,( .)),

125

and Y, = [y!],

< [ a ( u ( ~ ~ , t ) , t ) + b ( u ( x , , t )dt, t )+] Y,dz,.

Given y,, a solution of Eq. (4.5.6), it follows that

dui(wti,t ) - dyti < {[ai(ui(w,i,t ) , t)--a,(y,', t ) ] + [bi(u(x,,t ) , t)-b,(y,, t ) ] } dt. Now since by assumption composite system (9') satisfies the conditions of ) sample Theorem 4.2.2 and since u ( - ) is continuous, it follows that v ( x , , ~ is continuous with probability one. Likewise, it follows from hypotheses (iii) and (iv) and from Theorem 4.2.2 that y , is sample continuous with probability one. A sufficient condition therefore to guarantee that u(x,: t ) I y, with probability one for all t E T, given u(x,,O) I y o , is that dui(w,',t)-dy,' < 0 whenever ui(w;, t ) = yIi.But since bi E K', then this condition is satisfied and therefore u ( x , , t ) Iy , with probability one for all t E T. Equation (4.5.7) follows immediately since

s sup

u(x,t)
c I

i, j = 1

lVWiVi(Wi,t)'a&j,

t)12 I p ( y , , t ) .

This completes the proof. 4.5.8. Remark. Hypothesis (i) and the conclusion of Theorem 4.5.2 imply that Jy,l2Iu(x,, t ) l > $ i ( ~ w ~ ~ ) 2 ] 12 ' 2 $(Ix,l), where the existence of a function $ E K R follows from the continuity of the $i,i = 1, ...,I, and the norm 1 . I (refer to Hahn [2, pp. 97-99]). Therefore, since (x,I 5 $-'(ly,l), and since $ - I E KR, the stability of composite system (9')is implied by the stability of comparison system (4.5.6).

[x:=,

Inequality (4.5.7) is useful in determining the stability of Eq. (4.5.6) and this fact is used to obtain the following result. 4.5.9. Theorem. Assume that for comparison system (4.5.6) the following conditions hold. (i) The hypotheses of Theorem 4.5.2 are true; and (ii) there exists a continuous function u : R' x T + R in the domain of -E", (the 9 operator with respect to (4.5.6)), a function h: R'x T-. R and two functions $, E KR and $2 E K such that for all t E T, q E R', and y 2 0, 4'CVyy4Y, ?)I4 5 h(Y9 t ) 1412,

W , t ) 2 $l(lYl),

u(O,t> = 0,

(4.5.10) (4.5.1 1)

Iv

126

STOCHASTIC DIFFERENTIAL EQUATIONS

and such that )uI is decrescent and v,u(y,t)

+ v,u(v,t)TCa(y,t)+b(y,t)l

+ 4h(Y,t)P(Y,t)5 -$2(lYl).

(4.5.12)

Then comparison system (4.5.6), and hence composite system (9), is ASL w.p.1. Proof. With u ( y , t ) as given in the theorem, u is positive definite, decrescent, and radially unbounded for all y 2 0, and

Y,u(y,t)

= v,u(y,t)

+ v,u(y,t)TCa(y,t)+b(Y,t)l + 3trCY,TVyyu(Y,t)Ytl

I v,u(y,t)

+ v,u(y,t)TCa(Y,t)+b(y,t)l + th(Y,t)llY,ll2

5

v, u ( y , t ) + V,U(Y, f>TCa(Y,l)+bcY, t)l + 3h(Y, t ) P ( Y , t )

5

-$2(IYI).

Thus, g y u ( y ,t ) is negative definite for all y 2 0. Since y , 2 u(x,,t ) 2 0, as a consequence of hypothesis (i), it follows that stability conditions given for y 2 0 imply the stability o f y = 0 for all solutions such that yo 2 0. Therefore the above conditions on u and Z , u imply that Eq. (4.5.6) with the specified initial condition is ASL w.p.1 (refer to Kushner [ I ] and Arnold [l]) and hence as a consequence of Remark 4.5.8, the equilibrium of composite system (9) is also ASL w.p.1. This follows because fJ

(

sup lxt12&) I P(

OSt<W

SUP I Y t l 2 W )

OSt-=aO

for some $ E K . The remaining relationships are shown similarly, completing the proof.

4.5.13. Corollary. If in Theorem 4.5.9, t,b2(lyl) is replaced by cu(y,t) for some constant c > 0, then comparison system (4.5.6), and hence composite system (Y),is ESL w.p.1. Proof. The corollary strengthens the condition on Z,,u ( y , t ) to

Z,u(y,t) I -cu(y,t),

c

>0

resulting in ESLw.p.1 for system (4.5.6) and system (9’). Since the conditions on u( .) in the above results need to be satisfied only for y 2 0, it is possible to choose u in a particularly simple form and thereby simplify the hypotheses. The following result takes advantage of this fact.

4.5.14. Theorem. Assume that for comparison system (4.5.6) the following conditions hold. (i) The hypotheses of Theorem 4.5.2 are satisfied;

4.6

127

EXAMPLES

(ii) there exists an a > 0, a E R', and a function Y 2 0, ~ T C a ( Y , t ) + b ( Y01 , 5 -Il/z(lYl>.

E

K such that for all

(4.5.15)

is ASL Then comparison system (4.5.6), and hence composite system (Y), w.p.1.

4.5.16. Corollary. If in Theorem 4.5.14 t,b2(lyI) is replaced by ca'y for some constant c > 0 then the conclusion of Corollary 4.5.13 holds. The proofs of Theorem 4.5.14 and Corollary 4.5.16 are identical to the proofs of Theorem 4.5.9 and Corollary 4.5.13, respectively, with the exception that u ( .) is required to be of the form u ( y , t ) = a'y for some a > 0. This yields 9 , 4 Y , t ) = a'Ca(Y,t>+b(Y,t)l

from which the respective conclusions follow.

4.5.17. Remark. When the functions a ( . ) and b ( . ) in Theorem 4.5.2 assume special forms, then inequality (4.5.15) may be expressed in terms of conditions on some test matrix S. For example, if there is a wT = ('pl (I y' I), .. .,'p, (I y'l)) for some 'pi E K , i = I , ..., I, and a constant matrix S such that a T [ a ( y , t ) + b ( y , t ) ]I -aTSw

(4.5. 18)

then condition (4.5.15) may be replaced with the condition that S have positive successive principal minors and s, 2 0, i # j . Alternatively, if there is a wT = ( q 1(I $I), ..., q,(l $I)), 'pi E K , i = 1, ..., I, and constants a,, i, j = 1, ...,I, such that (4.5.19) and if we let S = [s,],

s, = aia,,

i, j

=

I , ..., 1, then we have

aTCa(y,t ) + b ( y , t ) ] I W T S W .

(4.5.20)

Condition (4.5.15) may then be replaced with the condition that S+STbe negative definite. The dependence of S on a in this case is useful since it allows the manipulation of S to improve results. Clearly, the stability conditions involving inequalities (4.5.18) and (4.5.20) are very similar to corresponding stability conditions obtained in Section 4.4. However, the results of the present section and those of Section 4.4 do not appear to be completely equivalent.

4.6 Examples To demonstrate applications of the preceding results, we consider some specific examples.

Iv

128

STOCHASTIC DIFFERENTIAL EQUATIONS

4.6.1. Example. Consider the indirect control problem

+ bf(wt2)dt + dz,' + dWt2 = [ - p ~ , ~ - r f ( w ~ ~dt) + ] aTw,'dt + dw,'

=

Aw,'dt

~11(~,')

dz;

(T~~(W,')

b21(wt') dz,'

+

622(wt2)

dz,' (4.6.2)

where w,' E R"', A is a stable n , x n , matrix, b E R"', r > 0 is a constant, a E R"', {z,', t E T } is an mi-dimensional normalized Wiener process with independent components, a&:R"' -+ R"i'mj,and the function f:R 4 R is assumed to have the following properties: (i)f(w') is continuous for all w' E R , (ii) f ( w ' ) = 0 if and only if w ' = 0, and (iii) 0 < w 2 f ( w 2 )< klw'l' for all w2 E R where k > 0 is a constant. Assume that for each aij(d) there is a constant d.,. > 0 such that \lo.,.(wi)/I 21 dijlwj12. System (4.6.2), which is clearly a special case of composite system (SP), may be viewed as a nonlinear interconnection under disturbances of two isolated subsystems

dw,'

= Aw,ldt

+ 0'

1

(w,') dz,'

dt

d ~ l= , ~[-pw,'-tf(w,')]

(91

+ u 2 2 ( w t 2 )dz,'

1

(92)

with interconnecting structure under disturbances specified by

+ a12(wt2) dzt2 2 bf(w,') dt + a12(w,') dzt2 g2(xt)dt + azl(w,') dz,' 2 aTw,'dt + uZ1(w,') dz,', gl(xt)dt

where the notation of Eqs. (4.3.1) and (4.3.2) has been used. Since A is stable, there exists a positive definite matrix P such that the matrix A T P + P A = - Q is negative definite. Choosing ul (w') = ( w ' ) T P w l , we have for (Yl), using the notation of Theorem 4.4.2, el = 2A,(P),

A,"(P)Iw112I u l ( w l ) I A,(P)Jw')', and LYU,(W')

+ tr[all(wl)TPa,l(w')] (PI di 111 w1 I '-

=

(w')~[A~P+PA w1]

5

C - 1, ( Q )+ 1 ,

Choosing u 2 ( w 2 )= lw2I2, we have for (SP'),

e , = 2, and

+

2 u 2 ( w 2 ) = -2plw21' - 2rw2f(w2) [022(w2)12I (-2p+d22)1w212.

Thus, isolated subsystems (9' and ) (9') both possess Property B, with $ i i ( r ) = A m ( f ' ) r 2 > $iz(r)=J-,w(P)r', $ i 3 ( r ) = r 2 , 6 1 =-L(€?)+A,(P>dii, i,hzl(r) = $ 2 2 ( r ) = $ 2 3 ( r )= r 2 , and o2 = -2p+d2,. For the interconnections we have

V , , , ~ V , ( W J ) ~ ~=~ (2X( )~ ' ) ~ P b f ( wI ' ) 2A,(P)k(bl lw'l

1~')

4.6

and

129

EXAMPLES

Vw2u2(w2)Tg2(x)= 2w 2aTw 1 I21al l w l l

Iw2I.

In the notation of Theorem 4.4.2 we now have b , , = b,, = 0, b , , = = 21~1. Choosing a1 = l/A,(P) and a2 = 1, matrix S of either Theorem 4.4.2, 4.4.3, or 4.4.5 assumes the form

2klblA,(P), and b,,

s=

[

+ d i i + d21 k 161 + 14

(-L(Q)/n,(P))

-2P

+ d22 + dl2

This matrix is negative definite if and only if P

and

'

(4.6.3)

3(42+d22)

k < ( l/lbI){ [ ( A m (Q)/AM (P))- d1I - d2 1 I ' I 2 C

~P d~2 - d221' I 2 - .I I}.

(4.6.4)

Therefore, if inequalities (4.6.3) and (4.6.4) are true, then composite system (4.6.2) is ESL w.p.1 and ESL q.m. If in Eq. (4.6.2) the Wiener processes { z t i ,t E T } , i = 1,2, are replaced by Poisson processes {qti,t E T } , i = 1,2, an identical analysis as the one given above yields again, in view of Theorem 4.4.5, the inequalities (4.6.3) and (4.6.4) as conditions for ESL w.p. 1 and ESL q.m. 4.6.5. Example. To illustrate the applicability of the results of Section 4.5, we reconsider the indirect control problem of Example 4.6.1. Choosing the and (9; as) before, we have same Lyapunov functions u1 and u2 for (9,) 9 1 u1 ( w ' )

I

C-nm(Q)

+ AM (PI d, 11I w1 I

5

[-(Am(Q)/AM

( P I )+ d11Iu1 (w')

and Y 2 u 2 ( w 2 ) 5 (-2p+d22)/w2\2 = ( - 2 p + d , , ) u 2 ( w 2 ) .

Thus, each isolated subsystem (yi) possesses Property E, and hence, hypothesis (i) of Theorem 4.5.2 is satisfied. Also,

vwl01 (w1lTg,(4+ 3 tr [a12 (w2)T V d =

w1

( w ' ) 0 12 (w"1

2 ( ~ ' > ~ P b f ( w+' )3 tr[2a12(w2)TPa12(w2)]

I 2A,(P) Ibl k Iw'

I I w21 + A,(P)d12 l w 2 I 2

IA,(P)1blk(lw'12+1w212)

5 IIn,(P)IbIk/Am(P)Iul(w')

+ A,(P)d121w212 + A,(P)(IbIk+d12)u2(w2).

(4.6.6)

Iv

130

STOCHASTIC DIFFERENTIAL EQUATIONS

Furthermore vw2 02 (w2ITg2 ( X I

=

+ t tr [a2 1 (w')'

( w 2 )6 2 1 (wl)l

V W ~ u2 W ~

2w2aTw' + ~ t r [ 2 a , , ( ~ ' ) ~ o , ~ ( w 'I ) ]2 1 a l ) ~ ~ 1 I ~ ~ I + d ~ ~ 1 ~ ~

I la1(1Mq2 + l w 2 I 2 ) +d2,1w112

I [(I4 + d21)/~,l(P)Iul(wl)+ I4 u 2 ( w 2 ) .

(4.6.7)

Since each coefficient in inequalities (4.6.6) and (4.6.7) is positive, hypothesis (ii) of Theorem 4.5.2 is satisfied. In addition, the linearity of the bounds in (4.6.6) and (4.6.7) ensures that hypothesis (iii) of Theorem 4.5.2 holds as well. Finally we have

+ 1 2(w1)TPo, (w"1 + 12w2cT21(w1)12+ 12w2022(w2)121

sup [I2(w')TPa V
11

(w') 12

2

I 4sup{Cn,(P)I2d,, l~114+C~'M(P)12d,21wL121w2~2 U
+d2,1

A

iw2i4}

~ ~ 1 1 2 1 ~ 2 1 2 + d ~ ~

P(Y17

and thus, hypothesis (iv) of Theorem 4.5.2 is also satisfied. Letting uT = ( l/AM( P ) , 1) in Theorem 4.5.14, we have uTC 4 Y >+ b(Y)l =

{(~~~(P)/~M(P))C-(A~(Q)/AM(P))+~III + IbIk + IQI + d21} x (yl/Am(P))

+ c - 2 ~+ d22 + k IbI +21'

+ lull y 2

(4.6.8)

which satisfies inequality (4.5.15) for y 2 0 if and only if

It now follows from Theorem 4.5.14 that system (4.6.2) is ASL w.p.1 if inequality (4.6.9) is true. In fact, since (4.6.8) is a linear function ofy, application

4.6

131

EXAMPLES

of Corollary 4.5.16 yields ESL w.p.1 for system (4.6.2) when condition (4.6.9) is satisfied. Although stability conditions (4.6.3)-(4.6.4) and (4.6.9) are similar, note that these conditions are not equivalent. 4.6.10. Example. Consider the system

1, = [ A (X,)+N(t)]X I ,

xo

= x,

(4.6.1 1)

where t E T, x, E R', A ( x ) is an I x I array of continuous bounded scalar functions as(x), i , j = 1, ..., I, and N ( t ) is a random matrix of independent wide-band, zero mean, Gaussian random processes 5,n,(t), i, j = I , . ..,I. System (4.6.1 I), which may be considered as a nonlinear system with random parameters, can be used to represent a very large class of physical problems. For example, the transistor circuit model considered in Example 2.8.41 is a special case of the deterministic version of Eq. (4.6.1 I). Letting N ( t ) approach a white noise matrix, Eq. (4.6.1 I ) may be transformed into an equivalent Ito differential equation as follows. Let a(x) be the I x l2 array of 1 x I submatrices oc(xj),where a , ( x j ) = ( O ,..., O , ~ , x j ,,..., o O),

i , j = 1,..., I

(4.6.12)

(the nonzero term occurring in the ith position), and let v ( t ) be the l2 x 1 vector given by

...,

V ( t I T = Cnll (I) n2, , (I), 4,(t>,n12(t>,. - . , n n ( t ) l .

Then Eq. (4.6.11) can be rewritten equivalently as f, = A(x,)x,

+ o(x,)v(t).

(4.6.13)

Following rules of transformation (see Wong [1, p. I62]), Eq. (4.6.13) can be replaced by the Ito differential equation dx,'

=

"

.]

C U ~ ( X , ) X , ' + + ~ ~ X , dt +

j= 1

1

J=1

oc(x,'>dz,',

(4.6.14)

i = 1, ...,I, where zi E R', j = 1, ..., I ({z),? E T } is a normalized Wiener process). In what follows we use the notation of Theorem 4.4.2. From Eq. (4.6.12) we obtain IIoii(xj)llm 2 - a# -2 IxjI2,

i, j = 1, ..., I,

and thus we have d, = Oi, provided that +hj3( r ) = r 2 , Choosing the isolated subsystems as

(x.)

dx,'

=

+aix,'dt

+ oii(x,')dz,'

Iv

I32

STOCHASTIC DIFFERENTIAL EQUATIONS

(x)

and choosing for the Lyapunov function v i ( x i )= 4Ixi12, we have (in the notation of Theorem 4.4.2) e, = 1 and P i v i ( x ' )= + ~ i l x+~4-ltr[aii(xi)Taii(xi)l ~ 5 5;Ixilz, i = I , .. .,1. Therefore, each subsystem

In addition, since

gi(xy V,c uj (xi> =

(x)possesses Property B with

ai = 5;.

9aij(x) xjxi

J=

1

we have (in the notation of Theorem 4.4.2)

bii = supaii(x), R'

i , j = 1 , ..., 1. The test matrix S is now given by

b,

=

= [sG]of

suplaij(x)l, R'

i #j,

Theorem 4.4.2 (and Theorem 4.4.3)

Thus, if matrix S is negative definite for some choice of clT = (al, ...,a,) > 0, then the trivial solution of Eq. (4.6.1 1) is ESL w.p. I and ESL q.m. by Theorem 4.4.3. If in particular ifu = 0, i #.j, i , j = 1, ..., 1 (i.e., only the diagonal elements of A ( x ) are subject to disturbances), then Theorem 4.4.1 may be applied. In fact, since supRlla,(x)l 2 0, the M-matrix results of Chapter I1 are applicable as well. In analogy with Theorem 2.5.1 I , we obtain from Theorem 4.4.1 the stability condition

i = 1, ..., I , for some constants Ai > 0, i = 1, ..., 1. In the present example, as well as in the preceding ones, the weak coupling conditions and the degradation of stability due to noise are clearly apparent. 4.7

Notes and References

A standard reference on probability theory and stochastic processes is the text by Doob [I]. For an exposition of the Ito calculus, refer to the book by Wong [I]. The primary sources for Section 4.2, dealing with the stability of

4.7

NOTES AND REFERENCES

133

stochastic differential equations are the books by Kushner [I] and Arnold [l]. Additional references on stability of stochastic differential equations (and stochastic difference equations) include Kushner [2], Kats and Krasovskii [I), Bertram and Sarachik [I], and Kozin [i]. The results of Section 4.4 are based on work by Michel [S, 10, 111 and Michel and Rasmussen [l-31. Section 4.5 is based on papers by Rasmussen and Michel [ 1, 31. The examples of Section 4.6 were considered by Michel and Rasmussen [3], Rasmussen and Michel [33, and Rasmussen [I]. Using the approach presented in this chapter, it is also possible to establish stability results for discrete parameter stochastic systems (see Michel [lo]) and for composite systems endowed with independent increment Markov processes other than Wiener processes (see Michel and Rasmussen [I, 31). In addition, it is possible to establish results similar to the present ones for other stability types not considered herein.

CHAPTER V

Infinite-Dimensional Systems

In the present chapter we extend the method of analysis advanced thus far to deterministic large scale dynamical systems described on Hilbert and Banach spaces. The motivation here is primarily systems represented by partial differential equations, although the method of analysis also applies to functional differential equations and includes many of the earlier results for ordinary differential equations. In addition, the results of this chapter can be used to analyze hybrid dynamical systems (i.e., systems described by a mixture of equations) in a systematic fashion. There are numerous difficulties that may be encountered when working in a setting of infinite-dimensional abstract spaces. For this reason we first present pertinent background material from the theory of linear and nonlinear semigroups Then we present several important examples of semigroups. In particular, we discuss at some length how nonlinear partial differential equations can be used to construct semigroups in Hilbert space. Finally, we address ourselves to the subject on hand, that of analyzing large scale systems described on Hilbert and Banach spaces. A brief overview of the present chapter is as follows. In the first section we establish the notation required throughout this chapter. The second section I34

5.1

NOTATION

135

contains selected pertinent results from the theory of linear semigroups while the third section deals with required notions from the theory of nonlinear semigroups. In the fourth section we present stability definitions for semigroups, Lyapunov theorems, and selected special results for determining stability of linear systems. (Some extensions of the Lyapunov theory, which constitute background material, are also considered in the ninth section.) The fifth section contains several examples of how one constructs linear and nonlinear semigroups. In the sixth section we develop the main stability, instability, and boundedness results for large scale systems (which as before, we also call composite or interconnected systems) defined on infinitedimensional spaces. To demonstrate the usefulness of the method of analysis advanced and to illustrate applications of individual results, we consider specific examples in the seventh section. These include an example of a hybrid dynamical system which may be viewed as an interconnection of two subsystems, one of which is described by an ordinary differential equation while the second subsystem is represented by a partial differential equation. Also, in this section we consider the point kinetics model of a coupled core nuclear reactor. In the eighth section we present special stability results for interconnected dynamical systems described by functional differential equations while in the ninth section we apply comparison theorems to vector Lyapunov functions in the stability analysis of composite systems. We conclude the chapter in the tenth section with a discussion of the literature cited. 5.1

Notation

As in the preceding chapters R = (-m, co), R + = [0, co), and R" denotes n-dimensional Euclidean space with norm 1 . I. Banach spaces are denoted by X or Z , with appropriate subscripts if necessary, and norms on Banach spaces are denoted by I/. Ii, with appropriate subscripts as required. Also, Hilbert spaces are denoted by X,Z , or H , with inner product ( . , .). In this case the norm of x E X is given by llxl]= (x, x ) l i Z . Let A be a linear operator defined on a domain D(A) c X with range in Z , i.e., A : D(A) + Z , and assume that D(A) is a dense linear subspace of X. We call A closed if its graph Gr(A) = {(x, A x ) E X x Z : x E D(A)} is a closed subset of X x Z . Also, we call A bounded if it maps each bounded set in X into a bounded subset of Z , or equivalently, if it is continuous. Subsequently, I : X + X will always denote the identity transformation. Given a closed linear operator A : D(A) X , D(A) c X,the resolvent set p ( A ) consists of all points A in the complex plane such that the linear transformation ( A -A/) has a bounded inverse R(A,A ) ( A -A/)-' : X+ X. The complement of p ( A ) is called the spectral set or simply the spectrum and is denoted by o ( A ) .

136

v

INFINITE-DIMENSIONAL SYSTEMS

Given a bounded linear mapping A : D(A) defined by ll'4ll

=

--f

Z , D(A) c ,'A its norm is

suP{//Axll:llxll =I}.

Finally, Lp(G,U ) , 1 ~p 5 03, denotes the usual Lebesgue space of all Lebesgue measurable functions with domain G and range U . The norm in Lp(G,U ) will be denoted by 11. 1, (or 11. IIL, if more explicit notation is needed). When the range U does not need emphasis, we utilize the notation L,(G) in place of Lp(G,U ) . If in particular G = R+ and U = R", we find it convenient to simply write Lpm= L p ( R + R"), , and when m = 1, we simply write L, = L , ( R + , R). I f f : R + R", 9 : R R", and p = 2 , we note that L," is a Hilbert space with inner product --f

rm

and norm llfllL2 = ( , f , f > ' l 2 On . the other hand, ifp = co,and f E L,", f i s an essentially bounded function with norm llfllm =

IlfIlL,

=

then

ess supIf(t)l.

120

5.2 Co-Sem igroups

Consider a process whose evolution in time t can be described by a linear differential equation i(t) =

Ax(t),

x(0) = xo E D(A)

(L)

for f E R + . Here A : D(A) + Xis assumed to be a linear operator with domain D(A) dense in X . Moreover, A is always assumed to be closed or else to have an extension A which is closed. By a strong solution x ( t ) of (L) we mean a function x : R + + D(A) such that 2 ( t ) exists and is continuous on R + -+ X and such that (L) is true. The abstract initial value problem (L) is said to be well posed if for each xo E D(A) there is one and only one strong solution x ( t , xo) of (L) defined on 0 I t < co and if in addition x(t, xo) depends continuously on (I,xo) in the sense that given any N > 0 there is an M > 0 such that /Ix(t,x,)ll I M when 0 5 t 5 N a n d l/xoIj5 N . If (L) is well posed, there is an operator T defined by T ( t ) x o = x(t,xo) which is (for each fixed t ) a bounded linear mapping on D(A) to X . We call T ( t ) x o= x ( t , x o ) a trajectory of (L) for xo. Since T ( r ) is bounded, it has a continuous extension from D(A) to the larger domain X . The trajectories

5.2 C,-SEMIGROUPS

137

x ( r , x,) = T ( t )x,, for x, E X but x, 4 D(A), are called generalized solutions of

(L). The resulting family of operators { T ( t ) :t E R'} is called a Co-semigroup or a linear dynamical system on X .

5.2.1. Definition. A family T ( t ) of bounded linear operators on X to X is said to be a C,-semigroup if T(O)x = x,

T(t+s)x = T(t)T(s)x

for all t and s E R+ and if T ( t )x is continuous in ( t , x) on R f x X . Every C,-semigroup is generated by some abstract differential equation of the form (L).

5.2.2. Definition. Given any C,-semigroup T ( t ) , its infinitesimal generator is the operator defined by A x = lim t - ' [ ~ ( r ) x - x ] t+0+

where D(A) consists of all x E X for which this limit exists. We have the following theorem. 5.2.3. Theorem. If (L) is well posed and generates a C,-semigroup T ( t ) ,then the infinitesimal generator of T is A (or A if it is already closed). Conversely, if T ( t ) is a C,-semigroup and A is its infinitesimal generator, then D(A) is dense in X , A is closed, and for each x , E D(A) the function T ( t ) x , is a strong solution of (L). The next result, called the Hille--Yoshida-Phillips theorem, provides necessary and sufficient conditions for a given linear operator A to be the infinitesimal generator of some C,-semigroup.

5.2.4. Theorem. A linear operator A is the infinitesimal generator of a C,semigroup T ( t ) if and only if D(A) is dense in X , A is closed, and there exist two real numbers M > 0 and w such that whenever A > w one has A E p ( A ) and IIR(A,A)"II I M(A-w)-" for n = 1,2,3, . _._In this case 1 T ( t ) /5 / Mew'. The following result will also prove useful.

5.2.5. Theorem. If A : X -+ X generates a C,-semigroup and if B : X -+ X is a bounded linear operator, then the operator A + B also generates a C,semigroup. A Co-semigroup of contractions is a C,-semigroup T ( t ) which satisfies I I T ( t ) l l ~ (i.e., l M = 1 and w = 0 in Theorem 5.2.4). Such semigroups are of particular interest in Hilbert spaces.

138

v

INFINITE-DIMENSIONAL SYSTEMS

5.2.6. Definition. A linear operator A : D(A) + H , D(A) c H , on a Hilbert space H is called dissipative if Re(Ax, x) 5 0 for all x E D(A).

For C,-semigroups of contractions we now have the following. 5.2.7. Theorem. If A is the infinitesimal generator of a C,-semigroup of contractions on a Hilbert space H , then A is dissipative and the range of ( A - 21) is all of H for any I > 0. Conversely, if A is dissipative and if the range of ( A - U )is H for at least one constant I , > 0, then A is closed and A is the infinitesimal generator of a C,-semigroup of contractions. The above theorem is particularly useful in connection with parabolic partial differential equations.

5.3 Nonlinear S e m i g r o u p s A (nonlinear) semigroup or dynamical system is a generalization of the notion of C,-semigroup. In arriving at this generalization, the linear initial value problem (L) is replaced by the nonlinear initial value problem i(t) =

A(x(t)),

x(0)

=

xg,

(N)

where A : D(A) + Xis a nonlinear mapping. If A is continuously differentiable (or at least locally Lipschitz continuous), then the theory of existence, uniqueness, and continuation of solutions of (N) is the same as in the finitedimensional case (see Dieudonne [I, Chapter X, Section 41). If A is only continuous, then (N) need not have a solution (see DieudonnC [ I , p. 287, Problem 51). Since we wish to have a theory which includes nonlinear partial differential equations we must allow A to be only defined on a dense set D(A) and to be discontinuous. For such functions A the accretive property replaces (and generalizes) the Lipschitz property. 5.3.1. Definition. Assume that C is a subset of a Banach space X . A function T(t): C 4 C is said to be a nonlinear semigroup on C if T ( t ) x is continuous in ( t , x ) on R + x C, T ( 0 ) x = x, and T(t +s)x = T ( t ) T ( s ) xwhen t and S E R + for each fixed x in C. A function T ( t ) is called a quasicontractive semigroup if it is a nonlinear semigroup on C and if there is a number w E R such that IIT(t)x - T(t)yll< ewtIlx-ylI for all t E R+ and for all x, y E C . If w I 0, then T ( t ) is called a contraction semigroup. Note that C = X i s allowed as a special case. The mapping A in (N) will sometimes be multivalued (i.e., a relation) and i n general must be extended to be multivalued if it is to generate a quasicontractive semigroup. Thus we shall assume that A ( x ) is a subset of X and

5.3

139

NONLINEAR SEMIGROUPS

identify A with its graph Gr(A)

{ ( x , y ) : X E X and y ~ A ( x ) c} X x X .

=

In this case the domain of A , written D(A), is the set of all x A ( x ) # @, the range of A is the set Ra(A) =

u

{ A ( x ) :x

E

E

X for which

D(A)},

and the inverse of A at any pointy is defined as the set A-'(y)

=

{xEX:~EA(X)}.

Let 1 be a real or complex scalar. Then 1.4 is defined by GMx)= and A + B is defined by A(x)+B(x)

=

{hJ: Y E A (4)

{ y + z : y E A ( x ) ,ZEB(X)}.

5.3.2. Definition. A multivalued operator A is said to generate a nonlinear semigroup T ( t )on C if T ( t ) x = lim ( 1 - ( t / n ) A ) - " ( x ) n-tm

for all x in C. The infinitesimal generator A, of a semigroup T ( t )is still defined by

for all x such that the limit exists. The operator A and the infinitesimal generator A , are generally different operators.

5.3.3. Definition. A multivalued function A on X is called w-accretive if l l ( x ~ - ~ ~ l ) - ( x 2 - ~ ~22()1l-l~ w ) l l x 1 - ~ 2 I l

(5.3.4)

forall/Z20andforallxiED(A)andyiEA(xi),i=1,2.

If X is a Hilbert space, then inequality (5.3.4) reduces to ( ( w x 1 - ~ l ) - ( w x 2 - ~ 2 ) 7x l - x 2 )

2 O'

This property for the nonlinear case is analogous to ( A - wl) being dissipative in the linear symmetric case. 5.3.5. Theorem. Assume that A is w-accretive and that the range of ( I - 1 A ) 2 C = D(A), the closure of D(A), for each 1 in the interval (0,1,).

v

140

INFINITE-DIMENSIONAL SYSTEMS

Then A generates a quasicontractive semigroup T ( t )on C with llT(t)x-T(t)Yll

for all t E R+ and for all x, y

E

ewtIlx-YII

C.

In general the trajectories T ( t )x determined by the semigroup in Theorem 5.3.5 are generalized solutions of (N) which need not be differentiable. Indeed an example is discussed in Crandall and Liggett [I, Section 41 where w = 0, D(A) = X , A generates a quasicontraction T ( t )but the infinitesimal generator A , has empty domain. This means that not even one trajectory T ( t ) xis differentiable at even one time t. If the graph of A is closed, then A is always an extension of the infinitesimal generator A , . So whenever x ( t ) = T ( r ) x has a derivative, then i ( r ) must be in A ( x ( t ) ) . The situation is more reasonable in the setting of a Hilbert space H . If A is w-accretive and closed (i.e., its graph is a closed subset of H x H ) , then for any x E D ( A ) the set A (x) is closed and convex. Thus, there is an element A o ( x )E A(x) such that Ao(x) is the element of A ( x ) closest to the origin. Given a trajectory x ( t ) = T ( t ) x , the right derivative

o + x ( r ) = Iim h+O

+

x(t+h)-x(r) h

must exist at all points t E R + and be continuous except possibly at a countably infinite set of points. The derivative i ( t ) exists and is equal to D ' x ( f ) at all points where D + x ( t )is continuous. Furthermore, D+x(r) = ~ O ( x ( t ) )

for all r 2 0. These results can be generalized to any space X which is uniformly convex. (Refer to Dunford and Schwartz [l, p. 741 for the definition of a uniformly convex space. In particular, any L, space for 1 < p < 00 is uniformly convex.)

5.3.6. Definition. A trajectory x ( t ) = T ( t ) x , is called a strong solution of (N) if x ( r ) is absolutely continuous on any bounded subset of R + (so that k ( t ) exists almost everywhere), if x ( t ) E D(A) and if i ( r ) E A ( x ( f ) )almost everywhere on R+ . We also have 5.3.7. Definition. Initial value problem (N) is called well posed on C if there is a semigroup T ( t ) such that for any x, E D(A), T ( t ) x , is a strong solution of (N) and if D(A) = C . ~

Thus, if X is a Hilbert space or a uniformly convex Banach space and if A is w-accretive and closed, then initial value problem (N) is well posed on C = D(A) and i ( r ) = A o ( x ( t ) )almost everywhere on R+.

5.4 5.4

I41

LYAPUNOV STABILITY OF DYNAMICAL SYSTEMS

Lyapunov Stability of Dynamical Systems

Throughout the present section T ( t ) denotes either a linear or a nonlinear semigroup on a subset C of a Banach space X . We shall always assume that the origin 0 (the null vector of X ) is in the interior of C and that T ( t ) has the trivial solution T ( t )x = 0 for all t E R + when x = 0. 5.4.1. Definition. The trivial solution of T ( t ) is said to be stable if for any d > 0 there is a 6 > 0 such that IIT(t)xll< d for all t E R+ whenever I/x//< 6 and x E C.

In the above definition 6 = 6 (8) depends on the choice of &. In contrast to earlier stability definitions (see Chapter 11) there is no mention here of an initial time to. This is because semigroups (as defined above) correspond to time-invariant (autonomous) processes. The initial time can always be taken to be to = 0. Since stability and uniform stability are equivalent for timeinvariant systems, we will not need to give a separate definition for uniform stability. Indeed this lack of need for uniformity will occur again when results concerning boundedness are considered. 5.4.2. Definition. The trivial solution of T ( t ) is said to be asymptotically stable if it is stable and there is a 6, > 0 such that ilT(t)x/l+ O as t + co

whenever

I(x(l<6,.

5.4.3. Definition. The trivial solution of T ( t )is uniformly asymptotically stable if it is asymptotically stable and if ( \ T ( tx// ) + 0 as t co uniformly in //x/1< 6,. --f

5.4.4. Definition. The trivial solution of T ( t ) is exponentially stable if it is

stable and if in addition there are positive constants M and IIT(t) xi1 5 M e - a t for all t 2 0 and all x satisfying llxll< So.

CI

such that

5.4.5. Definition. The trivial solution of T ( t ) is unstable if it is not stable. In this case there is an & > 0, a sequence {x,} in C and a sequence { t , } such that llx,ll--f0, t , -+ co, and ~ ~ T ( r , ) 2 x ,d ~ (for all n.

When the trivial solution of T ( t ) is asymptotically stable, then the domain of attraction of the trivial solution is the set {x E C : T ( t ) x+ 0 as t --+ a}. The trivial solution of T ( t )is called asymptotically stable in the large if C = X and if the domain of attraction is all of X . It is uniformly asymptotically stable in the large if it is uniformly asymptotically stable with domain of attraction X and if for any N > 0, ( ( T ( t ) x ( ( - + as O t + co uniformly on the set ((x(( < N. It is exponentially stable in the large if it is exponentially stable with domain of attraction X and if for any N > 0 there are constants c1 and M (depending only on N ) such that IIT(t)xII IM e p a ffor all t 2 0 and all x such that ] ( X I ( < N .

142

v

INFINITE-DIMENSIONAL SYSTEMS

5.4.6. Definition. The semigroup T ( t ) is said to be uniformly bounded if given a > 0 there is a constant M such that ~ ~ T ( t<) M x ~for ~ all t E R f whenever llxll < a.

5.4.7. Definition. The semigroup T ( t ) is uniformly ultimately bounded, with bound M , if given a > 0 there exists N = N ( a ) > 0 such that /(XI] < a implies that I/T ( t )XI/ < M for all t 2 N .

In the subsequent results we will find the following convention convenient. 5.4.8. Definition. A function v is called a Lyapunov function if there is an open neighborhood U c X of the origin such that v : U 4 R is continuous and such that v ( 0 ) = 0. For such a function

is the upper right Dini derivative along trajectories. We now summarize the basic theorems of Lyapunov's direct method for semigroups defined on Banach spaces. As in the preceding chapters, we phrase these results in terms of comparison functions of class K and class K R .

5.4.9. Theorem. If v is a Lyapunov function, if cpl E K satisfies v ( x ) 2 (pl(llxll)in a neighborhood U of the origin, and if D v ( x ) 5 0 for all x E I/, then the trivial solution of T ( t )is stable. 5.4.10. Theorem. If there exists a Lyapunov function zi and three functions cpl, v2,(p3 E K such that for all x in a neighborhood U of the origin cpl(llxll) 5

44 5 v2(11xl1~9

Dvtx) 5 -v3(ll4)

then the trivial solution of T ( t )is uniformly asymptotically stable.

5.4.11. Theorem. If in Theorem 5.4.10 U = C = X and if cpl E K R , then the trivial solution of T ( t )is uniformly asymptotically stable in the large. 5.4.12. Theorem. I f in Theorem 5.4.10 cp1,'pz,'p3 E K are of the same order of magnitude, then the trivial solution of T ( t )is exponentially stable. 5.4.13. Theorem. I f in Theorem 5.4.10 U = C = X , if vl,c p 2 , ( p 3 E KR, and if (pl, ( p 2 , ( p 3 are of the same order of magnitude, then the trivial solution of T ( t )is exponentially stable in the large. 5.4.14. Theorem. Suppose v is a Lyapunov function and cp E K is a function satisfying - D u ( x ) 2 cp( lixil) on some neighborhood U c C of the origin. If in every neighborhood W c C of the origin there is at least one point xo E W for which v ( x o ) < 0, then the trivial solution of T ( t )is unstable.

5.4

LYAPUNOV STABILITY OF DYNAMICAL SYSTEMS

143

The next results yield conditions for uniform boundedness and uniform ultimate boundedness.

5.4.15. Theorem. Assume that C = X and let S = { x E X : IJxJ/ 2 R } (where R may be large). If there exists a continuous function u : S + R and two functions (pl, 40, E K R such that 40l(llXlI)

I u(x) I (P2(llxll)

for all x E S and if Du(x) I 0 for all x bounded.

E

S , then the semigroup T ( t )is uniformly

5.4.16. Theorem. Assume that the hypotheses of Theorem 5.4.15 hold and that there exists a function q3 E K such that Du(x) I - (p3 (llxll) for all x E S. Then T ( t )is uniformly ultimately bounded. The computation of Du(x) can be a significant problem in some cases. However, if the semigroup T ( t ) is a C,-semigroup or a quasicontractive semigroup on a Hilbert space or on a uniformly convex Banach space, then the infinitesimal generator A , of T ( t )exists on a set D(A,) which is dense in C. For such cases the computation of Da can be simplified.

5.4.17. Definition. A pair (u, T ) is called admissible if u is a Lyapunov function for T ( t ) ,if the infinitesimal generator A , of T ( t )is defined on a set Do c D(A,) dense in C, and if there is a function Vu defined on { U n Do} x X with values in R such that (i) u ( y ) - u ( x ) I V u ( x , y - x ) + o ( l ~ y - x ~ /for / ) all x , y ~ D n , U , and (ii) V u ( x , h )is, for each fixed x , a bounded linear operator in h E X . We have in mind using the Frechet derivative of u at x for the operator Vu(x, .) (see DieudonnC [ l , Chapter 81 for the definition and properties of such derivatives). We now state and prove a result which we require in our subsequent development.

5.4.18. Theorem. If (u, T ) is an admissible pair and if there exists a function (p3 E K such that V u ( x ,A , x ) I - - q ~ ~ ( ~ ~for . x \all ~ ) x E Do n U , then Du(x) I -(p3(llxll) for all X E U . Proof. If x

E

Do n U then

D V ( X ) = lim sup(i/t) ( u ( T ( t ) x ) - u ( x ) f 1-O+

I lim s u p ( l / t ) ( V u ( x , T ( t ) x - x ) + o ( I ! T ( t ) x - x l J ) ) f-+O+

=

Iim s u p v v ( x , ( T ( t ) x - x ) / t ) = V v ( x , A , x ) 2 -q~~(llxll).

r+o+

v

144

INFINITE-DIMENSIONALSYSTEMS

If x # Do, choose a sequence {x,) in Do such that x, each x, E Do we have by the result proved above, y(T(t)x,)

- +),

I -

+x

as n -+ co. Since

sd

4n,(llm)xnll) ds

for all t E R'. Since all of the functions involved are continuous, this inequality remains true in the limit, i.e.,

ds. @ ( t ) x ) - u(x> I -s'4n3(llT(s)xll) 0 Therefore, lim sup(l/t) {v(T(t)x)-v(x)} I lim sup(- I / t ) I+,+

f'O+

= -cp3(llxll).

l

q,(llT(s)xll) ds

Thus, Du(x) I - q ~ ~ ( ~for l x all ~ ~x )E U . W For linear semigroups with generator A one can often establish stability by determining the spectrum of A . When X = R", A must be an n x n matrix whose spectrum is the set of eigenvalues {A} of A . In this case Re2 < 0 for all I E o ( A ) is certainly sufficient for stability. Slemrod [2] has studied the analogous problem for systems defined on infinite-dimensional spaces. He points out that there are C,-semigroups of the type characterized by the following theorem. 5.4.19. Theorem. Given any two real numbers c1 and j? with c1 < there exists a C,-semigroup T ( t ) on a Hilbert space H such that R e I I c1 for all I E o ( A ) and in addition llT(t)II= 2'for all t 2 0. Slemrod [2] shows that a generalization of the stability result for the finite-dimensional case is possible for the following class of semigroups. 5.4.20. Definition. A C,-semigroup T ( t ) is called differentiable for t > r if for each x E X , T ( t ) x is continuously differentiable on r < t < co. For example, a system of functional differential equations with delay

[ - r, 01 (as discussed in the next section) determines a semigroup which is differentiable for t > r . Also, systems of parabolic partial differential equations

(as discussed in the next section) normally generate semigroups which are differentiable for t > 0. Following Slemrod [2] we have the following result.

5.4.21. Theorem. If T ( t ) is a C,-semigroup which is differentiable for t > r, if A is its generator and if Re I 5 - cq, for all I E o ( A ) , then given any positive tl < CI,, there is a constant K(c1) > 0 such that l\T(t)ll I K(a)e-"' for all t E R'.

5.5

EXAMPLES OF SEMIGROUPS

145

5.5 Examples of Semigroups

At this point it is appropriate to consider several examples of important semigroups which arise in applications and to provide related background material. Because of the great diversity of the material involved, our presentation is by necessity brief. However, we point to several references at the end of this chapter, where the proofs of the results cited in this section can be found. A. Ordinary Differential Equations

A simple example of a semigroup is the solution of an autonomous (i.e., time independent) ordinary differential equation defined on R". Thus, if g: R" --f R" is smooth enough so that the initial value problem x ( 0 ) = xo

i = g(x),

has a unique solution cp(f,x,) defined for all f E R + , then T ( t ) x o = cp(t,x,) is a semigroup on X = R". If g satisfies a Lipschitz condition \s(x)-g(Y)l

-= L l x - v l

for all x, y E R", then g is w-accretive with w = L . In this case T ( t )is a quasicontractive semigroup. B. Functional Differential Equations

Differential equations with time delays and functional differential equations can also be used to generate semigroups. We have in mind equations such as i(t) =

g ( x ( t ) ,x ( t - 4 )

or

for t > 0 with x ( t ) equal to a given initial function $, i.e., x(t) = $(t)

on

-r

I t 5 0.

Initial value problems of this type are special cases of a more general class of problems, discussed below. Let C, be the set of all continuous functions cp : [ - r, 01 + R" with norm defined by licpll = max{lcp(t)l: - r I t I O}.

v

146

INFINITE-DIMENSIONAL SYSTEMS

Given a function x ( f ) defined on --Y 5 t < t l , let x, be the function determined by xf(s)= x ( t + s ) for - - Y < s < 0. Then clearly, x, is a mapping of the interval [0, tl) into C,. A functional differential equation (with delay r ) is an equation of the form xo = $ (F) where g : C, R" and where $ is the initial condition. We denote a solution of (F) with this initial condition by q ( t , $ ) . We assume here that for (F) solutions exist, are unique, and depend continuously on ( t , $ ) (a sufficient condition for this is that g(q) be locally Lipschitz continuous), and that all solutions are defined for all t E R + . In this case T ( t )$ = q,($),or equivalently ( T ( t ) $ ) ( s )= q(t+s,$), defines a nonlinear semigroup on C,.. If g is Lipschitz continuous, then T ( t ) is a quasicontractive semigroup. For such a g define A : D ( A ) -+ C, by i ( t ) = dx,),

--f

A$ = t,h,

D(A)

=

{$ E C, : $ E C,. and $(O) =g($)}.

Then the domain D(A) is dense in C,, A is the generator (and also the infinitesimal generator) of T ( t ) ,and T ( t ) is differentiable for c > r . (For proofs of the above assertions, see Webb [I].) The most general linear functional differential equation has a right-hand side of the form d r p ) = s_sdB(.)p(.)

where B(s) = [be(s)]is an n x n matrix whose entries are of bounded variation on [ - r , 01. Such a functional is Lipschitz continuous on C, with Lipschitz constant L less or equal to the variation of B. In this case the semigroup T ( t ) is a C,-semigroup and its stability can be ascertained by determining the spectrum of the generator. This spectrum consists of all solutions of the equation (5.5.1) If all solutions of Eq. (5.5.1) satisfy the relation Re2 5 - y for some y > 0, then the semigroup T ( t )is exponentially stable. In the case of functional differential equations (F), the Lyapunov theorems can assume a special form. We give one example (see Yoshizawa [ l , p. 1911). 5.5.2. Theorem. In Eq. (F) assume that g(0) = 0 and assume that g($) is locally Lipschitz continuous on C , . Suppose there is a locally Lipschitz continuous function u ( $ ) and three comparison functions rp,, cp2, q3E K such that rp1(1$(O)I) 5 u($> 5 4p2(ll$ll) and DU(F)(ICI) -(P3(I$(O)I)

5.5

147

EXAMPLES OF SEMIGROUPS

for all $ in a neighborhood of the origin, where D v ( ~denotes ) the derivative of u with respect to (F). Then the trivial solution of (F) is uniformly asymptotically stable. Modifications can be made in an obvious way to the above result to ensure uniform asymptotic stability in the large, exponential stability, uniform boundedness, uniform ultimate boundedness, and the like.

C. Volterra Integrodifferential Equations Research on this topic is still in progress. Some work has been done on nonlinear equations but we shall restrict our attention to the linear case. Volterra integral equations can be thought of as functional differential equations of the form (F) where the delay [- r , 01 is replaced by the delay interval (-a, 01. In this case the space X is a fading memory space as defined in Coleman and Mizel [l, 21. If h 2 0 is a given constant and p ( r ) a positive, continuously differentiable function such that p ( t ) 2 0 on --03 < t < - h , then X should consist of all functions ~p : (-03,0]+ R" such that q~is continuous on - h I t 5 0 and such that

Ilcp11 = sup{lcp(t)l: - h I t 5 0 ) +j-::P(t)lCp(f)l

dt

is finite. Another possible norm is IIcpII = sup{lcp(t)l: - h

If h

= 0, this

I t 20)+ js_bhp~t~lB~t~l~dt)i'2.

is equivalent to the inner product

llCpIIZ =

(9,Cp> = (Cp(O), Cp(0))

+

J

0

-m

P ( t ) (cp(t), cp(r)>

dt.

Most of the linear theory in the last section is also true for functional differential equations with infinite delay on a space of fading memory (see Hale ~3~41). We shall develop an alternate approach for Volterra equations of the form

or equivalently

PO

i(t) =

Ax,(O) + J

--m

K(s)x,(s)ds

(5.5.3)

for t 2 0 with x ( t ) = $ ( t ) given on -a < t < 0 and x(0) = z. Given p E [l, oo), the space X is in this case the set

X , = { ( z , $ ) : z E R "and $EL,((-CO,O),R")}

v

148

INFINITE-DIMENSIONAL SYSTEMS

with the norm defined by ll(z,

$111 = I4 +

{

-m

Iv(s)l~dt)ln

The matrix A in Eq. (5.5.3) is a given constant n x n matrix and K ( t ) is an k, E L , (-a, 0). If x ( t ) is the solution of Eq. (5.5.3) for a given pair of initial conditions ( z ,$1 E X,, we Put

n x n matrix whose entries

W)(Z?$)

= (x(t),.t)

for t 2 0, where as usual x,(s) = x ( t + s ) on -a < s I 0. Thus, T ( t ) is a Co-semigroup on X (see Barbu and Grossman [l] and Burns and Herdman [13) with infinitesimal generator

and domain

(z,$):

4 E L,,” and $ ( t )

$(s) ds

= z+

If Re l > 0 then 1 E o(2) if and only if det (A

+

-O /m

e”“K(s) ds - lZ

If R e l I 0, then l is always in ~(2)). Solutions of the Volterra equation i(t) =

Ax(t)

+

B(t-u)x(u)du JO

1

=

for all t

s0

0.

+ $(t),

(5.5.4)

x(0) = z

(5.5.5)

can also be used to construct Co-semigroups (see Miller [3]). Let Xu = { ( z ,$): z E R”,$ bounded and uniformly continuous on [ O , o o ) }

with norm ll(z9$)ll

= IzI

+ sup{l$(t)l: 0 5 t < 03).

If x ( t ) is the solution of Eq. (5.5.5) and if y‘(s) = p ( r + s - u ) i ( u )

du

+ $&)

then

W)(z,$)

= (X(t),

Yt>

5.5

149

EXAMPLES OF SEMIGROUPS

determines a Co-semigroup on Xuwith infinitesimal generator

A(z,$) = (Az+$(O), B ( . ) z + $ ( . ) )

and

D(A)

= { ( z , $ ) :$

is the integral of $ and B ( t ) z + $ ( t )

is bounded and uniformly continuous on [0, a)}. It can be shown (see Burns and Herdnian [l]) that this semigroup is the adjoint of the one defined earlier on X p for Eq. (5.5.3) if p = I and if B ( t ) = K ( - t )

o n O I tcco.

D. Partial Differential Equations In our discussion of partial differential equations we require the following additional notation. A vector index or exponent is a vector aT = (a1,a 2 , ...,a,) whose components are nonnegative integers, la1 = aj, and for any xE

x-J=

R",

xa = (XI,xz, ...) x,)a = x " 1 x ... ~.:x

Let D, = i(a/ax,) for k (D,,D,, ..., D,) so that

=

1,2, . . . , H ,

where i

= (-

1)"'

and let D =

Da = D;l D y ... 02.

In the sequel G will be a fixed closed bounded subset of R" with boundary aG. We will assume that aG is smooth, at least piecewise k-times continuously differentiable for some large enough k . This smoothness is easily seen to be true for the type of regions which normally occur in applications. Also, H ' ( B ) will denote all functions cp E L 2 ( B )such that for la1 I I the distribution derivatives Dacp E L , ( B ) (see Zemanian [l]) with norm

If C,'(B) is the set of all I-times continuously differentiable functions cp on B such that Dacp = 0 on aB for all la1 I I then Ho'(B) is the closure of Co'(B) in H'(B). This construction builds "zero boundary conditions" into the space H,'(B). Given m x m constant square matrices A , , let and consider the differential equation

v

150

INFINITE-DIMENSIONAL SYSTEMS

for the unknown vector valued function u(t,x). Here t 2 0, x E R", and I) E L , (R")is a given initial function. Proceeding intuitively for the moment, apply L,-Fourier transforms to Eq. (P) to obtain aii ( t , at

--

- A(o)ii(t,o),

ii(0,o) = $(o)

where A(w) = C , a , _ c r A a ofor a all w E R". In order to have a solution such that u(t, x ) and au(t,x)/at are in L , over x E R", it is necessary that A (0) ii ( t ,o) be in L, over w E R". This places some restrictions on A(w). For the proof of the next result, refer to Krein [ l , p. 1631. 5.5.6. Theorem. The mapping T ( t )I) = u(t, -) defined by the solutions u(t, x ) of Eq. (P) determines a C,-semigroup on X = L, (R")if and only if there is a nonsingular matrix S(o)and a constant K > 0 such that for all o in R" (i) IS(o)l I K and IS(o)-'l I K , (ii) S(w)A(o) S(o)-' = [C,(o)] is upper triangular, (iii) ReCmm(o)5 ... 5 ReC,,(o) IReC,,(w) 5 K , and for k = i+ 1 , i+2, ..., m. (iv) ICik(w)II K ( 1 [ReCii(o)I)

+

Parabolic equations often satisfy these conditions while hyperbolic equations do not. For example, the equation

au -+ 7 aZu + u-+ au at ax2 ay ax

- _-

yields m

=

1, r

= n = 2,

A(o) =

au

b-+ ay

cu

o = (01,02), and 2

-0,

Clearly Re A (0) = - oI2 -

- 02, + iuw, + ibo,

+ c = C,,(o).

+ c I c on R2.On the other hand the equation at2

-

ax,

or equivalently, the set of equations

au1 _ - 242, at

with u ,

=u

au, at

and u2 = &/at, yields m = 2, r

,

aZu,

---

ax, '

= 2,

n = 1, and

The eigenvalues of A ( w ) are C , (w) = iw and C,, then

(0) = - iw. If

SAS- = C,

5.5

151

EXAMPLES OF SEMIGROUPS

A straightforward computation yields

and

x

C G 2 (w) x1

(4- 2iwx,(w) Y l ( 4 1 -

l.

If the hypotheses of Theorem 5.5.6 were true, then there would exist a constant K 1 such that ~ C l z ( w ) ~ < Kand l all components of S ( o ) and S(w)-' are bounded by K,. In particular I ~ , , ( ~ ) x l ( ~ ) - - i O y l ( ~ )Kl l and IwI IClz(~)xt(~)-2~~Yi(~)I-' fK

can be combined to yield

l4/Kl I

I~,2(~)Xl(~)--~Yl(~~I+l-~Yl(~)I I K1 + l i W y l ( W ) l .

The first and last of these four inequalities imply that

(lwl/K1) - 2Kl I I

~

l

Z

~

~

~

~

l

~

~

~

I

I K, and Ix,(w)l I K , for all w E R, this is impossible. Thus Since ~Clz(m)~ no matrix S as asserted above exists. Many hyperbolic problems can be treated by choosing X in a different fashion.

5.5.7. Theorem. In order that there exist integers li 2 0 such that Eq. (P)

is well posed on the product space X =XT= H"(R") it is necessary and sufficient that there is a constant C such that Re A(w) I C for all o E R" and all eigenvalues A(w) of A(w).

For example, if X = H 1(R")x L, (R") for 3%

z = u 2 >

-=Ze at

a2ul

i= 1

then the above theorem applies. Next, let us consider Eq. (P) on the bounded set G,

a@

- = A (0) u,

at

u(0, x) =

*

(x)

.

v

152

INFINITE-DIMENSIONAL SYSTEMS

for t > 0, x E G and for II/ E L,(G). Let A , ( D ) = Z , a , = r A m D" be the part of A ( D ) with the highest order derivatives. Then Eq. (P) will be called strongly parabolic if A is strongly elliptic. Here, A is strongly elliptic means that r = 2s is an even integer, s 2 1, and Re ( A , ( o ) z, z ) < 0 for all w E R" and all z E R". Let A , be the operator defined by A , u = A (D) u on the domain

D(A,)

=

{

au a%

u E L , (G) : the normal derivatives satisfy u = - = an an2 = ... -

I F 1 U ~

ans-

=OondG

1

.

Here n ( x )is the outer normal to a point x E aG. Then A , can be extended to a closed linear mapping A on L,(G) with domain H'(G) n H,"(G) and ( A -21) will be dissipative on L 2 ( G ) when A 2 0 is chosen large enough. In general we have 5.5.8. Theorem. If A ( D ) is strongly elliptic and dG is smooth, then Eq. (P) is well posed on L,(G).

Other boundary conditions can sometimes be used in Eq. (P). For example, if a,, b,, and c are real, if

is strongly elliptic and if for some

r~

> 0,

au - -- nu an

on aG

then Eq. (P) is well posed on L, (G). Many nonlinear partial differential equations generate quasicontractive semigroups. We shall mention only one example here. Consider the partial differential equation

au - = A ( D ) u -f(u), at

u(0,x) = $(x)

(5.5.9)

for t > 0 and x E G and with boundary conditions

u = -au = - - azu an an2

-

... -

I F 1 U ~

ans-

=o

on dG. Here A ( D ) = A , ( D ) is an r = 2sth-order strongly elliptic linear operator and f is a nonlinear Lipschitz continuous function which maps L 2 ( G )into itself.

5.5

153

EXAMPLES OF SEMIGROUPS

We shall now consider the problem of constructing a Lyapunov function for Eq. (5.5.9) when A ( D ) = C;= a2/axi2= A is the Laplacian and zf(z) 2 fizz for all z # 0. We have

au - = Au -f(u)

and

at

in

R+

x G,

u ( 0 , x ) = $ ( x ) in

G,

u ( t , x ) = 0 on R+ x aG.

From the divergence theorem (also called Gauss’s theorem, see e.g., Apostol [ l , p. 3391 or Buck [I, p. 3501) it follows that

for w E H,’ (G) n H 2 ( G ) so that wAw dx

=

-LlVwl’dx

where dx = d x , d x , ... dx,. For smooth u we define u(u) =

( lu(x)j2dx.

JG

Using Theorem 5.4.18 we obtain

By PoincarC’s inequality (see Bers, John and Schechter [ l , p. 194, Lemma 51) we have

where the constant y can be chosen as S/Jn when n is the dimension of the X-space and G can be put into a cube of length 6. We therefore have,

Dtl(u) I -(/?+y-’)J

G

\ul2dx = - ( P + ( l / y 2 ) ) u ( u ) .

E. Stochastic Differential Equations Stochastic differential equations (see Chapter IV) can also be used to generate semigroups. Here we consider random processes { x , ( o ) E R”, t E R’} which are defined on a probability space (Q, d, 9). In this case 51 is the event

154

v

INFINITE-DIMENSIONAL SYSTEMS

space, d is a o-algebra of events in SZ and 9 is a probability measure on d , (Henceforth we simply write x, in place of {x,(w) E R", t E R+j . ) The random behavior of x, is characterized by the distribution function P ( t , B ) = B { x ,E B }

and the transition function P,(t,B)

=

~'(x,+,EBJx,=aj,

where the latter expression represents a conditional probability. The evolution of the distribution function is completely determined by the transition function via the relation

P(t+z,B)

=

S,

R" p

a ( t , B ) P (7,da).

Now let us define the operators T ( t ) , t E R + ,on the functionals of R", as the conditional expectation

J

n

u,(u) = T ( t ) U ( a )4 E,U(x,)

=

baR"

U(b)Pa@,db).

If x, is a homogeneous Markov process, then T ( t ) will be a semigroup. If A is the infinitesimal generator of T ( f ) then ,

au

- = Au, at

uo(a) = U ( a )

is called the backward diffusion equation of x, . In practice there are several specific types of Markov processes of interest. For example, x,(o) may be defined as the solution of the It0 differential

equation

dx, = f(x,) dt

+ a(xJ d5,

where f: R" + R" and (5, E R", t E R + ) is a normalized Gaussian random process with independent increments. If in particular 5, is also a normalized Wiener process, then the infinitesimal generator A can be computed as

A U ( a ) = , f ( ~ ) ~ V ( i ( a+) 4 tr{o(a)TVa, U ( a ) o ( A ) } where the symbol V,, is as defined in Chapter 1V. This corresponds to the case when the disturbance is "white noise." Other types of stochastic disturbances can also be considered, as well as certain other types of nonlinear stochastic differential equations. For further details on stochastic processes and stochastic differential equations, refer to the books by Wong [l], Arnold [l], Doob [l], and Kushner [1, 31.

5.6 STABILITYOF SYSTEMS DESCRIBED ON BANACH

SPACES

155

F. Hybrid Systems It is possible to construct examples of semigroups which are generated by a mixture of different types of equations. Indeed systems which may be viewed as an interconnection of subsystems modeled by different types of equations (e.g., partial differential equations, ordinary differential equations, integrodifferential equations, functional differential equations, and the like) are quite common in practice. Such systems, consisting of distributed parameter and lumped parameter components are called hybrid systems. The class of sampled-data systems considered in Chapter 111 is an example of a hybrid system described on finite-dimensional spaces. For examples of hybrid dynamical systems described on infinite-dimensional spaces, see Wang [3,4]. In Sections 6 and 7 of the present chapter we consider some theorems and examples pertaining to hybrid systems. G. Other Types of Systems As a final note, we point out that the theory of semigroups can be modified to include also discrete-time systems (such as those considered in Chapter 111) and time-varying systems (i.e., nonautonomous systems) of various types. Since such modifications are even more mathematically complicated, while applications in the setting of infinite-dimensional spaces appear to be few, these modifications are not considered here.

5.6 Stability of Large Scale Systems Described on Banach Spaces

Now that all the required background material has been established, we are in a position to consider the qualitative analysis of large scale systems described on Banach spaces. We first formulate the class of composite systems which we will consider. Whereas the problem of well posedness presents no difficulties in the case of dynamical systems described on finite-dimensional spaces, great care must be taken in insuring existence and uniqueness of solutions of interconnected dynamical systems described on infinite-dimensional spaces. For this reason we present some well posedness results for the classes of problems considered herein. Finally, we state and prove several results for uniform asymptotic stability, exponential stability, instability, and uniform ultimate boundedness.

156

v

INFINITE-DIMENSIONAL SYSTEMS

A. Interconnected Systems

We begin by considering I isolated subsystems of the form i i

(q)described by equations

(a

= .h(zi),

i = 1, ... ,1, which are assumed to be well posed on Banach spaces Zi with respective norms 11. \Ii. By this we mean that there is a C,-semigroup Ti(t) or a nonlinear semigroup T , ( t ) defined on a set Ci c Zi where the origin is contained in the interior of Ciand Ci= Ziis allowed. The domain D(J) must be dense in Ciand the functionf, must be the generator of the semigroup T ( t ) .We note that for a given integer i system (g) might represent a system of ordinary differential equations, a system of delay differential equations, a system of linear partial differential equations, or perhaps a nonlinear partial differential equation. In the case where T i ( t )is a nonlinear semigroup recall that we must allow the possibility thatf, is multivalued, i.e., a relation. In this case system (y;’) should actually be replaced by

ii E f , ( Z i ) , Next, we interconnect these subsystems to form a system (Xi)described by equations of the form i i

= f,(zi)

+ gi(x>,

(Xi)

i = 1, .. .,I, where the operators gi, representing the interconnecting structure of (Xi) are defined on D(gi) c X and have range in Z i . The hypervector xT= (z1, .. .,zI) is a point in the product space 1

x= xzi i= 1

with the norm specified by

Letting f(x>’= [fl(zl), ..., fi(zJ1, g(x)T= Cgl(x), ...,gl(x>l,we can express Eq. (&) equivalently as

1

= f(x)

+ g(x) p

A (x).

(9)

System (9) is clearly a special case of system (N). As in the preceding chapters, we speak of composite system (9’) or interconnected system (9’) with decomposition ( X i ) . Note that D(f+g) = D ( f ) n D(g) = D(f) n D(gl) ... n D(gJ is the domain for (9’).Henceforth, each subsystem as well as composite system (9) is assumed to have the trivial solution z i ( t ) E 0 and x ( t ) = 0,

(x.)

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

157

respectively, as trajectories. In addition, the composite system (9) is assumed to be well posed, that is, f + g generates a semigroup T(1). Moreover, Do 4 D(f+g) n D(f,) n D((f+g)J is dense in X .

B. Well Posedness In most cases the well posedness of composite system (9) must be verified, using the type of theory outlined in Sections 2, 3, and 5 of this chapter. However, in the next two theorems we present results for two special cases where the well posedness of the isolated subsystems and the form of the interconnecting structure guarantee the well posedness of system (9). 5.6.1. Theorem. Let A i : D(A,) + Zi be a linear operator with domain D(Ai) dense in Z i . If fi(zi) 4 A,zi generates a C,-semigroup on Zi for each i= 1, ..., 1 and if D(gi) = X and if g i : X + Ziis a bounded linear operator for each i, then composite system (9) with decomposition (Xi) is well posed on X . Proof. Since each subsystem (q)determines a C,-semigroup on Z i , it follows from Theorem 5.2.4 that there are constants M iand w such that

IIR(I,Ai)nlli I M i @ - w)-” for all I > wand for i = 1,2, .,.,1. If A x is defined by

(Ax)’

= [(,41Zd,(,42Z2)>

...,(A,z,>l

on D(A) = Xf= D(Ai), then A is closed and D(A) is dense in X.Moreover, when A > w ,we have

c 1

IIR(A,AYII

i = 1 IIR(I,

AiYIli 5 ( i l M i ) ( A - w ) - n +

From Theorem 5.2.4 it now follows that A generates a Co-semigroup on A’. If ( B x ) = ~ [gl(x), g 2 ( x ) ,...,g[( x)], then B is a bounded linear operator on X . It follows from Theorem 5.2.5 that A + B generates a Co-semigroup on X. -

5.6.2. Theorem. If fi(zi) is w,-accretive on Z i , if Ra(Z-IL) = Z i = D(A) for 0 < I < A,, if D(gi) = X with gi Lipschitz continuous, and if gi(0) = 0, i = 1, ...,1, then composite system (9) with decomposition ( X i ) is well posed and generates a quasicontraction on X .

v

158

INFINITE-DIMENSIONAL SYSTEMS

= (1-Aw)~~x1--xz~~.

Thus, f is w-accretive. Clearly the range Ra(Z- Af) = X for 0 I 1 < 1, since this is true by coordinates. By Theorem 5.3.5 it follows that f generates a quasicontractive semigroup on C = X . Since g(x) is Lipschitz continuous with some constant L, we have, given (xi, Yi +9 (xi)) ~ f 9, + (ICXl --A(YI

2

+g(x,))l-

IKXl -AYd

[xz--I(~z+g(xz))lII

- (Xz-~Y2)II

- A Il~(~l)--S(YZ)Il

2 (1 - Aw)11x1- xz I - AL. 11x1-x2 =

[l-A(w+L)]

II

llX1-XzII.

Therefore,f+ g is ( w + L)-accretive. To see that the range Ra(f- A(f+g)) = X , fix z E X and A E (0, A,,). The equation [I-A(f+g)] x = z for x E D(f+g) is equivalent to (K'1-f)~ =

ZA-,

+ g(x)

or

x = R(A-',f)(zA-'+g(x))

where R ( p , f ) = ( p I - f ) - ' is the resolvent operator for$ Since 1 =f(x) is well posed and accretive, i t follows from Eq. (5.3.4) that IR(p,f)l I(p- w)-' when p > w = maxi w i . This means that the right-hand side of the above equation is Lipschitz continuous, i.e.,

1 R (A-

1,

f)(ZA -

+g (x)) - R (A-

1,

f)(ZA-

+g (X)) 11 I(A-

11

- w)- ,L x - 2 11.

By the contraction mapping theorem this means that the equation

x-A(f(x)+g(x)) = z has a solution x for 1 < (L+w)-' = A,, that is, for (Ap1 - w)-'L < 1. Since z E X is arbitrary, we have Ra[f-A(f+g)] = X if 0 < A < A,. By Theorem 5.3.5, it now follows that (f+g) generates a quasi-

contractive semigroup on X .

W

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

159

In addition to the well posedness assumption at the end of Section A, the domain hypothesis that the set Do 4 Dcf+g) n D(f,) n D((f+g),) be dense can be verified in the settings of the last two theorems. The linear case is completely trivial since A = A,, A f B = (A B), (see Theorems 5.2.3 and 5.2.4), while D(A) = D(A +B) since B is bounded. In the nonlinear case it is necessary to make the additional assumption that all of the spaces Zi are uniformly convex. If we use the equivalent norm

+

for all x E X , then X will also be a uniformly convex Banach space. (The proof of Theorem 5.6.2 is easily modified to accommodate this change in norm.) By the last statement of Section 5.3, D ( f ) = D(f,) wherefis the closure off. Since D(g) = X , it follows that Do Li D(f,) n D((f+g),) n D(f+g) = D(f) is dense in X. C. Uniform Asymptotic Stability In characterizing the qualitative properties of subsystems following convention.

(x),we use the

5.6.3. Definition. Isolated subsystem (q)with corresponding semigroup Ti (t) is said to possess Property A if there exists a Lyapunov function ui such that (ui,TJ is admissible in the sense of Definition 5.4.17, if there exist three functions (pil,(pi2, (pi, E K, and if there exist real constants oi and mi > 0 such that (i) C P ~ ~ ( ~ \ I Z ~ui(zi) \ \ ~ )I (piz(\\zilJi) for all zi E Zi such that \ \ z i / J
(x)

is said to possess Property B if it has Property A with mi = + co and if qil,(pi2 E KR. If cri < 0 in Definition 5.6.3, then it follows from Theorem 5.4.10 that the trivial solution of (q.) is uniformly asymptotically stable. If oi < Oin Definition 5.6.4, then in accordance with Theorem 5.4.11, the trivial solution of (Yi) is uniformly asymptotically stable in the large. In the latter case we always assume that the semigroup generated by (q) is defined on the entire Banach space Zi. If cri > 0 then T , ( t ) may possibly be unstable. As in the preceding chapters, oi may be viewed as a measure of the degree of stability of subsystem As such, oi will be useful in studying the qualitative effects of the isolated subsystems on the behavior of the entire interconnected system (9).

(x).

v

160

INFINITE-DIMENSIONAL SYSTEMS

We are now in a position to state and prove several stability results for composite system (Y). 5.6.5. Theorem. Assume that for composite system (9) with decomposition ( X i ) the following conditions hold. (i) Each isolated subsystem (q) possesses Property A ; (ii) given the Lyapunov functions vi for and the corresponding comparison functions ( p i 3 E K,i = 1 , . .., I, there exist real constants b, i, j = I, ...,f, such that

(x) 1

Vvi(zi, gi(x)) 5

(pi3 (IIziIIi)1’2

C b,

j= 1

I

q j 3 (IIzj j)1’2

for all xT = (zl, ..., z,) E D(f+g) with IIzilli < m i ; and (iii) there exist positive constants a i , i = 1, ..., I, such that the test matrix R = [ r i ] defined by

is negative definite. Then the trivial solution of composite system ( Y )is uniformly asymptotically stable. Proof. Let Q

=

{x’= (zl, ..., z,) :iizilli < mi for all i}. On Q we define v(x) =

C ai vi(zi),

i=l

where the constants ai > 0 are as given in hypothesis (iii). Clearly, v(x) is continuous and v (0 ) = 0, since each vi(zi) satisfies these conditions. Since each subsystem possesses Property A, it follows that

(x)

for all x E Q. The two summations above are both positive definite, decrescent functions, i.e., there are functions q1,cp2E K such that I

Thus, for all x E Q.

1

5.6

STABILITY OF SYSTEMSDESCRIBED ON BANACH SPACES

161

To compute the derivative of u along solutions of ( Y ) ,note that for Do

XE

- u(x)

u(x+h)

c 1

=

f.%i{Ui(Zi+hi) - v i ( z i ) }

i =1 1

1

I ai {Vui(zi hi) 9

i=1 1

1 aiVvi (zi

=

i= 1

+ 0 (//hiII

i)}

hi) + 0 (IlhI]),

9

so that Vu(x,h) =

I

1 aiVui(zi,hi).

i= 1

Since each Vui(zi,hi) is continuous and linear in hi it follows that Vu(x,h) is continuous and linear in h for each fixed value x. By this linearity and in view of hypothesis (ii), we have Vu(x, AX)+9 (x)) =

=

I

1 m i Vui (zi

9

i= 1

f(x) +g(x))

U~RU

where R is the test matrix given in hypothesis (iii) and UT = C'p13(llz1

/I

'p23(llzzl12)1'2>

... 'p13(11~1111)1/21. )

Since R is a negative definite symmetric matrix, its largest eigenvalue A(,R) is negative and we have Vu(x,f(x)+g(x))

< uTRu < A,(R)\uI2.

Also, since 1

Iu12 =

for some function

i= 1

(~i3(IIziIIi) 2

(P~(IIxII>

v3E K , it follows that VU(X9

f(x) +9(x))

I , 1 ( R )' p 3 (I/ x I l l

(5.6.7)

for all x E Q n Do. Applying Theorem 5.4.18 to inequality (5.6.7) we obtain DU(Y)(X)

~M(R)'p3(llXIl).

(5.6.8)

v

162

INFINITE-DIMENSIONAL SYSTEMS

Jn view of inequalities (5.6.6) and (5.6.Q the hypotheses of Theorem 5.4.10 are satisfied and the trivial solution of composite system (9’)is uniformly asymptotically stable. H

For global asymptotic stability we have the following result. 5.6.9. Theorem. Assume that for composite system (9) with decomposition (Xi)the following conditions hold.

(i) Each isolated subsystem (Yi)possesses Property B ; (ii) given the Lyapunov functions ui for (q)and the corresponding comparison functions qi3E K , i = 1 , ..., I, there exist real constants b,, i, j = 1, ..., I, such that I

j=1

hold for all x E D(f+g) where xT= (zl, ..., z,); and (iii) there exist positive constants cli for i = 1, ..., I such that the test matrix R = [ r g ] defined by

4

r.. =

mi (ci

+ bii)?

$(cliby+cljbji),

is negative definite.

if i = j if i # j

Then the trivial solution of composite system (9) with decomposition (Xi) is uniformly asymptotically stable in the large. Proof. As in the proof of Theorem 5.6.5, we choose u(x) = Xf=,aiui(zi). It is an easy matter to show that 1

I

ql(llxll) 5

C “iqil(llzilli) i= 1

and

~ Z ( l l x l l )2

CI aiqi2(I/ZiIIi)

i=

can be chosen so that q l ,q z E K R , and that the inequalities

~l(llXIl)5 are satisfied for all x E X

=

Xf=

u(x) 5 qz(l1xIl)

Z i . In addition, the inequality

4 Y , ( x ) 5 L(R)cp3(llxll)

with

is satisfied for all x E X . It now follows from Theorem 5.4.1 I that the trivial solution of composite system (9’)is uniformly asymptotically stable in the large. H

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

163

D. Exponential Stability For exponential stability of interconnected system (9) we have the following theorem. 5.6.10. Theorem. Assume that for composite system (9) with decomposition

(Xi) the following conditions hold.

(i) Each isolated subsystem (q) possesses Property A ; (ii) all functions in the set {cpij: i = 1,2, ..., /, and j = 1,2,3} are of the same order of magnitude; and (iii) there exist real constants b,, i, j = 1, ..., I, such that the inequalities Vui (zi ,gi <XI>

I

( ~ i (3

I zi II J1” j1 bij ( ~ j (3IIzj I j)’” =1

hold for all x E D(f+g) with I(zi(Ii< mi; and (iv) there exist positive constants ai, i = 1, ..., f, such that the test matrix R = [r”] defined by

4

r.. = is negative definite.

ai(0i + bii),

if i = j

&(aibij+ajbji),

if i # j

Then the trivial solution of composite system ( Y )is exponentially stable.

Proof. As in the proof of Theorem 5.6.5, we choose

so that

1

and

where I, ( R ) < 0 denotes again the largest eigenvalue of the test matrix R. To complete the proof we must show that there are functions pl, (pz, (p3 E K which are of the same order of magnitude such that

(Pl(llXll) 5

44 5 402(11x11)

for all x E Q = {x’= (zl, ...,zI) : IIziJli< mi for all i}, and

for all x E Q .

D’(9)

(x)

I M ( R )(P3 ( Ilx!)

v

164

INFINITE-DIMENSIONAL SYSTEMS

By hypothesis, there is a function cp, e.g., cp k, such that

=

c p l l , and positive constants

2 ki, ~ ( r ) ,

Pi2 (r) 5 ki2 c ~ ( r ) , cpi3(r) 2 ki3 q( r) for all r in a region 0 5 r < ro and for i = 1,2, ..., 1. We define 4Oi1 ( r )

t l ~ l l m=

max llzilli I

~ 1 ( r= ) min(aikil)q(r) i

and Clearly, each yi E K . Moreover, since cp is strictly increasing, we have

Also,

and

1

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

165

By Theorem 5.4.12 it follows that the trivial solution of composite system (9) is exponentially stable in the norm 11 . llm. Since

2 IIziIIi 5 1

11x11 =

i= 1

~ I I X I I ~

we conclude that the trivial solution of composite system (9) is exponentially stable in the norm IIxII. For exponential stability in the large we have the following result. 5.6.11. Theorem. Assume that for composite system (9) with decomposition (Xi) the following conditions hold. (i) Each isolated subsystem (g) possesses Property B ; (ii) all functions in the set (9,:i = I, 2, ...,1 and j = 1,2,3} belong to class K R and are of the same order of magnitude; and (iii) there exist real constants b,, i , j = 1, ..., 1, such that the inequalities j= 1

hold for all x E D(f+g); and (iv) there exist positive constants ai,i = 1 , . ..,I, such that the test matrix R = [rJ defined by (ai(oi +bii), if i = j r.. = ' [f(aib,+ajbji), if i # j is negative definite. Then the trivial solution of composite system (9) is exponentially stable in the large.

Proof. The proof is similar to the proof of Theorem 5.6.10. Consistent with the method of analysis advanced in the preceding chapters, in the above results express stability properties of composite system (9') terms of the qualitative properties of the lower order subsystems (q) and in terms of the characteristics of the interconnecting structure. As before, the confining relationships between these properties are determined by the test matrix R. Concerning test matrix R, several observations are once more in order. Since a necessary condition for the negative definiteness are the relations o,+ bii < 0, i = 1, ..., 1, each subsystem must either be stable or else the interconnecting structure must provide local stabilizing feedback for each unstable subsystem. The nature of the bounds on the interconnecting structure expresses the strength (and the direction) of coupling relative to the degree of stability of

166

v

INFINITE-DIMENSIONAL SYSTEMS

each subsystem. The negative definiteness condition on the test matrix R has the effect of limiting the degree to which the interconnecting structure is allowed to affect the qualitative behavior of system (9’)by constraining the size and/or sign of the coupling terms. As in the finite-dimensional case, the above observations suggest a systematic procedure for the stabilization of unstable composite systems by utilizing local stabilizing feedback for the subsystems, and also, by using local feedback associated with various interconnecting terms which has the effect of decreasing the strength of coupling. E. Application of M-Matrices In all of the results above it is necessary to find weighting factors cci > 0 such that matrix R is negative definite. Although the choice of such constants is not unique, it is not always evident that these constants exist. In the next results we show that in the special case when b, 2 0 for all i # j , the necessity of choosing a weighting vector uT = (al,...,a,) can be eliminated. It is emphasized that these results are at best only equivalent to the ones presented thus far in this section. Indeed, since no sign restrictions on the b,, i # j , exist in the above theorems, these stability results remain important due to their greater generality. 5.6.12. Theorem. Consider the I x I matrix S = [s,] s.. =

’

-((ai+bii), (-by,

defined by

if

i

if

i#j

=j

(5.6.13)

where ( a i , i = 1, ..., 1, and b,, i, j = 1, ..., I, are as defined in Theorems 5.6.5, 5.6.9, 5.6.10, and 5.6.11. (i) Assume that hypotheses (i) and (ii) of Theorem 5.6.5 are satisfied with b, 2 0 for all i # j . If all successive principal minors of test matrix S are positive, then the trivial solution of composite system (9’)is uniformly asymptotically stable. (ii) Assume that hypotheses (i) and (ii) of Theorem 5.6.9 are satisfied with 6 , 2 0 for all i # j . If all successive principal minors of test matrix S are positive, then the trivial solution of composite system (9) is uniformly asymptotically stable in the large. (iii) Assume that hypotheses (i), (ii), and (iii) of Theorem 5.6.10 are satisfied with b, 2 0 for all i # , j . If all successive principal minors of test matrix S are positive, then the trivial solution of composite system (9’)is exponentially stable. (iv) Assume that hypotheses (i), (ii), and (iii) of Theorem 5.6.1 1 are satisfied with 6 , 2 0 for all i # J . If all successive principal minors of test

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

167

matrix S are positive, then the trivial solution of composite system (9) is exponentially stable in the large. Proof. Since the proofs of the four parts follow along similar lines, we prove only part (i). As in the proof of Theorem 5.6.5, we choose

c tiiUi(Zi) I

u(x) =

i= 1

where the constants ai > 0, i = 1 , ..., I, will further be specified later. As before, there are functions ‘pl, (p2 E K such that (Pl(llXll)

5

4x1 5

(P2(llxll>

(5.6.14)

for all x E Q . Also if x E Q, then

=

-+u’(AS+

S T A ) u,

where S = [Is,] is the matrix defined in (5.6.13), UT

=

( ( P i 3 (11Zi

11 I ) ” ~ , 4023 (11% /I z.)”~, ...) 4013 ( I I Z I I I I ) ~ ’ ~ ) ~

and A is the diagonal matrix specified by A = diag[al, ...,ti,]. Now by Corollary 2.5.6, the condition on the successive principal minors of matrix S is equivalent to the existence of a diagonal matrix A with positive elements such that the matrix ( A S + S T A ) is positive definite. Thus, if 1, is the largest eigenvalue of the matrix -(AS+STA)/2, then 1 , < 0 and Du(,,(x) 5 1 ,

I

C (~i3(ltzilli)5 ~ ~ ( P ~ ( ~ I x I I(5.6.15) ~ i= 1

for all x E Q , where (p3 is some function of class K . It now follows from inequalities (5.6.14) and (5.6.15) and from Theorem 5.4.10 that the trivial solution of composite system (9’)is uniformly asymptotically stable. H It is emphasized that the condition on the test matrix S in the above results can be replaced by any of the other equivalent M-matrix conditions discussed in Chapter I1 which insure that - S is a stable matrix. In particular, S-pZ will also be an M-matrix if and only if p < min Rel(S ) . This means that any modification of the subsystems (q), or their local feedback, which increases each diagonal term oi+bii by less than p will leave composite system (9’) asymptotically stable (or exponentially stable). In this case p can again be interpreted as a margin of stability for system (9’).

168

v

INFINITE-DIMENSIONAL SYSTEMS

F. Instability In discussing instability results for composite system (Y),we find it convenient to employ the following convention. 5.6.16. Definition. Isolated subsystem (8) is said to possess Property C if there exists a Lyapunov function ui(zi) such that (ui,T,) is admissible in the sense of Definition 5.4.17, two functions q i 2(pi3 , E K and real constants oi and mi > 0 such that (pi2(11zilli) 5 -ui(zi> vui(zi?As(zi)) 5 oi(pi3(IIzilli)

for all zi E D(f;:,), where D(J;.,) denotes the domain of the infinitesimal generator of T , ( t ) , with ~ ~ < z im~i . ~ i 5.6.17. Definition. Isolated subsystem (Sq) is said to possess Property C’ if there exists a Lyapunov function ui(zi) with (ui, Ti)admissible in the sense of Definition 5.4.17, a function ‘pi3 E K , and real constants ci and mi > 0 such that Vui (2;

9

A s

(Zi)) 5 oi (pi3 ( llzi /Ii)

for all llzi~li< mi, zi E D(Js). If in Definition 5.6.16 oi < 0, then the trivial solution of subsystem unstable (in fact, completely unstable). We now have the following instability result.

(q)is

5.6.18. Theorem. Let N be a nonempty subset of L = {1,2, ..., I}. Assume that for composite system (9’)with decomposition ( X i ) the following conditions hold. possesses Property C and if i 4 N , i E L, then (i) If i E N , then (3) possesses Property C‘; (ii) there exist real constants b, such that the inequalities

(x)

hold for all xT = (zl, ..., z,) E D(f+g) with lizilli < mi for all i s L ; (iii) there exist positive constants txi for all i E L such that the test matrix R = [ r i ] defined by M i ( G i + bii), if i = j

4

y.. =

f(aiby+ajbji),

if

i#j

is negative definite. Then the trivial solution of composite system (9’)is unstable.

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

169

Proof. Given the constants cii > 0, i E L, choose

u(x)

c 1

=

i= 1

ciiUi(Zi).

As in the proof of Theorem 5.6.5, we can show that u is a Lyapunov function such that (5.6.19) 4 9 , ( x ) 5 &Am (P3(llxll) for all x E Q, where Q = {x' = (zl, ...,z,) : /Izi/li< mi for all i E L } , where &(R) < 0 is the largest eigenvalue of matrix R = [r,] and where q3 is a function of class K such that

Now let Xu= {x E Q : xT= (zl, .. ., zl) and zi = 0 whenever i q! N } . Since u(0) = 0, then for x E Xu,x # 0, we have i.e., in every neighborhood of the origin there is at least one point xo for which u(xo) < 0. It now follows from inequalities (5.6.19), (5.6.20), and Theorem 5.4.14 that the trivial solution of composite system ( Y )is unstable. The proof of Theorem 5.6.18 can be modified along the lines outlined in the proof of Theorem 5.6.12 to show that the following result is true. 5.6.21. Theorem. Assume that for composite system ( Y )with decomposition ( X i ) the following conditions hold. (i) Hypotheses (i) and (ii) of Theorem 5.6.18 are satisfied with b, 2 0 for all i # j ; (ii) all successive principal minors of the test matrix S = [s,] are positive, where -(ai+bii), if i = j 3.. = rl

(-a,,

if

i # j.

Then the trivial solution of composite system (9) is unstable. G. Uniform Ultimate Boundedness Next, we consider the boundedness of solutions of system (9). 5.6.22. Definition. Isolated subsystem (8) is said to possess Property D if there exists a Lyapunov function ui defined for all zi E Zi with ( u i , T,) admissible in the sense of Definition 5.4.17, three functions q i l ,q i z ,qi3E KR,

V

170

INFINITE-DIMENSIONAL SYSTEMS

a constant oiE R, and positive constants M i and R , , such that the inequalities vi1(llzilli)

Vi~(llzilli)

ui(zi)

and Vui(zi, &s)5

bi Vi3 (llzilli)

hold whenzi E D(hs)and llzilli > R i , andif lui(zi)l I Miand IVui(zi,&,
(g) is uniformly

5.6.23. Theorem. Assume that for composite system (9’) with decomposition ( X i ) the following conditions hold. possesses Property D; (i) Each isolated subsystem (q) (ii) there exist real constants b,, i, j = 1, ..., I, such that the inequalities

hold for all xT = (z,, ..., z,) E D(f+g); and (iii) there exist positive constants ai,i = 1, ..., I , such that the test matrix R = [ r J defined by

is negative definite. Then composite system (9) is uniformly ultimately bounded. Proof. Given positive constants ai,i = 1, . ..,I, we choose

v(x)

c I

=

i= 1

.iVi(Zi).

Let Bi(Ri) = {z, E Zi : llziI/i5 R i } . Following the procedure in the proof of Theorem 5.6.5, it is clear that I

1

C1aivi2(IIziIIi)

2 Eivi1(IIziIIi) 5 v(x)

i= 1

i=

and 1

D ~ < Y ~ (5x )A M ( N for all x E X-Xi=

E m .

i= 1

Vi3(llzilli)

5.6

STABILITY OF SYSTEMS DESCRIBED ON BANACH SPACES

171

To complete the proof we must consider the situation where some but not all zi satisfy the condition llzilli 5 Ri. First we suppose that llzilli > Ri for i = 1,2, ..., r and lzil 5 Ri for i = r + 1, ...,I and xT = (zl, ..., zI) E D(f+g). For such values of x we have

or

jj

i= 1

c 1

aiqi1(lIziIIi)-

i=r+ 1

aiMi 5 u(x) 5

i= 1

aiqiz(llziIIi> +

c 1

UiMi. i=r+1

The functions on the left and right of the above inequality are dominated by two functions ql*,qz* E KR such that

and

Similarly, for such values of x we also have

172

v

+

INFINITE-DIMENSIONAL SYSTEMS

I

1

i=r+ 1

mi~i3(Ri)1’2

1

j=r+ 1

We thus have an inequality of the form Vu(x,f(x)+g(x)) I UTR*U

+

b,i40j3(Rj)1‘2.

UTL,

+ L,

for some constant L , > 0 and some vector Lo E R‘. Since the test matrix R is negative definite, the submatrix R* is also negative definite. Thus, there is a constant A,(R*) < 0, a real constant r*, and a function ‘p3* E KR such that V V ( X , f ( x ) + g ( x ) ) 2 LM(R*)IuI2

+ uTLo + L1

I +A,(R*)/u/2 =

+A,(R*)

i;

i = 1 qi3(11zijli)

t

5 + A M ( R * ) ~ ~i =*1( ~ ~ z i ~ ~ i )

for all xT = ( z , , ..., z,) E D ( f + g ) such that llzilli > r* for i = 1, ..., r and / / z i / _< J i R, for i = r + 1, ..., 1. Above we have assumed that IIzilli > Ri for i = 1,2, ..., r and ~~~~~~~~2 R, for i = r + 1 , ..., 1. The identical argument works for any other combination of indices. In view of Theorem 5.4.18 and the above, we can now apply Theorem 5.4.16, which completes the proof. Using the modifications outlined in the proof of Theorem 5.6.12, we can also prove the following result. 5.6.24. Theorem. Assume that for composite system (9) with decomposition

(Xi)the following conditions hold.

(i) Hypotheses (i) and (ii) of Theorem 5.6.23 are satisfied with b, 2 0 for all i # j ; and (ii) all successive principal minors of the test matrix S = [si] defined by

-(ai+bii),

if i

=j

if i # j are positive. Then composite system (-9) is uniformly ultimately bounded.

5.7 SOME EXAMPLES AND

173

APPLICATIONS

5.7 Some Examples and Applications

In the present section we consider two specific dynamical systems to demonstrate the applicability and usefulness of the results of Section 5.6. The first of these is a simple hybrid system, while the second involves the point kinetics model of a coupled core nuclear reactor.

5.7.1. Example. (Hybrid System.) Consider a hybrid system described by the set of equations

with boundary conditions

z,(t, y )

=

for all ( t , y ) E R f x dG

0

(5.7.3)

and initial conditions z1 (0) = z l 0 given and z,(O, y ) A ~ ( ygiven ) for y E G. Here A is an n x n matrix, b and c are given n-vectors, t~ and L are given positive numbers, A is the Laplacian in m-space R", G is an open subset of R" with smooth boundary aG, H I and H , are given functions which satisfy the conditions H,(y,O)

=

forall y E G,

0

I Hl ( Y ,z ) - Hl ( Y , z*)I

H,(O)

=

0

5 Ihl (Y)l Iz - z* I

for all y E G , and z , z* E R, and IH,(u)-H,(w)I

5 Lb-wl

for all u, w E R,and hi E L , (G), i = 1,2, are given functions. Hybrid system (5.7.2) may be viewed as an interconnection of two isolated subsystems (Yl), (9, described ) by the equations i l ( t > =f1(z1) = A z , az2

-=

at

f 2 ( ~ 2 ) = ctAz2

(91)

- Hz(z2).

The interconnecting structure is specified in this case by g,(z,,z*)

=

b~HI(Y.Z2(Y))

and S2(Zl,Z2) = h2(Y)CTZl.

(9?2>

v

174

INFINITE-DIMENSIONAL SYSTEMS

In the present case Z , = R",Z, = L,(G), and X = R" x L, (17). As usual, the norms for R" and L, (G) are denoted by I . I and 11 . 1 ,, respectively. Since (Sp) is an ordinary differential equation, it is certainly well posed on Z,. Moreover, if A is a stable matrix, there exists a positive definite symmetric matrix P such that ATP+ PA

=

-Q

is negative definite. Choosing = Z,TPZ,

V,(Z,)

and using the notation of Section 5.6, we obtain

v,,(IzII)

U,(Z~)5

= L(P)Iz112

J-M(p)Iz1I2 =

(~12(IzlI)

and

V U , ( Z , , ~ I ( Z I )= ) -zl'Qzl

5 -L(Q>Iz,12

so that C T ~= -A,(Q) and q I s ( r )= r 2 . Subsystem (Y2)is the type of system discussed at the end of Section 5.5, Part D (see Eq. (5.5.9)). The Laplacian A is a second order, strongly elliptic operator while H , is Lipschitz continuous. Thus, subsystem (9, determines ) an w-accretive operator on L,(G)and by Theorem 5.6.2 is well posed. If H , satisfies the condition

for all u

uH,(u) 2 flu'

E

R

for some fixed constant f l 2 0, then we can choose r

In the notation of Section 5.6 we have in this case (P21(llz2/12)

= u 2 ( z 2 ) = (P22(Ilz21/2).

Also, using the notation of Section 5.5, Part D,

r

P

(5.7.4)

5.7

175

SOME EXAMPLES A N D APPLICATIONS

Using the notation of Section 5.6 we have o2 = -(aT+p) and (p32(r)= r 2 . The constant r = y-’ can be estimated by y s 8/n’I2, where

6

=

sup{(u-ul:u,vEGandforsomej(l~jsn), u i = ui for i = 1, ..., n, i#j}.

Thus it is clear that isolated subsystems ( Y 1 ) and (9,) are well posed and that they satisfy Property A as well as Property B if matrix A is stable and if condition (5.7.4) is true. The interconnecting terms are Lipschitz continuous. Indeed,

5 Ibl l l ~ l l l 2 l l ~ z - ~ z l l 2

by the above assumptions and the Schwarz inequality. Also, 19,(z1, z2)-g2(u,, u2)l2 dY

=

s

SG

Ih2(Y)

(llh2llz

- u1)l’ dY

I4 Iz1-u11)2.

It now follows from Theorem 5.6.2 that composite system (5.7.2) is well posed on X = R” x L2(G). Next, we check hypothesis (ii) of Theorem 5.6.5. We have VVl (z1, 91

(4)= 2zlTPgl(x)

with the last inequality following from the Schwarz inequality. Using the notation of Theorem 5.6.5 we have b , , = 0 and b , , = 2A,(P) Ibl Ilhll12. We also have n

In the notation of Theorem 5.6.5 we have b,, = 0 and b,, test matrix R of this theorem assumes the form R =

[

-mi

L(Q>

Ca,lhf(P>Ibl llhllI,+~,

[El llh2112

141

=

2 llh2112lcl. The

M P ) Ibl IP, II 2 + a2 IIh, - a2 @+ar)

I12 143

I*

v

176

INFINITE-DIMENSIONAL SYSTEMS

If we choose a1 = b;; and u2 = by:, matrix R is negative definite if and only if (5.7.5) Therefore, if matrix A is stable and if inequalities (5.7.4) and (5.7.5) are satisfied, the trivial solution of composite system (5.7.2) is uniformly asymptotically stable, by Theorem 5.6.5, and exponentially stable, by Theorem 5.6.10. As a matter of fact, since subsystems (9,) and (Y2) satisfy Property B under the above conditions, it follows that the trivial solution of composite system (5.7.2) is uniformly asymptotically stable in the large by Theorem 5.6.9 and exponentially stable in the large by Theorem 5.6.1 1. 5.7.6. Remark. Since b I 22 0 and b,, 2 0 we can also apply Theorem 5.6.12. In this case the test matrix S is specified by

Matrix S has positive successive principal minors if and only if inequality (5.7.5) is satisfied. We have thus arrived at the same stability conditions as before. 5.7.7. Remark. Next, let us replace condition (5.7.4) by the condition u H 2 ( u ) 2 flu2

if

(5.7.8)

Iu( 2 R , .

Then Theorem 5.6.23 can be applied. Indeed, if matrix A is stable and if conditions (5.7.5) and (5.7.8) are true, then composite system (5.7.2) is uniformly ultimately bounded, by Theorem 5.6.23. 5.7.9. Remark. Next, let us assume that H , ( y , z 2 ) and H , ( z 2 ) are linear, having the special form H,(Y,Z,)

=f1(v)z2,

ffZ(Z2) =

Dz2

(5.7.10)

where > 0 is a constant andf, is a real function such that llf, 11 < co.In this case, composite system (5.7.2) determines a C,-semigroup on X = R" x L , ( G ) . If in particular G is the one-dimensional interval G = [ a , b ] ,then r in (5.7.5) can be replaced by ( b - ~ ) - ~ . 5.7.11. Remark. For composite system (5.7.2) boundary conditions different from (5.7.3) can also be used. For example, let us consider for Eq. (5.7.2) the boundary conditions dz2 ---(j,Y)

an

=

0

for all ( t , y ) E Rt x dG

5.7

SOME EXAMPLES AND APPLICATIONS

I77

where n denotes the outward unit normal on dC. For these boundary conditions we can compute

so that o2 = -p, r = 0, and ( ~ ~ = ~ r(2 r. The ) rest of the analysis is the same as before. Thus, if for composite system (5.7.2) the boundary condition (5.7.3) is replaced by the last boundary condition above, then the stability condition (5.7.5) is replaced by the condition (5.7.12)

5.7.13. Remark. Similarly, if we replace C by R", Z , = L,(R"), and if we assume (5.7.4), then condition (5.7.12) will still be the stability condition for (5.7.2). 5.7.14. Example. Let us reconsider hybrid system (5.7.2) with boundary conditions (5.7.3). However, in the present case we assume that A is an unstable matrix. Specifically, we assume there is a nonsingular real matrix B such that (5.7.15) where - A l is a stable ( k x k ) matrix, A , is a stable ( j x j ) matrix, k + j = n, and 1 i k 5 n. This transformation is always possible if A has at least one eigenvalue with positive real part and has no eigenvalue with zero real part. We allow the possibility that k = n , j = 0, and B = I, i.e., that A is completely unstable, and will discuss it separately later. Let w = Bz, so that

If we put

(5.7.16)

v

I 78

INFINITE-DIMENSIONAL SYSTEMS

with w 3 ( t , y ) = 0 for all ( t , y ) E R’ x aG. System (5.7.16) may now be viewed as an interconnection of three isolated subsystems of the form

aw3

- =f3(w3) at

=

~Aw, H2(w3).

(93)

Since the trivial solution of (9,) is completely unstable, there exists a symmetric positive definite matrix P such that the matrix

is positive definite. Let

Then

AITP1

+ PIAl = Q ,

u1 (w,)

= -WITPI

2 L(P1)lw1l2

-vl(wl)

and

=

vui(I.t’1,f1(wi))

w1.

-wiTewi 5

(P12(Iw,I)

-Arn(Qi)/wil2

A

OicP,3((wi/).

Also, since the trivial solution of (9, is) asymptotically stable, there exists a positive definite symmetric matrix P2 such that the matrix AzTP2

is negative definite. Let u,(w,)

Then

Vu2(~2,f2(~2))

+

P 2 A2 = -Q2

= W2=P2W2.

-L(Qz)Iwi12

Next, choose u3 (w3) =

0 2 ?23(IN’2I)-

(II w3 II 21,.

Then the same calculation as in Example 5.7.1 can be used to see that Vu3(w3,f3(w3))

I - a / G I v ~ 3’ \dY

5

-/isc Iw312dY ki

-(ar+P)(llw31!2)2

~3(P33(Ilw31!2).

Using the notation of Section 5.6 we have the estimate Vu1 (W!,>

91

(4)= 2W’lTPlbl

SGH( 1

Y , w3 ( Y ) )

I ~ II

5 2 1 %’I1 AM (PI)

1 G

lh t (Y)lI% (Y)i &

5 2 I w r 1I h ( P 1 ) lbl I llhl112 II ~ ~ 3 l l 2

so that b,,

= b,, = 0

and

613

= 2/l,,,(P1)~61~ Ilhll12.

5.7

179

SOME EXAMPLES AND APPLICATIONS

Similarly, we have the estimate

so that b,, = b,, = 0 and b23 Also, we have the estimate

= 2A,(P2)

lbzl llhl 112.

L

sothatb31 =211h21121~lI,b32=211h21121~21, and633 = O . We are now in a position to apply either Theorem 5.6.18 or Theorem 5.6.21. The test matrix of the latter assumes the form Arn(Q1)

s=[

0

-2Ilh2llzlC1l

0 Arn(Q2)

-2ll~2ll2lc2l

-2~,(P1)Ib,I

IlhIIl2

- ~ A M ( P ~ )llh1ll2 I~~I

p+ar

1

The successive principal minors of matrix S are positive if and only if the inequality Arn(Ql>Arn(QJ(P+a')

'4IIhllI2 l l ~ ~ l I ~ C ~ ~ ( ~ l ) ~ ~ ~ Q ~ > I ICiI

+ ~ M ( P z ) A ~ ~ ( QIczII I)I~zI

(5.7.17)

is satisfied. Thus, it follows from Theorem 5.6.21 that the trivial solution of composite system (5.7.2) is unstable if inequality (5.7.4), transformation (5.7.19, and inequality (5.7.17) hold. Finally note that in case A = A,, so that B = I in (5.7.19, then there is no w2-component and inequality (5.7.17) reduces to inequality (5.7.5). 5.7.18. Example. (Point Kinetics Model of a Coupled Core Nuclear Reactor.) In the present case we consider a somewhat more complicated example, motivated by a physical problem. Among other items, this problem illustrates the fact that the choice of subsystems and interconnecting structure is often obvious in applications. We consider the point kinetics model of a coupled core nuclear reactor with 1 cores (see Akcasu, Lillouche, and Shotkin [l], Plaza and Kohler [l])

v

180

INFINITE-DIMENSIONALSYSTEMS

described by the set of equations of the form

i

i k i ( t ) = nki[Pi(t)-c,i(r!)],

k

= =

1,2, . . . , I , 1,2,..., 6 ,

where pi: R + R and cki:R + R represent the power in the ith core and the concentration of the kth precursor in the ith core, respectively. The constants A i , gi,P k j , & j i , Pio, and n k i are all positive and

The functions hji E L , (R', R). They determine the coupling between cores due to neutron migration from thejth to the ith core. The function pi represents the reactivity of the ith core which we assume to have the form pi(t)

=z

m :J

w i ( t - s ) P i ( s ) ds

(5.7.20)

where wi E L , (R', I?). The functionsp,(t) and c k i ( t )are assumed to be known, bounded continuous functions on --03 2 t 5 0. The problem is to determine these functions and their stability for t > 0. If we make the physically realistic assumption that C k i ( t ) e Z k i ' + O as t + -00, then we can solve for ckiin terms ofp,, obtaining (5.7.21) Using Eqs. (5.7.20) and (5.7.21) to eliminate pi and cki from Eq. (5.7.19), we are left with I Volterra integrodifferential equations for pi([), i = 1, ...,I. In order to write these equations in a more convenient and compact form, we use the notation

K i

= AT'[&i++J,

ni(t) =

Af' wi(t),

5.7

181

SOME EXAMPLES AND APPLICATIONS

and z ' ( t ) = p i ( t ) on -a < t < 00. We have fii(t) = - K i p i ( t )

+

c

j=l,i#j

+

F i ( t - s ) p i ( s ) ds + p i ( ? )

J

~ ~ ( t - s ) p ~ds, (s)

i

=

-m

c'

n i ( t - s ) p i ( s ) ds

1, ...,I,

for t 2 0 and p i ( t ) = q i ( t ) given on -a < t < 0. Now let Zi be the fading memory space of all measurable functions that = Iqi(o)l2

IIqi112

+J

0

-m

Iqi(s)lZeL'sds

where Li > 0 is some constant which will be specified later. Let X Then Eq. (5.7.19) can be expressed as

+1

0

2 ( t ) = -Kiz,'(o)

ro

+ C J i

j=l,i#j

'pisuch

Fi( -s)z,'(s) ds

+ Z,'(O)

1"

i

1 ,...,I,

~,(-s)z,j(s)ds,

=

Hi(

-m

=

Xf=

Zi.

-s)zl'(s) ds

(5.7.22)

with zoi = 'pi E Zigiven. (Refer to Section 5.5, Part C , for a brief discussion of Volterra integrodifferential equations.) Composite system (5.7.22) may be viewed as an interconnection of I isolated subsystems (q) described by equations of the form

[

0

-KizI'(0) + J - m F i ( - s ) z ; ( s ) ds

with interconnecting structure characterized by

For each subsystem

(q) we choose

v

182

INFINITE-DIMENSIONAL SYSTEMS

and

+

zi(s)fi(zi)eLiSds.

Vv,(z',f;-(z'))= 2z'(O)fi(z') 2Ki

O m -!

If 9,E D (A),then 4, E Z ; , and Iim

'pi(t)'eLi' =

0.

t--a,

Integrating by parts, we obtain rn

J

rn

z'(u)fr(zi)(u)eLiudu= J

-a,

zi(s)2(s)eLi"ds

/

-a,

= +z'(O)' - (Li/2) =

+

0

[z'(s)l2eLis ds

--OD

[Z'(O)]2

- (Li/2)biz

where b, is defined in an obvious way. Now assume that Li > 0 can be chosen so that ci

TIZ

(

c[Fi(u)]2eLi"du <

00

and

The definitions for ci and di, integration by parts, and the Schwarz inequality can now be used to show that

V V , ( Z ' , ~ , ( Z=' )2) ~ ' ( 0 )-Kiz'(O)+

e

F ~ ( - s ) z ' ( s )ds

+ z i ( o )/-oa,ni(-s)zi(s)ds

1

+ 2Ki [ + ( ~ ' ( O ) ) ~ - + L i b i ' ] 5 -Ki[z'(0)J2+ 2ciz'(0)bi - KiLib: + 2 d i [ ~ ' ( 0 ) ] Z b i -

(5.7.23)

5.8

FUNCTIONAL DIFFERENTIAL EQUATIONS

I83

The first three terms in (5.7.23) form a quadratic form which is negative definite if Kid% > ci. (5.7.24) The fourth term in (5.7.23) is cubic so that in a neighborhood of the origin, VVi(Z',fi(Z')) < Oi[(zi(0))'+bi'] for 0.= - Ki(Li+

1)

+

[K.(L;- 1)'

(5.7.25)

2

(c

Next, we consider the interconnecting structure. We assume that

cii = Then

[Gii(s)]'eLjsds

< co.

>"'

Thus, the hypotheses of Theorem 5.6.12, part (iii), are satisfied, provided that the test matrix S = [sJ has positive successive principal minors, where

and where oi is specified by Eq. (5.7.25). For example, if I = 2, then the trivial solution of composite system (5.7.19) is exponentially stable if 0102

>4~12~~1.

Thus, if in a two core coupled nuclear reactor each core is exponentially stable and if the coupling between cores via neutron migration is sufficiently weak, then the reactor is exponentially stable. 5.8 Functional Differential Equations-Special R esuIts

Functional differential equations with finite delays determine semigroups in the space C,, as outlined in Section 5.5, Part B. As such, the general stability results for composite systems, presented in Section 5.6, can be applied

v

184

INFIN1TE-DIMENSIONAL SYSTEMS

to systems described by functional differential equations. However, equations of this type have a special form and as such, they have special Lyapunov theorems and comparison theorems which are not applicable to general dynamical systems. For further details and examples, refer t o Yoshizawa [l, Chapter 81, Lakshmikantham and Leela [2, Chapters 6-81, and Driver [ 11. In the process of characterizing the qualitative properties of isolated subsystems, we give two examples of Lyapunov theorems for functional differential equations, which are useful in studying composite systems. Then we present two sample stability theorems for interconnected systems. In the present section the notation of Section 5.5, Part B, is employed. In particular we let Cyr denote the set of all continuous functions $i: [ - r , 01 + R"' with norm defined by

il$ili

=

max{l$'(t)l: - r I t 5 0).

We consider systems which may be described by equations of the form

i'(t) =f;(z,')

+

j = l ,i # j

gu(zr'),

i

=

I , ..., 1

(5.8.1)

where Letting

x:=ni = n, I

I

and system (5.8.1) may be rewritten as k, ( t ) = f ( X , >

+ S ( X J LL

(X,).

(5.8.2)

Composite system (5.8.2) with decomposition (5.8.1) may be viewed as a n interconnection of I isolated subsystems (q) described by equations of the form

2(t)

= f,(z,i)

with interconnecting strucure characterized by

(a

5.8

185

FUNCTIONAL DIFFERENTIAL EQUATIONS

Henceforth we assume that the right-hand sides of Eqs. (5.8.1), (5.8.2), and

(.q)are sufficiently smooth so that their initial value problems have unique solutions. We also assume that these equations possess the trivial solution.

5.8.3. Definition. Isolated subsystem (q) is said to have Property F1 ifthere is a , i 2qi3 , E K, continuous functional ui($') defined on C:,, three functions q i l q and constants ci > 0 and Li > 0, such that the inequalities

(i> qil(II$iII) U i ( I c l i ) 5 qi 2(I I $'I I )> (ii) DUi(Yij($') 5 - ci (pi3 (I/ IIj,and (iii) I ~ i ( $ 1 ~ ) - u i ( $ 2 9 I 5 Li II$11-$2Lll hold for all I)",t,kl',

$2i

in a neighborhood

il$ill

< Hi of C:,,

If isolated subsystem (q)possesses Property FI, then, as shown in Yoshizawa [1, pp. 189-1 921, the trivial solution of subsystem is uniformly asymptotically stable. (Note that if composite system (5.8.2) possesses Property FI, where this property is rephrased for the appropriate space Cr, then the trivial solution of system (5.8.2) is uniformly asymptotically stable.)

(z)

5.8.4. Theorem. Assume that for composite system (5.8.2) with decomposition (5.8. I), the following conditions hold.

(i) Each isolated subsystem (q) possesses Property F1; (ii) for each i , j = 1, ..., I, i Zj, there are constants k , 2 0 such that Ig,($j)ts~,qj3(II$jII) for all Il$jII
if i = j -Lik,,

if i f j

are all positive. Then the trivial solution of composite system (5.8.2) is uniformly asymptotically stable. Proof. Given ui of hypothesis (i), let tli > 0, i = 1, ..., I, be arbitrary constants and choose as a Lyapunov function for system (5.8.2),

v

186

INFINITE-DIMENSIONAL SYSTEMS

Moreover, the derivative Du($) along solutions of Eq. (5.8.2) can be estimated as (see Yoshizawa [1, pp. 186-1 891) 1

5

i

i

j = l , i # j~ i k , j q j 3 ( 1 1 ~ 1 1 )

i = 1 ai {-ciqi3(ll+il\)+

= -ciTSU,

where ciT = ( a l , ...,@/I,uT = ( q 1 3 ( ~ ~ ...,qt3(ll$rll)), $1~~), and S is the test matrix given in hypothesis (iii). The results in Chapter 2 on M-matrices imply that ci can be picked in such a way that ciTS> 0 (see in particular the proof of Theorem 2.5.15). Thus, there is a function q3E K such that D4$) 5 -gTSu 5

-v3(ll$ll).

(5.8.6)

It now follows from inequalities (5.8.5) and (5.8.6) (and from the remark following Definition 5.8.3) that the trivial solution of composite system (5.8.2) is uniformly asymptotically stable. 5.8.7. Definition. Isolated subsystem (Sq) is said to possess Property F2 if there is a continuous function ui(i,hi), three functions q i l ,q i 2qi3 , E K , and positive constants ci and L i , such that the inequalities (i) ' ~ i (I$'I> l 5 ui(Gi> qi~(II$iII)~ (ii> Dui(yt?($')5 -ciqi3(I$i(?>I>> and (iii) Iui($lI)-ui($zi)I Li l I $ ~ ' - $ z ' l l hold for all $ l i , $ 2 i , $ i

in a neighborhood

\l$ill

< Hiof C,?.

If isolated subsystem (q) possesses Property F2, then the trivial solution of (q) is uniformly asymptotically stable (see Yoshizawa [ l , pp. 189-1921). (When rephrased for the space C,", Property F2 implies the uniform asymptotic stability of composite system (5.8.2).) 5.8.8. Definition. The interconnections g , in Eq. (5.8.1) are said to be delay free if there are functions G, such that

9,w = Gij($j(O))? i,j = I , ..., I , i # j , for all

l\$jl\

< Hi.

5.8.9. Theorem. The trivial solution of composite system (5.8.2) with decomposition (5.8. I ) is uniformly asymptotically stable if the following conditions hold.

5.9 APPLICATION OF COMPARISON THEOREMS

187

(a

(i) Each isolated subsystem possesses Property F2; (ii) all interconnections gii are delay free and satisfy the inequalities Igij($j)l = lG~($j(O))l kgqj3(I$j(O)I)

for 11$' 1 1< H i ; and all successive principal minors of the test matrix S (iii) s.. IJ =

are positive.

(

= [sJ

defined by

if i = j

-Lik,,

if

i #j

Proof. The proof is essentially the same as the proof of Theorem 5.8.4, once Theorem 5.5.2 is noted. 5.9 Application of C o m p a r i s o n T h e o r e m s t o V e c t o r Lyapunov F u n c t i o n s

Comparison theorems can be applied to vector Lyapunov functions in the qualitative analysis of dyanmical systems defined on infinite-dimensional spaces. Since these comparison theorems can be formulated to make use of the maximum principle for parabolic partial differential equations, their use has the potential to yield significant improvement over the scalar and vector Lyapunov theory outlined thus far. However, at this time it is not yet clear whether the full power of the theory involving this maximum principle can be used to advantage in the qualitative theory of interconnected systems. Indeed, a good deal of research will be required in this area. For this reason we treat this subject very briefly. For further information on the maximum principle for parabolic partial differential equations, refer to Matrosov [ I , 21, Walter [ I , especially p. 2691, and Lakshmikantham and Leela [ I, 21. In the present section the notation established in Section 5.5, Part D, is used. Let G be a bounded open subset of R" with smooth boundary dG. Given any twice continuously differentiable function u : R i x G + R', with components uk,we define u,k and u:, as u,"

= (u:',u:2'

...)u:-),

u, = ( U , l , ...,u,I)

and u:, = [u:~,,], i , j = 1, ...7n,

u,,

=

[u;,],

k = 1, ..., I ,

where uti denotes the first partial derivative of uk with respect to xi while utiXiis the second partial derivative of uk with respect to x i . For a vector

v

I88

INFINITE-DIMENSIONAL SYSTEMS

Lyapunov function u ( x , u ( x ) ) the u,k and u;, are similarly defined via the chain rule. We now consider a partial differential operator (5.9.1)

T(w) = wt - F ( x , W , w,, w,,)

where x E G and M' E R' while F has components F i ( x , w,w;, w h ) mapping G x R' x R" x R"' into R. 5.9.2. Definition. The function F ( x , w, w,,w,,) is called elliptic in G if for any x E G, any w E R', any P E R", and any symmetric matrices R and Q, where R - Q is positive semidefinite, each component Fi of F satisfies the inequality

Fi ( x , W , P , Q ) I Fi (x, W , P,R). The operator T i n Eq. (5.9.1) is called parabolic if F is elliptic. Notice that these definitions differ from the earlier definitions of strongly elliptic and parabolic, given in Section 5.5, Part D. We now consider the class of problems characterized by the set of equations for

ut = F(x,u,u,,u,,) u(0,x) =

(t,x) E

for x

$ l ( 4

E

G,

for (/,x)

u(t,x) = 0

E

R+ x G E

(5.9.3)

C(G)

R C x aG.

We also consider the ordinary differential equation

P.

=

dY),

(5.9.4)

Y(0) = Yo

where g: R' R' and g is sufficiently smooth for existence and uniqueness of solutions of (5.9.4). For the proof of the following comparison theorems and further details, see Lakshmikantham and Leela 12, Chapter lo]. ---f

5.9.5. Theorem. Assume that q l , q z E K , that u ( t , x ) solves Eq. (5.9.3), that y ( t ) solves Eq. (5.9.4), and that the following hypotheses hold.

(i) F and H are defined and continuous on G x R' x R"" x R'."' into R'; (ii) H is elliptic on G, u : G x R' R', and ---f

where d v j d u = [ d v i / d u j ] ,i,,j = I , ...,I ; and (iii) y is quasimonotone, g(0) = 0, H ( x , u, 0,O) I g ( v ) for all (x, v) c' 2 0, and u(x, $ 1 (x)) I yo for all x E G.

E

G x R',

Then u(x,u ( t , x)) I y ( r ) for all t 2 0. Moreover, if in addition to hypotheses (i)-(iii) we assume that

5.9

APPLICATION OF COMPARISON THEOREMS

189

(iv) ( p l ( l u l ) _< u(x, u) I (pz(lul) for all x E G, E R', u 2 0; and (v) the trivial solution of Eq. (5.9.4) is uniformly asymptotically stable, then the trivial solution of Eq. (5.9.3) is uniformly asymptotically stable on the space C(C). Similar results, as the above, can be proved for the case when the trivial solution of Eq. (5.9.4) is uniformly asymptotically stable in the large or exponentially stable. Moreover, the boundary condition in Eq. (5.9.3) may be replaced by the conditions u(t,x) =

0

for x E

ac - aG,

and a(x)

au

( t ,X)

=

Q ( x ,U )

on aG,

where CI E C(BG), aG, = { x E i3G : ct > 0}, n denotes the outer normal to a point x E aG,, and Q is quasimonotone in u. For these boundary conditions, we need in addition to (i), (ii), and (iii) of Theorem 5.9.5, the condition

As a simple example, let Gk be a bounded domain in R" with smooth boundary aG, and define two operators

2 ej + z b j * E n

azu

Lk(U)

i,j=l

xi

j= 1

for k = 1,2, where a$ E R , bj* E R , and Lk is strongly elliptic. Next, consider isolated subsystems described by equations of the form auk/at = Lk(uk) - ck uk

(O,

=

$k

uk(t,x)

=

0

uk

on Rf x Gk

(%I

on Gk

(x)

on R + x aGk

with $k E c ( G k ) . We now form an interconnected system by considering interconnecting terms of the form gk(U1,uz) =

Bkuj,

j # k

where Bi E R.Choosing uk(uk)

= 3(uk)z

and

k,j

=

1,2,

v

190

INFINITE-DIMENSIONAL SYSTEMS

the derivative of u k for the resulting composite system can be obtained by the following series of computations. DVk

=

Llk

Lk(uk) - Ck U k 2

+ Bk U k U j

In accordance with Theorem 5.9.5, we can take g(v) = - 2

c1u1 -

P I I Ju, Ju2 - -

-1B2IJduz+c2u2

1.

JG,

For this g the transformation w i= i = 1 , 2 , linearizes d = g ( v ) . The trivial solution of Eq. (5.9.4) is asymptotically stable if c, > 0, c2 > 0, and c1 c2

>1 41 1&1.

The same analysis works if one or both of the boundary conditions u k ( t ,x) = 0 on aGk is replaced by

where

(Tk

> 0 is a constant and n is the outward normal on aGk. 5.10

Notes and References

For the proofs of the results given in Section 5.2 and for additional results on linear semigroups refer to the book by Hille and Phillips [I] which is encyclopedic. Refer also to the book by Krein [ l , Chapter 13 which has an especially good exposition from the point of view of abstract differential equations. Consult also Pazy’s notes [ I ] which have many useful results including material on differentiable and on compact semigroups not found elsewhere in book form. For a more complete discussion and for proofs of the results given in Section 5.3, see Crandall [l], Crandall and Liggett [ l ] , Brezis [l, 21, and Kurtz [l]. For further results on existence and nonexistence of solutions of

5.10

NOTES AND REFERENCES

191

initial value problems (N), when A is continuous, see Godunov [l] where it is shown that for any given infinite-dimensional space X there is a continuous A ( x ) and an initial condition xo such that (N) has no solution. Also, see Lasota and Yorke [11 where it is shown that for “most A (x) and most xo” there must be a solution. General references on stability results presented in Section 5.4 include Hahn [2], Yoshizawa [l], and Krasovskii [l]. Some interesting material on stability theory in a Banach space setting can also be found in Lakshmikantham [l], Massera [l], and Massera and Schaeffer [l]. Webb [11 studied functional differential equations as quasicontinuous semigroups. Proofs and additional results on the Lyapunov theory for functional differential equations (Section 5.5, Part B) can be found in Driver [l] and in the books by Yoshizawa [I], Hale [2], and Krasovskii [I]. Hale [3,4] studied functional differential equations with infinite delay when Xis a fading memory space. The results on Volterra integrodifferential equations in Section 5.5,. Part C , are based on the references by Barbu and Grossman [I], Burns and Herdman [l], and Miller [l]. For more information and examples of well posed linear partial differential equations (see Section 5.5, Part D), refer to Hille and Phillips [I], Krein [1, Chapter 1 , Section 81, and Pazy [ 1, Chapter 51. For nonlinear problems described by partial differential equations, see Crandall [11, Crandall and Liggett [l], Brezis [I, 21, and Kurtz [l]. For examples and hints on how to construct Lyapunov functions for partial differential equations, see Infante and Walker [I], Walker [I], Chaffee and Infante [I], Matrosov [2], Sirazetdinov [l], Zubov [ I , Chapter 51, Wang [l, 21, Pao [l], and Plant and Infante [I]. Some results on combining Lyapunov functions and the idea of invariant sets can be found in Walker and Infante [l] and Defermos and Slemrod [I]. For general references on stochastic processes see Doob [13 and Wong [I11. For stability results of systems described by stochastic differential equations and stochastic difference equations (Section 5.5, Part E) see Kushner [I-31, Arnold [l], and Kozin [I]. The stability results for interconnected systems in Section 5.6 and the examples in Section 5.7 are expansions and modifications of work reported b y Rasmussen and Michei [2, 41 and Rasmussen [I]. Matrosov [a] has surveyed some related literature. Theorem 5.6.2 is good enough for many applications, however, it would be interesting to obtain a theorem of this type which allows a mixed group of quasicontractive subsystems and linear Co subsystems. Note that Theorems 5.6.1 and 5.6.2 are stability theorems for composite system (9) in the sense that well posedness is a weak type of stability.

192

v

INFINITE-DIMENSIONAL SYSTEMS

The results in Section 5.8 are modifications of work by Michel [S, 71. Many other modifications are possible for composite systems described by functional differential equations, along the lines of the results of Chapter 11. Specifically, global asymptotic stability, exponential stability, instability, boundedness, as well as results for time varying systems can readily be established. Note also that Theorem 5.8.4 can be proved for a general dynamical system in a Banach space.

CHAPTER VI

of Large Scale Systems

In the present chapter we consider the input-output stability and instability of time-varying nonlinear interconnected systems. As in the preceding chapters the objective is still to analyze large scale systems in terms of their lower order subsystems and in terms of their interconnecting structure. By the term input-output stability we mean boundedness and/or continuity of the relations (describing a dynamical system) which connect system inputs to system outputs. Here we note that the motivation for input-output stability has its roots in systems engineering, where frequently systems are viewed as “black boxes” (i.e., relations) which connect input signals to output signals. In this context, stable systems are those for which bounded input signals result in bounded output signals, and/or small changes in input signals result in small changes in output signals. Of particular interest in the present chapter are stability results in the setting of &-space, I,-space, &,-space, and I,-space. Wherever possible, we emphasize results which allow analysis based on graphical methods. The first section contains required background material. Specifically, we I93

194

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

establish in this section the necessary notation and recall several useful general results from systems theory. In the second and third sections we present the main stability and instability results, respectively, in a rather general form. In the remainder of this chapter, as well as in the next chapter, we establish several special stability and instability results and we consider specific examples to illustrate applications and usefulness of the method of analysis advanced. Specifically, in the fourth section we combine a special transformation with some theorems from Section 6.2 to obtain several specific useful results involving sector and conicity conditions in a Hilbert space setting. The fifth section contains Popov-type results for composite dynamical systems while in the sixth section stability results for systems defined on L,- and 1,-spaces are presented. Finally, in the seventh section a design and compensation procedure, using the viewpoint of treating dynamical systems in terms of lower order subsystems and interconnecting structure, is proposed. The chapter is concluded in the eighth section with a discussion of references. 6.1

Preliminaries

The present section consists of five parts. First we present several classes of useful function spaces. Next, we discuss properties of relations on extended spaces. Then we consider an important class of nonlinear operators on extended spaces. This is followed by a discussion of an important class of linear operators on extended spaces. The section is concluded with the statement of two types of input-output stability theorems. The first type involves gains of relations (small gain theorem) while the second involves sector conditions. A. Extended Function Spaces

In this chapter we again let L, = L,(R+, R ) denote the space of Lebesgue measurable functionsf: R + + R with norm

while L," = L , ( R + , R") is the corresponding space of vector valued functions f : R + + R" with norm /If ItLp. The spaces L , and L," are defined in a similar manner (refer to Section 5. I). The truncation of a function f E L, or a function f E L,"' at time T is denoted by fT and is defined by

6.1

195

PRELIMINARIES

The extended space Lre is defined as

Lre = {f: R+ -+ R" :fT

E

Lp" for all T > O } .

Given f E Lre, we let llfllT

=

~ l f T ~ ~ L p ~

Note that llfllT = 0 does not mean that f ( t ) = 0 on R + since in this case If(f(t)lcan be nonzero for t > T . Therefore, the function 11. JIT is a pseudonorm or seminorm (see Hille and Phillips [l, p. 1371) rather than a norm. Let C denote the space of all continuous functions f : Ri -+ R" and let BC = C n L," denote the subspace of bounded and continuous functions. If we put

Ilf

for any f

E

/IT

=

sup{/f(t)l:

1 5

C and 119Ilm = sup{lg(t)l: 0 5 t <

..>

for g E BC, then it is clear that C can be thought of as an extension of BC. Other subspaces of C are also of interest. For example, the space of continuous functions which have a finite limit at infinity, C , , is defined as

C, = { f C~: f ( t )-+f(co) as t -+ co,f(co)

E R"};

the space of continuous functions which tend to zero, C,, is given by

c, = {fEC,:f(..)=O}; the space of asymptotically (or ultimately) periodic functions, A , , is specified by A , = { f = p + e : e E C O , p E C ,a n d p ( t + o ) = p ( t ) forall Z E R } ;

or, the space of asymptotically (or ultimately) almost periodic functions, A P , is defined as

AP

=

{f= p + e : e E C,, p

E

C , and p is almost periodic}

(see Besicovitch [I] or Fink [I] for a discussion of almost periodic functions). All of these spaces have the same norm as BC, namely )I , . Also of interest will be weighted spaces. Let X be any of the spaces Lpm,BC, C , , C,, or A , and let g E C be a given positive scalar function. Then the weighted space X , consists of all$ R + -+ R" such thatflg E X . In particular, letf/g be defined by ( f l g ) ( t ) =f(t)/g(t) and let n=

{ f Lye ~ :f/SE Lp"}

with norm (6.1.1)

196

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Also,

(BC), =

{f€

c :flg E BC}

with norm

ll.flI, = sup{lS(t)l/g(t):

0 5 t < a>.

Weighted spaces for the cases when X = C , , Co, or A , are defined similarly. The natural extended space for the spaces (BC),, (CJ,, (C,),, o r ( A & is the space ( C ) , , where ( C ) , is the space C with the new seminorms defined by IIfilT =

sup{I.f(t)lMt): 0 5 f

TI-

The natural extended space for (L,”), is the space L;e with the new seminorms defined by

IISllT = ~ ~ [ l f ( ~ ) l / g ( r ) l , d r } l ’ p .

(6. I .2)

Note that in the above, g ( r ) 3 1 is allowed and that for this special case, = 2, (LP”),has a Hilbert space structure, i.e., both the seminorms (6.1.2) and the norm (6.1. I ) are induced by

X, = X. Also note that for p ( f i J 2 ) T

A

pi 0

(r)Tfi(r)dr)-2dr

( A , ? h 2 ) , 4 r h , (r)Th2(r)g(r)-2dr while (ilfllT)2= ( f , f ) T and

g = e” is defined by

(Il f l , )’

=

( J f ) , . In the special case where

g ( t ) = exp(at)

for some real number a, (L2”),is the well-known shifted L,-space. Next, let S, denote the space of all Rm-sequences {a,,: n = 0,1,2, ...} = {a,} c R”.Then I,“ is the subset of S, for which the norm, defined by

is finite. Let g = {g,,} be a given real valued sequence with g,, > 0 for all n. We now define (/,”), as the set of elements of S, for which

The extended space of (l,”), is S, with seminorms given by

6.1

197

PRELIMINARIES

for T = 0,1,2, ... . Similarly, we define cl as cl =

and c,, as

{

{a,} E S, : lim a, = a, exists as n + co, a, E R"' n+m

co = {{a,} E c1: a, = O }

with norm given by II{a,>Ilm = sup{la,l: n = 0,1,2,

...>.

The extended space for cI and for co is S, with seminorms

II(Q,)I\~

=

max{la,l: n = 0,1,2 ,,.., T ) .

The spaces ( c ~ and ) ~ (cJScan now be defined in an obvious way. From the above discussion it is clear that the number of possible spaces which one may want to consider is rather large. This enables us to construct particular spaces to meet specific needs in various applications. For example, the space of measurable, asymptotically periodic functions

AM,

=

{ f = p + e : eELp"',pELp[O,co], andp( t+co)=p(t) )

is sometimes useful in applications, particularly when p = I , 2. We also remark that the idea of weighted spaces can be generalized by replacing the scalar function g ( t ) by a matrix weight function G ( t ) (see Gollwitzer [l] and Dotseth [13). Using the spaces discussed thus far as motivation, we characterize an extended space as follows. 6.1.3. Definition. An extended space X , is a linear space of functions f: {0,1,2, ...) R") and a family of seminorms T ) for any T E R + (or any T E {0,1,2,. . .}) such that { 11 ( X J T 4 {fT: f E X,} is a Banach space under 11. / I T , and the set (i) X , , lifilT is a nondecreasing function of T. for eachfe (ii)

f : R + --t R"' (or sequences

--f

Note that for all spaces discussed above we have the following property. 6.1.4. Definition. A subspace X of an extended space X , is called a completely compatible subspace of X , if the norm 11.11 of X is determined from the semi-

norms of X , by the relation

llfll = TIim IlfllT, +m These definitions allow for the possibility of mixing the basic extended spaces L;, and C. For example, X , could represent the four-dimensional vector valued functions whose coordinates (f,', f2T,,'3f f4T) E X , =

198

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INPUT4UTPUT STABILITY OF LARGE SCALE SYSTEMS

x C x C. If X I = (L,"),, X , = (Lqm),,, X , = BC, and X , = C,,, (L;,), x then X = XI x X , x X , x X , is a completely compatible subspace of X , . The spaces BC, C , , C,,A,, and A P could all be thought of assubspaces of Lre for some p E [l, a],while the space X = Lpmn BC with norm

llfll

=

sup(If(t)l:

fE

R+l+{~lf(t)lpdt}l'p

is a subspace of LFe or of C. These subspaces are not completely compatible but they d o possess the following property. 6.1.5. Definition. A subspace Xof an extended space X,is said to be compatible if the condition lim Ilf,-fllx = 0 n-tm

forf, andfin X , implies that lim

for all T > 0.

n-tm

Ilf-fllT

=0

B. Relations on Extended Function Spaces The ideas of a multivalued function H a n d of a relation H were discussed in Section 5.5 and we shall use the notation and definitions introduced there. In particular, if H is a relation on an extended space X , (i.e,, H i s a subset of the product space X , x X,) and if (x, y ) E H , we write x E D(H) and y E Hx c Ra(H). We often will need to consider the inverse of H, denoted by H - ' ( H - ' always exists) and we often find it convenient to identify H with its graph . A relation H defined on X , is called causal if (HX)T =

(HXT)7

for all T > 0 and all x E X,. A relation H is time invariant if it commutes with all time delays and memoryless if the value of Hx at time t depends only on the value of x at time t . Given a relation H on X , and given a subspace X , we let S ( H ) denote the stable manifold of H , defined as S(H)= (xEX:HxcX)

and we say that H is bounded with respect to X (or for short, bounded) if S ( H ) = x. The conditional gain of H (if it exists) is defined by g , ( H ) = inf{M: llHxllT I MllxllT for all x E S ( H ) and all T > O}.

6.1

199

PRELIMINARIES

If H is bounded, then the gain of H (if it exists) is defined by

g ( H ) = inf{M: llHxllT

MllxllT for all

XE

X , and all T > 0).

In this case S ( H ) = Xand g,(H) = g ( H ) . We call a relation H on X , interior conic (c,r) if there are real numbers r 2 O and c such that ~ ~ y - c x ~ ~ Tfor~ all r ~T>O, ~ x ~all~ X , E X , , and all ( X , Y ) E H . InthiscaseHisboundedandhasgaing(H) I max{lc+rl,Ic-rl}. The numbers c and r are called the center and radius of H , respectively. Similarly, a relation H is called exterior conic (c, r ) if the inequality above is replaced by (1 y - cx 1 2 r llxll for all T > 0, all x E X , and all (x, y ) E H . Next, let us assume that the extended space X , has Hilbert space structure. We say that a relation H is inside the sector {a,b} if for all (x, y ) E H and all T > 0, the inequality (y-ax, y - b x ) , I 0

holds, where a and b are real constants. We say that H is outside the sector {a,b} if for all (x, y ) E H a n d T > 0, the inequality (y-ax, y-bx),

2 0

is satisfied. The relation H i s said to be positive if (x, y ) T 2 0 for all (x, y ) E H and all T > 0. In these definitions no particular ordering of a and b is implied. Thus, the statement “ H is inside the sector {a,b}” is equivalent to the statement “H is inside the sector {b,a}.” If in particular c = (a+b)/2 and r = lb-a1/2, it is easy to see that H i s inside (outside) the sector {a,b} if and only if it is interior (exterior) conic (c, r). It will sometimes be necessary to consider relations H with domain in one extended space XI, and range in a second extended space X,, (i.e., H is a subset of X , , x X,,). In this case we will assume X , and X , are compatible or completely compatible subspaces. The gain of such a relation H i s defined by

g(H)= inf{M: IIHxII,IMJ(x(lTforall T > O andall EX^,}. Moreover, H is called bounded with respect to (A’,,X,) if Hx E X , whenever

x E XI.

In order to discuss the continuity of dynamical systems, we introduce the corresponding incremental definitions, but only for relations which are single valued (Le., H must be an operator). Thus, the conditional incremental gain, g^,(H), is the smallest M 2 0 such that IIHx-Hyll, I Mllx-yll. for all T > 0 and all x, y in the stable manifold S ( H ) . If in particular S ( H ) = X , then g^,(H) = #(H) is the incremental gain. Also, H i s said to be incrementally interior conic ( c , r ) if r 2 0, C E R, and II(Hx-Hy)-c(x-y)ll, 5 rl\x-yllT for all T > 0 and all x, y E X,. Similarly, H is said to be incrementally exterior conic ( c , r ) if r 2 0 , C E R ,and II(HX-H~)-C(X-~)~~.~VI/X-~I/T for all

VI

200

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

T > 0 and all x,y E X,. If X , has Hilbert space structure, we define relation H to be incrementally inside (or outside) of the sector {a,b } if

( ( H x - N y ) - ~ ( x - y ) ,( H x - H y ) - b ( ~ - y ) ) , I 0

(or 2 0 ) for all x , y E X , and all T > 0. In this case we also define H to be incrementally positive if ( x - y , H x - H y ) , 2 0 for all x , y E X , and all T > 0. Note that if HO = 0 and H satisfies any of the incremental definitions, then it will also satisfy the corresponding nonincremental definitions. Now let H = [ H J be a matrix such that H , is a relation on X j e to Xi,(i.e., H , is a subset of X j , x Xi,)with gain g ( H @ < ) co. We define G ( H ) = [ g ( H & ] as the matrix of these gains. Similarly, we define e ( H ) = [ g ( H & ] as the matrix of incremental gains.

C. A Class of Nonlinear Operators Next, we characterize an important class of operators. 6.1.6. Definition. Let Jf denote the class of operators from X , into X , such that E .A"' means ( E x ) ( t ) N ( x ( t ) ,t)for allx E X , and all t 2 0, N ( 0 , t ) = 0 for all t L 0, and there is a constant F 2 0 such that lN(y,t)l 5

for all y

E

Flvl

(6.1.7)

R" and all t 2 0. (Subsequently we write N in place of N.)

Any operator N E JV is causal and memoryless. I t is time invariant if and only if the function N ( x , t ) is time invariant and linear if and only if N ( x , t ) is linear. If in particular X = Lpm,BC, or C,, then the gain with respect to X satisfies g ( N ) < F. If X = C , and N is time invariant, then N is bounded on X and g ( N ) < F. We call N E Jtr interior conic (c, r ) if IN(x, 1 ) - cxI 5 r 1x1 for all x E R" and all t 2 0, exterior conic (c, r ) if IN(x, t ) - cxI 2 r 1x1 for all x E R" and all t 2 0, and positive if x T N ( x ,t ) 2 0 for all x E R"' and all f 2 0. Replacing inequality (6.1.7) by the inequality lN(x,t)-N(v,t)l

Flx-yl

for all x , y E R" and all t 2 0, we define and characterize the incremental counterpart of class Jf, denoted by &, in an obvious way. For the discrete time case we define the class of operators q by replacing t 2 0 by j = 0,1,2, .. ., N by n, and inequality (6.1.7) by I n ( x J l i Flxl

for all x E R" and all j = 0, I , 2, ... . The incremental counterpart of class q , denoted by 4, is characterized by making obvious modifications.

6.1

201

PRELIMINARIES

D. A Class of Linear Operators We now consider a large class of linear, causal, time invariant operators defined on various extended spaces X , . We present sufficient conditions for boundedness of these operators and establish estimates of the gain of these operators defined on various spaces X , .

6.1.8. Definition. Let 9, denote the set of all operators H of the following form. There is a sequence { h j ) of m x r n real matrices, a sequence of real numbers { t i } , 0 = to < t , < t , < ... < t,-+ 00 as n + co, and an m x r n matrix valued function h ( t ) = [hii(t)] such that all h, E L , , and such that for all x E X , , (6.1.9) 6.1.10. Definition. The set 9 is that subset of 9, such that

5 ]hi\ < co

j=O

and

j)(r)ldt

< co.

(6.1.11)

The set Y l eis that subset of Yesuch that hi = 0 for i > 0 and Yo,is the subset of Yle such that h, = 0. Also, TIis defined as Y,, n 9and Y ois defined as 9o= 9 n Yo,. The subscripts in Tland Y orefer to the number of elements hj which are allowed to be nonzero. It will be convenient to assume that the functions x in any extended space X , are defined as x ( t ) = 0 for all t < 0. The function h ( t ) in Eq. (6.1.9) will also be taken as zero for all t < 0. With this proviso, the infinite sum in Eq. (6.1.9) reduces to a finite sum at any particular time t 2 0 and is thus always defined. Furthermore, in this case the integral in Eq. (6. I .9) can be written as

since x ( z ) = 0 if z < 0 and h ( t - z ) 6.1.12. Theorem. If x then H x E LTe.

E

=0

if z > r.

C and H E LYoe then H x E C. If x E L;, and H E 9,,

Proof Both results are known when H E Y o ,(see, e.g., Miller [I, pp. 165-167, 2731). To finish the proof assume that x E LTe and H E 9, with

1hjx(t-tj) + W

Hx(t) =

j=O

sb

h ( t - z ) x ( z ) dz.

In view of the first sentence of the proof we can now take h ( t ) E 0 so that only

202

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

x.j"=,

the sum h j x ( t - t i ) is left to consider. Since ti co as,j+ 00 then on any finite interval 0 _< t _< T only a finite number of terms of the infinite sum are nonzero. Since each term is in L,([O,T],R"), so is the finite sum and IIHxIIT 5 (Cj"=oIhjI)IIxIIT. w --f

The above argument can also be used to prove the next result. 6.1.13. Theorem. If H E 9 and 1 I p 5 co, then H is bounded on L," and

The proof of the next theorem can be found in Corduneanu [2, Section 2.71 or in Miller [I, Chapter V]. 6.1.14. Theorem. If H E 9l and X = BC, C , , C,, A , , or A P , then H is bounded on X and satisfies the estimate g(H) 5

Pol

+ [lh(t)l

dt.

The next result is a special case of a theorem due to Corduneanu (see Corduneanu [2, Section 2.71 or Miller [l, pp. 261-2621). 6.1.15. Theorem. If H E POe and XI is (BC),, and X , is the corresponding space ( B C ) , , then H is bounded with respect to (XI,X,)if and only if there is a constant M 2 0 such that

f d l h ( f - z ) / g ( r )dz I MC(t) for all t 2 0. In this case the gain of H satisfies the estimate g ( H ) I M . As a consequence of the above theorem, we have the following result.

6.1.16. Corollary. If H E 9,, and if X is (BC),, then H is bounded on X if and only if there is a constant M 2 0 such that /jh(t-z)lg(r)

5 Mg(t)

for all t 2 0. In this case g ( H ) 5 lhol + M . Proof. Since h , x ( t ) E X whenever x E X and since Ih, x(t)l _< lhol I x ( t ) l , it is only necessary to apply Theorem 6.1.15 to the integral part of operator H. w The next result follows from the above corollary.

6.1.17. Corollary. I f h = H E YOe and if X = BC, C,, or C,, then His bounded on X if and only if JhlE L , . I n this case g ( H ) I \\hllL,.

6.1

203

PRELIMINARIES

Proof. The proof follows from Corollary 6.1.16 with g ( r ) = 1. The arguments used in Corduneanu [2, Section 2.21 can be used to prove the next result. 6.1.18. Theorem. If X = A , for some w > 0 or X H i s bounded on X and g ( H ) I lhol +so" Ih(r)l dr.

= AP

and if H E 9,, then

Little is known about when an operator H E Teis bounded on (L,"),, except for the shifted case, i.e., except when g ( t ) = exp(at) for some a E R . Fortunately, the shifted L,-case is of most interest in applications. Let H * ( s) denote the Laplace transform of H, i.e., H*(s) =

2hi exp(-st,)

i=O

4-1,

+

J

O

h ( t ) exp( - s t ) dt,

where s = a+jw, j = and a, w E R . This representation is guaranteed to converge to the continuous function H * ( s ) , when R e s = a 2 a and H , = e-"H E 9, and may possibly be extendable to other complex values of s by analytic continuation. For s = jw and - co < w < co,the function H , * ( j w) = H* ( a+ j w) is essentially the Fourier transform or frequency response of the operator H , . The expression B ( j w ) = ReH*(jw)

+jw[Im H * ( j o ) ]

is called the modified frequency response of H . The graph of { H * ( j w ) : -co < w < co}in the complex plane is called the Nyquist plot of H. For the next three well-known results refer to Willems [l] and Desoer and Vidyasagar

PI.

6.1.19. Theorem. If X = (L2),, where g = e", and if e-"HE 9, then H is bounded on X and g ( H ) = ess supo R + I H*(a+jw)I. 6.1.20. Theorem. If X = (L2m)g, where g = e", and if e-"H E 9, then H is bounded on X and g ( H ) = esssuposR+A,[H*(a+j~)~H*(a+jw)]~/~. 6.1.21. Theorem. If e-"H E 9, if X = (L2)g,if X , if H , = e-"H, then

=

(L2e)gfor g

= e',

and

(i) H is interior conic ( c , r ) if IH,*(jw)- CI I r for all w E R + , (ii) H i s exterior conic (c,r ) if I H, *( jw ) - cl 2 r for all w E R+,and (iii) H is positive if Re H,*(jw) 2 0 for all w E R'. Since H i s a linear operator, it is not necessary to state separate incremental forms of any of the preceding definitions or theorems. For m > 1 a version of Theorem 6.1.21 can also be stated though it is not very useful, since in this case simple graphical interpretation is no longer possible. For example, if

204

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

m > 1, H is interior conic (c, r ) on (L,"),with g

= e" if

- c Z ] ~ [ H I * ( ~ W ) - C-Zr21 ]

is a negative semidefinite m x m matrix for all o E R+. We also wish to point out that if H , = e-"H with a # 0, then H , is called the shifted operator. Likewise, H , *(jm)= H *(a+ j w ) is called the shifted Fourier transform, the graph H,*(jw), -00 < w < 00, is termed the shifted Nyquist plot, and so on. 6.1.22. Definition. An operator H E Yeis said to have decaying L,-memory if there is a nonnegative, nonincreasing function m , E L , such that

IHx(r)12 I for all t 2 0 and all x

E

LTe.

lI X ( T ) ~ ~ M ~ ( ~ - T )

dz

For operators with this property we have the following result. 6.1.23. Theorem. If a > 0 and if H E 2ZOe with e"h E L2m,then H has decaying L,-memory. Proof. By the Schwarz inequality we have

=

~ ~ x ( r ) l ' e x p ( - o ( t - T ) ) dT

sd

lh(z)12exp(at) dT

5 ~lX(T)12m~(~ d7- T )

where m l ( t )= rn,(O) exp( - a t ) and rm

m,(O) =

J

0

lh(T)I2 exp(a7) dT <

00.

We also have the following theorem. 6.1.24. Theorem. If H has decaying L,-memory with memory function m , and if x ( t ) satisfies

Ix(z)12exp(ct) d7 I a

6.1 PRELIMINARIES

205

for some constants c > 0, and a > 0, then for all t 2 0

=yo

Proof. Let J ( t ) Ix(z)l'exp(-c(t-z))dz, Ix(t)12 and J ( 0 ) = 0. Then J ( t ) I a and

Ix(z)IZml(t-z) dz

that is ( d J ( t ) / d t ) + c J ( t )=

=

1

+ cJ(T) m,(t-z)

Using integration by parts we see that

I csdJ(z)m,(t-z)dz

dz.

+ J(t)ml(0)

For shifted L, spaces we have the following theorem.

6.1.25. Theorem. If X = (Lpm)gand g ( t ) = exp(at) for some p , 1 I p I co, and some real number a, if H E Yoe and if e-"H E Yo,then H is bounded on X with gain g ( H ) 4 % l h ( t ) l e-a'dt.

Proof. If x E X then e-"x E L,. Let F be defined as F ( t ) = exp(-at) = [{h(f-T)eXp(-U(t-T))/X(T)

exp(-az)dt.

By Theorem 6.1.13 it follows that F E L, and IIFIILp4 lle-ahllL, Ile-"xllLp.

Analogous to the above theorems we can also develop results for bounded operators for discrete time systems. In this case LYe is replaced by 1, = S, m , the space of m x m matrix valued sequences, and I is the space 1F2 for p = 1.

6.1.26. Definition. Let Ze be the space of all m x m matrix valued sequences H = {hi} = {ho,hl,h,, ...}. Given any sequence {xi} c S,, we define

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INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

The following results are straightforward modifications of the results stated above for the continuous time case. 6.1.27. Theorem. If H E I and if X = ",I for 1 I p I co, or X X = co, then H is bounded on X and has gain

= ct,

or

m

Next, for H EI, let H I = e-"H transform of H , as H,(z)

= {hie-"'}.

I f H I E I, we define the z-

CD

(hje-"j)z-j.

= j=O

This representation converges to a continuous function of z when IzI 2 1. As is well known, when X = ( / z m ) g , m = 1, g = e", and H , = e-"H E I, then H is bounded on X with gain maxIH,(z)l.

g(W Note that a

=0

121 =

1

or a < 0 is allowed.

6.1.28. Theorem. If H

E

I and X ,

=

IFe, then

r for 1.1 = 1, (i) H is interior conic (c, r ) if IH(z)- c JI H is exterior conic (c, r ) if I H ( z ) CI 2 r for IzI = 1, and (ii) H is positive if Re H ( z ) 2 0 for IzI = 1. (iii)

As before, in the case of shifted operators, H is replaced by HI = e-"H. Also, since the operators H E 1 are linear, it is not necessary to state separate

incremental forms of the preceding definitions and theorems.

6.1.29. Definition. A relation H E I, is said to have decaying C,-memory if there is a nonnegative, nonincreasing sequence M = {mi} E I, such that j = o , 1 , 2 )...)

for all { x i } E S,. As in the continuous time case, if e"H E I, and if a > 0, then H has decaying /,-memory. If H has decaying I,-memory, if c > 0, and if there exists a > 0 such that

C IxjlzecjIa j

ePcj

then

k=O

[

I(HX)~I' I a c

jyo

]

mi+mo .

6.1

PRELIMINARIES

207

E. The Small Gain and the Sector Theorems Let us now consider physical systems which may be represented by the familiar block diagram of Fig. 6.1 and which may be represented by equations of the form (6.1.30) a x + w 1 1

Figure 6.1 Block diagram of a single loop feedback system.

We call x an input, w , and w2 reference signals (in the case of linear systems w , and w2 are called biases), we call el and e2 errors and y , and y , outputs. For system (6.1.30), X , is some fixed extended space and X i s a completely compatible subspace. Also, H , and H , are two given relations (not necessarily operators) on X,, a , and a, are real constants, x E X , , w , , w2 E X , e l , e, E X,, and y,, y , E X , . Furthermore, for system (6. I .30), there are relations E l , E2, Y,, and Y, on X , which are defined by the errors and outputs corresponding to a given input, so that Eix = e, and yix = y,, i = 1,2. Subsequently, we shall avoid all questions of existence by assuming that solutions for system (6.1.30) belonging to X , exist. Also, we shall often avoid the question of uniqueness of solutions by the use of operators instead of relations. The next result is called the small gain theorem.

6.1.31. Theorem. If g ( H , ) g ( H , ) < 1, then the relations E l , E,, Y,, and Y, associated with system (6. I .30) are bounded. The next theorem is called the incremental form of the small gain theorem.

6.1.32. Theorem. If Q(H,)Q(H,) < 1, then the relations E l , E,, Y,, and Y , associated with system (6.1.30) are all continuous with respect to X . When X , has Hilbert space structure, the following result, called the sector theorem can be proved.

6.1.33. Theorem. Let y and & be real constants, one positive and the other nonnegative, such that (i) - H 2 is inside of the sector {a+y, b- y } , where b > 0, and (ii) H , satisfies one of the following conditions:

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INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

(a) if a > 0, H , is outside the sector { -u-' -8,-b-' +a}, (b) if a < 0, H I is inside the sector { - b - ' + € , - u - ' - S } , (c) if a = 0, H , +(b-' - & ) I is positive and if y = 0, then also

g(H1) <

Then the relations E l , E,, Y , , and Y, associated with system (6.1.30) are all bounded. In closing we remark that the incremental form of Theorem 6.1.33 is also true. 6.2 Stability of Large Scale Systems: Results Involving Gains

In this section we establish sufficient conditions for input-output stability of a class of large-scale systems. All results are phrased in terms of gains of relations, and as in previous chapters, all results make use of test matrices. We consider two sets of results. The first of these is concerned with general multi input-multi output systems. In the second set of results we view certain classes of multi input-multi output systems as interconnected systems. First we need to establish some additional notation and prove a preliminary result. Let Xi,be a given extended space with completely compatible subspace Xi for i = l , ..., f. Let X e = X l e x . . . x X I , , let X = X , x . . . x X , , and let M = [ M , ] denote an I x I matrix relation where M , is a relation on X j , x Xi,(i.e., M , is a subset of X j e x Xi,). We use ~~x~~ = ~ ~ x i ~as ~ a~ norm ) l ' z on X , or any other convenient standard norm. As before, if a matrix A has only real eigenvalues, then 2 , ( A ) denotes the largest eigenvalue of A .

(xi=,

6.2.1. Theorem. If M = [M,] is a relation on X , , if g(M,) < co for all i,j = 1,2, ..., I, and if G ( M ) a [g(M,)], then g ( M ) I (~,CG(M)Tw4>1)1'2. Proof. Fix T > 0, x E X , , and let y

i = 1, ..., 1. Then

= Mx,

i.e.,

6.2 RESULTS INVOLVING But since ( Y T ) T Y T (IIYIIT)2

5

= (IIyIIT)’

and

(XT)TXT =

(XT)TG(M)TG(M)XT

5

209

GAINS

(IIxIIT)’, it follows that

nMIG(M)TG(M)l

(//x//T)2.

Let us now consider a multiple input-multiple output feedback system governed by the set of functional equations I

yi =

j= 1

H,ej,

ei =

xi + wi + zi (6.2.2)

I

zi =

C BGA,

j= 1

f;: = u i + u i + y i

for i = 1, ..., 1. Here wi and ui are fixed reference signals in Xi, x i , ui E Xi,are inputs, and B, and H, are relations on X j , x Xi,. The error signals ei and fi and the outputs yi and zi are assumed to exist and belong to Xi,. Let xT= (xiT,..., x): with u, w , u, z, y, e, and f defined similarly. Let X = Xfzl Xi with X , defined similarly, and let H = [H,] and B = [B,] denote I x I matrix relations on A’,. Then system (6.2.2) can be expressed equivalently as y=He,

e=x+w+z

z=Bf,

f=u+u+y,

which is clearly of the same form as system (6. I .30). Finally, let Ei, Fi,&, and Zi denote the relations associating given inputs x, u E X , with errors ei,fi and outputs y i,z i , respectively, satisfying Eq. (6.2.2). According to the small gain theorem (Theorem 6.1.31) and the estimate in Theorem 6.2.1, system (6.2.2) is input-output stable (i.e., the relations E i , Fi, &, Zi, i = I , ..., I, are all bounded) if g ( B ) g ( H )< 1, i.e., if

A,[G(H)TG(B)TG(B)G(H)]< 1. Similarly, if

l , [ G ( H ) T e ( B ) T G ( B ) e ( H ) ] < 1, then system (6.2.2) is continuous. In the present section we obtain different and hopefully more readily applicable boundedness conditions and continuity conditions. Our first result is as follows.

6.2.3. Theorem. If all the gains g(H,) and g(B,) are finite and if the successive principal minors of the I x I test matrix R

=

I - G ( B )G ( H )

are positive ( I denotes the Zx I identity matrix), then all the relations E i , Fi, y i , and Zi associated with system (6.2.2) are bounded.

210

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Proof. From Eq. (6.2.2) it follows immediately that

(

'

j =1

Now for any T > 0 we have

1

(6.2.4)

k=I

so that

I

+ jC g(Bv)CIIujIIT + IIvjIITI. =1 Define E, = (/I e , (1 *, . .., l\ef1 T)T and define X,, W,, U,, and V, analogously. Then the above inequality can be expressed as { I - G ( B ) G(H)} E , I

X, + W,

+ G(B)[U,+VT].

Since the off-diagonal terms of matrix R = I - G ( B )G ( H ) are all nonpositive and since by assumption, all the principal minors of R are positive, it follows that R is an M-matrix, and furthermore, it follows from Theorem 2.5.2, that R-' = A exists and that all elements of R-' are nonnegative. Thus, E, I A[x,+w,+G(B)(U,+I/,)]. Recalling that X is a completely compatible subspace and letting T-+GO, we have llell 5 IIAII

[Illxll + ll4 + l

l ~ ~ ~ ~ l l+~ ll~ll>l l l ~ < l l GO.

The last inequality and Eq. (6.2.2) imply that lly\\ < g ( H ) l\ell< //uI/+ llull+ l l ~ l l <00, and I/zl/5 g ( B )llfll
GO,

llfll 5

By interchanging the roles of B and ff in the previous theorem we can prove the following result. 6.2.5. Theorem. If all of the gains g(B,) and g ( H v ) are finite and if the successive principal minors of the test matrix R = I - G ( H )G(B) are positive, then all the relations E i, F i , Yi,and Ziassociated with system (6.2.2) are bounded.

Our next result has essentially the same proof. 6.2.6. Theorem. If all gains g(B,) and g(H,) are finite, if either ui+ui = 0 for i = 1, ..., I or B, is linear for i,j = 1 , ..., I, and if the successive principal minors of the test matrix

S = Z- G ( B H )

6.2

RESULTS INVOLVING GAINS

211

are positive, then all the relations Ei,Fi, Yi,and Ziassociated with system (6.2.2) are bounded. Proof: From the hypotheses and from Eq. (6.2.4) it follows that

e,

=

xi+ w i+

1 Bii(uj+uj) + 1 1

j=

for i = 1,..., 1. Thus, for any T > 0 we have

[ I - G ( B H ) ] ET 5

x~+ w~+ c(B)[u,+v~],

as in the proof of Theorem 6.2.3. Since all the off-diagonal terms of matrix S are nonpositive and since all principal minors of matrix S are positive, it follows that S is an M-matrix. Therefore, from Theorem 2.5.2 it follows that S-' A exists and that A 2 0. Therefore

+

I A ( X T + w T ) AG(B)(U,+VT). To complete the proof we let T 4 00 and proceed as in the proof of Theorem 6.2.3. Interchanging the roles of B and H in Theorem 6.2.6, we obtain the following result. ET

6.2.7. Theorem. If all gains g(BJ and g ( H & are finite, if either xi+ wi = 0 for i = 1, ..., I, or if Hii is linear for i,j = 1, ..., I, and if the successive principal minors of the test matrix S = I- G ( H B ) are positive, then all of the relations Ei,Fi,Y i , and Zi associated with system (6.2.2) are bounded. We now show that the hypotheses in Theorem 6.2.3 imply that all eigenvalues I of G ( B )G ( H ) satisfy the condition \I1 < 1.

6.2.8. Lemma. If the hypotheses of Theorem 6.2.3 are true, then any (possibly complex) eigenvalue A of the matrix M = G ( B )G ( H ) must satisfy the condition 121 < 1. Proof: Since M 2 0, it is known (see, e.g., Gantmacher [2, p. 661) that M has a dominant eigenvalue I , which is real, positive, and simple with associated eigenvector u 2 0 such that 1 1 = 1. Now since I , is dominant, A,, > \A1 for any other eigenvalue 1, of M . Also, since A = ( Z - M ) - ' exists, A, # 1. Let r = & - i f O . Then ( I , Z - M ) v = O , 1120,and o f 0 imply that ( I - M ) u + ru = 0, or u = - rAu 2 0. Thus, r must be negative.

Thus, the hypotheses of Theorem 6.2.3 and the hypothesis

I,[G(H)TC(B)TG(B)G(H)] < i needed to apply the small gain theorem are evidently similar. However, these two stability conditions are apparently not equivalent, nor is one a special case of the other.

212

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

The type of condition assumed in Theorem 6.2.3 appears to offer several advantages over other existing results. For example, this condition leads to a linear constraint on the margin of boundedness parameter (which plays a similar role as the degree of stability in previous chapters) to be defined in Section 6.4, Part C, while the small gain condition

AM [ G ( H ) T G ( B ) T G ( B ) G ( H )< ] 1 leads to a corresponding quadratic constraint, The above remarks are also applicable to the matrices G ( H ) G ( B ) ,G ( B H ) , or G ( H B ) , encountered in Theorems 6.2.5-6.2.7. The stability results considered thus far (Theorems 6.2.3, 6.2.5, 6.2.6, and 6.2.7) have incremental analogs. 6.2.9. Theorem. Let all the incremental gains Q(B,) and @(If,) be finite, let the successive principal minors of the test matrix R be positive, and assume one of the following conditions: (i) R = I - G ( B )G ( H ) , (ii) R = I - G ( H ) G ( B ) , (iii) either u i + v i = 0 for all i or B = [B,] is linear and R = Z-Q(BH), (iv) either x i + wi = 0 for all i or H = [H,] is linear and R = I - @ ( H B ) . Then all of the input-output relations Ei,F i , Yi, and Ziassociated with Eq. (6.2.2) are continuous. Proof. We outline the proof of part (i). The proofs of the remaining parts are similar. Consider Eq. (6.2.4) for two sets of inputs x, u E X , and Z,U E X , with Sx Li x--X E X and 6u = u-ii E X . Let e and 2 be solutions of Eq. (6.2.4) corresponding to those inputs and let 6e 4e - 2 . In order to estimate Se we subtract the two copies of Eq. (6.2.4) and take seminorms to obtain

The rest of the proof follows the proof of Theorem 6.2.3, and we obtain 116eIl 5 IIAII (ll6-4 + I G(B)II Il6ull) < I l ~ Y l5 l Q(H)/l6ell

ll6fll

03,

<

116~Il+ ll6Yll < a,

and 116Zll 5 s^(B)II6f//<

03.

Next, we consider the special case of system (6.2.2) where H ,

=0

for all

6.2

RESULTS INVOLVING GAINS

213

i # j . Relabeling Hii = H i , we can rewrite Eq. (6.2.2) as 1

fi = ui + vi + y ,

1 By&,

zi

=

Yi

= Hiei,

j= 1

ei =

xi + wi + zi

for i = 1 , ..., 1. may be viewed as an interconnection of l isolated subsystems System (9) described by equations of the form zi

=

BiiJ,

fi = ui + vi + y i

yi

=

Hiei,

ei

=

xi + wi + z i ,

(8)

i = 1, ..., I, with interconnecting structure specified by relations Bii for all i # j . The isolated feedback system (3) and the interconnected system (9’) are depicted in block diagram form in Figs. 6.2a and 6.2b, respectively. x. + w. 1

-

l.?

e.

Yi

u. + v . l l

(b)

Figure6.2 (a) Block diagram of isolated feedback system (sPi). (b) Block diagram of interconnected system (9’where ) i = I , ...,I.

Alternatively, we may regard (9) as an interconnection of I isolated subsystems described by equations of the form y . = H I. e .

I )

(Hi)

214

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

i = 1, ..., I, with interconnecting structure specified by all of the B,, and

speak of isolated subsystem (Hi). For either case, the following additional comments are appropriate in the present context. In general, the term ui+vi may include external inputs as well as disturbances. If in particular u i + v i does not contain external input terms, then fi represents the available output of subsystem Hi under additative disturbances. Furthermore, in applications, the assumption ui = ui = 0 is frequently valid. In many of our subsequent results, we will address ourselves to this special case. Since all of the Theorems 6.2.3 and 6.2.5-6.2.7 can directly be applied to composite system (Y),the following results are seen to be true.

6.2.10. Theorem. If the gains g ( H i ) and g(B,) are finite for i,j = 1, ..., I and if the successive principal minors of the test matrix R = [vJ defined by

are all positive, then all of the input-output relations E i , F i , associated with system (9’ are ) bounded.

F,

and Zi

6.2.11. Theorem. If all the gains g ( H i )and g ( B , ) are finite, if either ui+ ui = 0 for all i or B = [B,] is linear, and if the successive principal minors of the test matrix R defined by 1 - g(BiiHi),

if

i

-g(Bij Hj),

if

i # j

=j

are all positive, then all of the input-output relations E i , F i , Y i , and Zi associated with system (9) are bounded.

In view of Theorem 6.2.9 the following result is also true for system (9’). 6.2.12. Theorem. Suppose that all of the incremental gains Q(Hi)and Q(BJ are finite, suppose the successive principal minors of the test matrix R = [rJ are all positive, and suppose that one of the following conditions hold:

(0

4

y.. =

1 - Y^(Bii>Q(Hi),

Q

Q

- (By) ( H j ) ,

, ~ : ) ~ + ( u..., ~ u:)’ ~ , (ii) either u f c = ( u , ~..., and (1 yii =

-

@(BiiHi),

j-Q(R,H,),

.

2

=J

.

i # j =0

i = j i # j

or B

=

[B,] is linear,

6.2 (iii) either x + w linear, and

215

RESULTS INVOLVING GAINS

= (xlT,..., X : ) ~ + ( W ~ ~..., , w : ) ~= 0

or H

= diag[Hi]

is

Then all the input-output relations E i , F i , Y i , and Ziassociated with system (9') are continuous. 6.2.13. Remark. If in Theorems 6.2.3, 6.2.5-6.2.7, and 6.2.9-6.2.12, the gains (incremental gains) are replaced by upper bounds for the corresponding gains (incremental gains), and if the altered test matrix still has positive successive principal minors, then all of the above theorems remain true. Next, we observe that Theorem 6.2.10 can be written in the following altered form. 6.2.14. Theorem. If all gains g(Hi) and g(B$ are finite, if g ( H i ) > 0 for all i, and if the successive principal minors of the test matrix S = [se] defined by

are positive, then all of the input-output relations E i , F i , q , and Ziof system (9) are bounded. If all gains are replaced by incremental gains, then the above relations are continuous.

A

Proof. The crucial point in the proof of Theorem 6.2.3 is to show that R-' = ( I - G ( B )G ( H ) ) - ' exists and is nonnegative. Since

=

R then A

=

I - G ( B )G ( H ) = [ G ( H ) - ' - G ( B ) ]G ( H ) 4 S G ( H )

= R-' = G ( H ) - ' S - ' .

G(H)-'

=

Now

diag[g(H,)-', ...,g( I f l ) - ' ]> 0

'

and the hypotheses of this theorem ensure (see Theorem 2.5.2) that S - 2 0. Thus, A 2 0 and the proof of Theorem 6.2.3 applies. The proof of the incremental form is essentially the same. The results and proofs given in the present section are intended to serve as models. More important than these specific theorems is the method of analysis developed. Thus, in specific applications it may be necessary to establish new results, using the present ones as motivation. Frequently, preliminary work (e.g., appropriate transformations of the system in question) can be used to great advantage before applying the present method. These remarks are also applicable to the instability results given in the next section. In subsequent sections of this chapter, the thrust of the above remarks will become further evident.

216

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

6.3 Instability of Large Scale Systems

In the present section we consider once more composite system (9) and isolated subsystems (Hi), i = 1, ..., I, encountered in Section 6.2. Throughout this section Xis assumed to be a completely compatible subset of an extended space X,, and X , is assumed to possess Hilbert space structure. Then for any T > 0, the seminorm 11 is defined by the inner product ( ,) T and ( X , Y > = iim

T-rm

<X,Y>T

determines an inner product on X . Given any subspace N of X , we define as usual the orthogonal complement of N as

N' Let

N

x2 E N'

= {x E

X :(x,y)

=0

for all y

E

N}.

denote the closure of N . Then for any x E X we can find x1 E = (N)*such that x = x1 + x 2 and such that

m and

llxll = 11x1I + llx2lI

(see, e.g., Dunford and Schwartz [I, p. 249, Lemma 41). Given a square matrix M , we define the spectral radius r ( M ) by r ( M ) = max { 1111:

A is an eigenvalue of M } .

Let us now assume that at least one subsystem (Hi) is unstable for i = io and let us consider composite system (9). 6.3.1. Theorem. Suppose that the gains g ( B & are finite for i,j = 1, ..., I and suppose that on Xi, the H i are continuous operators for i = 1, ..., 1. Let N , = S ( H , ) be the stable manifold of Hi with g , ( H i ) < 00 for all i and suppose that N i L # (0) for at least one value i = i,. For R = Cry] = [ g ( B & g , ( H j ) ] assume either r ( R ) < 1 or r ( R ) = 1 and rij > 0 for all i , j = 1, ..., 1. If ii = u = 0, if x i + w, E (NJL for all i, and if x i + w, # 0 for at least the one value i = i,, then y X . (In this case system (9) is said to be unstable, or inputoutput unstable.) Proof. For purposes of contradiction, assume that y E X , i.e., y i E Xifor i = 1, ..., 1. Since y solves (Y), then y i = Hiei E Xi if and only if ei = x i + poi + z i E N i . Then zi= e i - ( x i + wi) where e, E N i and x i + w, E N:. It follows now that jJziII= IIeill lixi+wiII and

+

1

Ileill 5 llziil =

C1 Bij(Hjej)

Ilj=

1

5

with strict inequality at least for i = i,.

C g(Bij)gc(Hj)llejll

i f 1

.x I

=

J=

1

rYlIejIl

(6.3.2)

6.3

INSTABILITY OF LARGE SCALE SYSTEMS

217

If r ( R ) < 1, replace g ( B i ) g c ( H j ) by g ( B i ) g c ( H j ) + A for all i , j . Choose A > 0 and so small that for the new matrix R thus formed, we still have r ( R ) < I . Now for this alternate R , we have ri > 0 for all i, j . So by the Perron theorem (see Gantmacher [2, p. 531) there are positive numbers ai such that aT = (al, ...,aI)> 0 is a row eigenvector corresponding to the dominant eigenvalue r(R) of R . Then uTR = r(R)aT or

I

2airi = r ( R ) a j ,

j = 1, ..., 1.

i= 1

Since all ai > 0 and since IIxi+ will > 0 at least for i 1

= i,,

(6.3.3) we have

1

This inequality and (6.3.2)and (6.3.3)imply that

i.e., However, since r ( R ) 2 1, we have arrived at a contradiction. This proves the theorem. In Theorem 6.3.1 it is assumed that when i = i,, Nil # ( 0 ) . We note that for some operators H the assumption that H is unstable does not imply that [S(H)]' # (0). This can be seen from the example X = L, and Hy(t) =

l

y(z) dz,

t 2 0,

y E X.

This H is clearly unstable but at the same time its stable manifold contains the set No = {xT:x E X } . Since No is dense in L,,it follows that [ S ( H ) ] = L, = X and [S(H)]' = (0). The next result will show when the condition that H is unstable implies [S(H)]' # (0). 6.3.4. Lemma. If H : X ,

-+X , is a continuous linear mapping, if H is unstable on X and if g , ( H ) < 00, then [S(H)]' # (0).

Proof. We first show that N A?+ S ( H ) is closed. In so doing, pick a sequence (xi} c N such that xj -+ x E X as j - t m. Now H i s continuous on X , means

VI

218

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

that for any T > 0 there is a number p ( T ) 2 0 such that llHx1lT I p ( T ) llxllT for all x E X , . Thus, Ilffxj-HxII.

=

llff(xj-X)llT 5 P ( T )IIx-xjllT.

Since X is completely compatible, it follows that /Ix- xi 11 < IIx- xi 1 -+ 0 as j + co. In particular, pick j such that p ( T ) Ilx-xjll < 1 . For thisj, llHxllT I / / H x j / / 1 . But g , ( H ) < co implies that IIHxjII I g , ( H ) llxjll < K for some K , which works for all j . Thus, IIHx(IT< K + 1 for all T > 0. In the limit, as T+co, we have IIHxIIT-'IIHxII]'.

+

Note that Lemma 6.2.8 and Theorem 6.3.1 can be combined to yield the following result. 6.3.5. Theorem. Suppose that the gains g(Bi)< 00 for all i , j = 1 , ...,I, suppose that H i : X i , -+ X i , is continuous and linear with stable manifold N i = S ( H , ) , i = 1, .. .,1, such that N i l # ( 0 ) for at least one i = i,,, and suppose g , ( H i ) < co,i = I , ...,1. Suppose the successive principal minors of the test matrix

R

=

Z - G ( B )G,(H)

are positive. Then system (9) is unstable with respect to X .

In the remaining results of this section we require the following concepts. 6.3.6. Definition. An operator H : X , + X , is called

(i) passive if for all x E X such that H x E X we have ( H x , x) 2 0; (ii) properly passive if for all x E X such that x # 0 and H x E X we have ( H x , ~ >) 0 ; (iii) strictly passive (with constant S) if there is a 6 > 0 such that when x E X and Hx E X , we have ( H x , ~ 2) S ( x , x ) . We now prove the following result. 6.3.7. Theorem. Let g ( B i ) < co for i # j , i , j = I , ..., 1. Let - B i i be linear, strictly passive operators with constants Si > 0, i = 1, ..., 1. Assume that for some i = i,, the stable manifold Ni = S(Bii) has nonzero orthogonal complement, N i # ( 0 ) . Let Hibe a bounded, properly passive operator such that H i O = O for i = l , ..., 1. If u = u = O , if x i + w i e N ; , i = l , ..., I, and if x i wi # 0 when i = i,, and if the successive principal minors of the test

+

6.3

219

INSTABILITY OF LARGE SCALE SYSTEMS

matrix R defined by if i = j if i # j

-g(B@),

are all positive, then any solution e of (9’)cannot be in X (and (9’)is inputoutput unstable). ProoJ For purposes of contradiction, assume that e E X , i.e., assume that

ei E X i , i

=

1, ..., 1. Using the equations describing (9’)we have

ei = ( x i + w i )

+

I j= 1

B,Hjej,

i = 1,

...,1.

{ 6.3.8)

Since all of the elements in this equation belong to X and since Hiei E X , it follows that Hiei E N i . Taking the inner product of Eq. (6.3.8) with H i e i , we have (Hiei,ei)

=

( H i e i ,x i + w i )

+

I

j= 1

( H i e i , BVH i e j )

Since Hiei E Ni and since x i + wi E N i l , we have (H i e i ,x i + w i ) = 0 and ( H i e i ,- B i i H i e i )

-(Hiei,ei)

=

+

1

j=l,i#j

( H i e i ,B V H j e j ) .

Since Hi is passive and since - Bii is strictly passive, we can estimate

I

or I

i = 1, ..., 1. These 1 inequalities can be written as RA I 0, where AT = [ 11 H , el 11, ..., IIHl el 111 2 0 and R = [ r s ] is the given test matrix. The hypotheses of the theorem imply that R - ’ exists and that R - ‘ 2 0 so that A I 0. This means H i e i = 0 for i = 1, ..., 1 and ( H i e i ,e i ) = 0. Since Hi is properly passive, we have ei = 0 for i = 1, ..., 1. This and Eq. (6.3.8) imply that xi + wi = 0 for i = 1, ..., 1, a contradiction. This completes the proof. The above theorem states that under suitable conditions, if at least one subsystem (Yi)is unstable, then the composite system (9’)is also unstable. In our next result, the instability can occur in a subsystem, in an interconnection term, or in both the subsystem and interconnecting structure.

220

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

6.3.9. Theorem. Let - B = [ - Bg] be a passive linear operator on X,. If N = S ( B ) is the stable manifold of B, let N' # (0). Let Hi be a bounded, properly passive operator with Hi 0 = 0 for i = 1, ...,1. If u = u = 0, x + w E N' and x+ w # 0, then e $ X . Proof. System (9) can be rewritten equivalently as

e

x

+ w + BHe

(6.3.10) where (He)T= [ H , el,..., Hl ellT. For purposes of contradiction we assume that e E X . Then He must belong to N . Thus, (He, e ) If e # 0, then

=

=

(He , x + w )

0 < ( H e ,e )

+
=

( H e , BHe) 5 0.

Therefore e = 0. Then He = 0 and it follows from Eq. (6.3.10) that x+ w = 0, which is a contradiction. This concludes the proof. H Finally, we note some of the conditions which imply that B (in Theorem 6.3.9) is unstable. (a) (b) (c) (d)

For some pair i,j , the entry B, is unstable. There are constants u isuch that the column sum m iB, is unstable. There are constants Bj such that the row sum C:.=,B,Bj is unstable. There exist constant matrices C and D such that CBD is unstable.

cf=

Note that statement (d) includes the other statements. Moreover, (d) can be verified by noting that if B is stable, then CBD is also stable. Any of the preceding conditions can be used, together with Lemma 6.3.4, to see that N' # ( 0 ) . 6.4 Stability of Large Scale Systems: Results Involving Sector and Conicity Conditions

The present section consists of four parts. In the first of these we discuss a special transformation with the property that the stability (i.e., boundedness and continuity) of the transformed system implies the stability of the original system. It will be seen that this transformation widens the applicability of the results of Section 6.2 considerably. In particular, we will be able to formulate stability results for interconnected systems involving sector conditions and conicity conditions, to be introduced in the second part of this section. In the third part we present stability results for composite systems phrased in terms of some useful design parameters, called margin of boundedness and gain factor. The present results are applied to two specific examples in the last part of this section.

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

222

A. Useful Transformation Let us restate composite system (9') first introduced in Section 6.2,

yi

=

+ wi + zi

ei = xi

Hiei,

i = I , ..., 1. We now rewrite system (9') in vector-matrix form. Let B = [ B J , H = [Hid,] where 6, = 0 if i # j and dii = 1 , zT = (zIT,..., z:), and f T = (flT, ...,AT) with u, v, e, y , x, and w similarly defined. Then (9) can be rewritten as z=BJ f=u+u+y (6.4.1) y=He, e=x+w+z. System (6.4.1) in turn can be rewritten as

e

= (x+w)

f

= (u+u)

+ Bf

+ He.

6.4.2. Theorem. Let Cand D be linear operators and define E Z+(B+ C )D. Assume that either B is linear or D = 0. If D # 0, assume that E is a one-to-one operator on X , with inverse E - ' . Then the transformation

e'

=

e

+ Cf - C(u+u),

J"

=

De

+ (I+ D C ) f

or

(T) e=(Z+CD)[e'+C(u+u)]-Cf',

f = f'-D[e'+C(u+u)]

takes system (9) to the new system

where B'

e'

=

E - ' ( x + w ) - C(u+u)

f'

=

( I + D C ) (u+ u)

= E-'(B+C)

and H '

+ H'e'

+ B'f'

V')

= (H-'+C)-'+D.

Proof. From f- u - u = He we get

H-'(f-u-u)

=

(6.4.3)

e.

Adding C (f - u- u) to both sides of Eq. (6.4.3) and recalling the defiiition of e' we see that (H-'+C)(f-u-u) Thus we have ( H -'

=e

+ C )He = e' and

+ C(f-u-u)

He = ( H - ' + C ) - ' e ' .

=

e'. (6.4.4)

222

VI

INPUT-OUTPUT

STABILITY OF LARGE SCALE SYSTEMS

Using the definitions of e' and f we obtain

e'

+ Cf- C(u+v) = ( x + ~ > + B f+) Cf - C(u+v) = x + w - C(u+v) + ( B + C ) f =

e

=x

+w

-

C(u+v)

+ ( B + C ) [ f ' - De'-

DC(u+u)].

If B is nonlinear, then D = 0 and we are done with e'. If B is linear then we can write

e'

+ ( B+ C )De'

=

x+w

-

+

[ I + ( B C )D]C(u+ v)

+ ( B+ C )f ' .

Recalling the definition of E we obtain

+ w - EC(u+v) + ( B + C ) f '

Ee'

=

x

e'

=

E - ' ( x + w ) - C(u+v)

or

+E-'(B+C)~'.

Thus, for the case when B is linear and D # 0, we are also done with e'. Using the definition of f ' and Eq. (9) we obtain after obvious manipulations,

j ' = f + De'

+ DC(u+v) = u + v + He + De' + D C ( u + u ) = (Z+DC) (u+ v) + He + De'.

Applying Eq. (6.4.4) t o the above, we see that

f"

+ ( H - ' + C ) - ' e ' + De' = ( I + D C ) ( u + v ) + H'e'. =

(I+DC)(u+v)

This completes the proof. The following result follows now immediately. 6.4.5. Corollary. Assume that the hypotheses of Theorem 6.4.2 are satisfied. If C and D are bounded with respect to Xand if u+ v E X , then e andjbelong to X if and only if e' and f'are in X . The above corollary states that the stability properties of the original system (9) and those of the transformed system (9') are equivalent. This from those of system allows us to deduce the stability properties of system (9')

(9'). B. Sector and Conicity Conditions Let us now apply the transformation (T) to a more specific model. Assume that B is linear and that X , has Hilbert space structure. Let Co and Do be disjoint subsets of integers such that C, u Do = { I , ..., l } . Then for each

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

223

i = 1,2, ..., I, pick constants di and ci such that di = 0 if i $ Do and ci = 0 if i $ C,. Define C = diagcc,, ..., cl] and D = diagcd,, ..., d J . Under the transformation (T), H' is determined in this case by H' = diag[H,', ..., H,'] where Hi di I, if i E Do

+

if i E C,.

(H;'+C~Z)-',

This means that a given subsystem (Hi) has feedback ci or feedforward di associated with it, but not both. 6.4.6. Definition. A relation Hi is called conic (incrementally conic) with constants (a, b) if for a < b the relation is inside (incrementally inside) the sector {a,b} and for a > b it is outside (incrementally outside) the sector {b, a}.

Our next result yields boundedness conditions for a multi-loop system (9') when certain sector conditions are met. 6.4.7. Theorem. Let B be linear and let X , have Hilbert space structure. Let C , and D, be the sets of integers defined above. Assume that Hi is conic with constants (ai,bi) where bi > a, if i E Do and aibi(bi-a,) < 0 if i E C,. Define

and ci =

1

- (bi+ai) 2ai bi

,

if i

E

C,.

Assume that I+ BD has an inverse and define B'

=

[I+(B+C)D]-'(B+C)

=

(I+BD)-'(B+C).

Define

?Ii

=

I.-

2ai bi , bi- ai

-

if i E C,.

If B' = [Bb] has finite gain (i.e., g ( B $ ) < a)and if the successive principal minors of the test matrix

R

=

I - G ( B ' ) diag[v,,q,, ..., ?IJ

are all positive, then all of the input-output relations Z i , Y i , Fi, and Ei associated with system (9') are bounded with respect to X .

224

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Proof: Given any relation M on X , note that the following results are true.

(i) If M is conic with constants (a,b), then for any real number k , M + k l is conic with constants (a + k , b + k ) ; (ii) if M is conic with constants (a,b) and k > 0, then k M is conic with constants (ka, kb) and if k < 0, then k M is conic with constants (kb, ka); (iii) if M is conic with constants (a,b) and if ab # 0, then M - ' is conic with constants (b- ',a - I ) ; (iv) if M + a l is positive and if a # 0, then M is conic with constants (0, a- I ) ; and (v) if - M + b l is positive and b # 0, then M - ' is conic with constants (b-',O).

-'

The proofs of these statements are straightforward. For example, for item (i) we note that M x - r x = ( M + k ) x - ( r + k ) x for r = a or r = b. It follows that ( M x - a x , M x - ~ x ) ,2 0 (or 2 0 ) is equivalent to ((M+k)x-(a+k)x, (M+k)x-(b+k)x), I 0 (or 20). To verify part (ii) it is only necessary to factor out the constant ( k k ) ' , i.e., ( k M x - akx, k M x - bkx), = k 2 ( M x - ax, MX - b x ) T .

In part (iii) we note that if (x,y) E M then ( y , x ) E M - I . If ab # 0 and if (y-ax, y-bx),

then

2 0

' ,= ab ( a - ' y -x , b

ab ( a - ' y - M - ' y , b - ' y - M - y )

=

-

l y - x)T

a b ( X - L ' y , x - b - ' ~ ) , 2 0.

Parts (iv) and (v) are similarly proved. We now use these statements with M = Hi. If i E Do then Hi' = H i + d l . Since Hiis conic with constants (ai,bi), it follows that Hi' is conic with constants (ai+ di, bi+ di). But di= - (b,+ ai)/2. So a, di = (a,- bi)/2 = - qi and bi+di = (bi-ai)/2 = q i . Thus, Hi' is interior conic (O,qi) (i.e., Hi' is conic with center 0 and radius q,). Now assume that i E Co so that Hi' = (H;' + ciZ)-'. Since Hi is conic with constants ( a i , b i )and aib, # 0, it follows that H Y 1 is conic with constants (b; ', a,: I ) , H i ' + ci I is conic with constants (b; + ci, a; + c,) and

+

'

6;'

+ ci = -(bi-ai)/(2aib,> = -(aI:'

+ci)

=

qI:'.

Using item (iii) above again we see that Hi'= (H['+c,Z)-' is conic with constants (- q i , qi). Thus, Hi' is interior conic (0, q i ) and g(Hi') 5 q i . The stability of (9) now follows by applying Theorem 6.2.10 and Corollary 6.4.5.

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

225

6.4.8. Remark. In Theorem 6.4.7 the conditions on a, and bi mean the

following. If i E Do, we must have bi > ai and Hi must be inside of the sector {ai,b,}. If i E C,, there are three possibilities. (i) bi > 0 > a, and Hi is inside the sector { a i ,bi}, (ii) a, > bi > 0 and Hi is outside of the sector { a i , b i } ,and (iii) bi < a, < 0 and Hi is outside of the sector { a , ,bi}. In (i) the limiting case b, + co can be used. In this case ci = (- 2ai)-' = (qi)-'. In (ii) ai co can be used with - Hi+biZ positive and ci = (-2bJ-I = (qi)-'. In (iii) the case bi-+ -ca can be used with Hi-a,Zpositive and q i = -2ai = ( c i ) - l . -+

When B is not linear, we can prove the following alternate form of Theorem 6.4.7. 6.4.9. Theorem. Assume that X , has Hilbert space structure, that Do is the empty set, and that C , = {1,2, ...,I } .Assume that Hi is conic with constants

(ai,b,) where aibi(b,- a,) < 0. Define (bi + ail c. = -~ 2a, b,

If g ( B & < co for all i,j test matrix

=

and

q, =

- 2ai bi ~

(bi- ai) '

1, ...,I and if the successive principal minors of the

I - G ( B ' ) diag[q,, . . . , q J

are all positive, then all the input-output relations E , , F,, Y , ,and Z , associated with system (9) are bounded with respect to X . Proof. Since in the present case the transformation (T) transforms system (9) to system (Y')with D = 0, the proof of Theorem 6.4.7 still works. W Remark 6.4.8 applies also to Theorem 6.4.9, with the understanding that Do is empty so that i E Do is impossible. Corresponding to the preceding two boundedness theorems, we can establish analogous continuity results. For example, we have the following result. 6.4.10. Theorem. Assume that X , has Hilbert space structure and that either B is linear or that Do is empty. Assume that Hiis incrementally conic with constants (ai,bi) where aibi(bi - a i ) < 0 if i E Co and bi > a, if i E Do. Define

di

=

[

-(bi+ai)/2,

if i

0,

if i E C O

E

Do

226

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Suppose that I+BD has an inverse. Define B‘ ( I + B D ) - l ( B + C ) , and qi

=

(

=

{I+(B+C)D}-’(B+C) =

(bi- ai)/2,

if

i e Do

(-2uibi)/(bi-ui),

if

i E C,.

If all incremental gains g^(B$)are finite and if the successive principal minors of the test matrix

I - G ( B ’ )diag[ql, . . . , q l ] are all positive, then all input-output relations E i , Fi,Y i , and Ziassociated with system (9) are continuous with respect to X. The proof of Theorem 6.4.10 is similar to the proof of Theorems 6.4.7 and 6.4.9.

C. Margin of Boundedness and Gain Factor We now present stability results which are phrased in terms of some useful parameters. One of these, called margin of boundedness, is somewhat analogous to the degree of stability (of the entire composite system or of individual subsystems) considered in the previous chapters. The second parameter, called gain factor, provides a measure of gain of the overall system (i.e., of the overall composite system or of individual subsystems). As in the preceding chapters, these parameters are especially well suited in providing qualitative trade off information between the various isolated subsystems and the interconnecting structure. Although we define these parameters only for system ( Y ) these , definitions are of course also applicable to isolated subsystems (using obvious changes in notation, if necessary), as well as to the more general multiple input-multiple output system described by Eq. (6.2.2).

(x.)

6.4.11. Definition. System ( Y ) ,viewed as a single loop system, is said to possess margin of boundedness 6 or 6’, where 0 < 6 < 1 and 0 < 6‘ < 1 , if (9’) is stable with respect to X and

(i) for u = u = 0, llell i 6 - l llx+ wIl for all x, w E X,or I ( 8 - l IIu+uII for all u,u E X . (ii) for x = w = 0 , ~~f~~ 6.4.12. Definition. System ( Y ) ,viewed as a single loop system, is said to possess gain factor p or p’ if (9’)is stable with respect to X and (i) for LI = u = 0, llylj I p ilx+ wIl for all x, w E X , or (ii) f o r x = w = O , ~ ~ z ~ ~ < p ’ f~o r~a ul l u+, u E~X~.

6.4

227

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

In the corresponding incremental definitions, 8, 8’ or 0, 0’ are called a margin of continuity and an incremental gain factor, respectively. Theorems 6.4.7, 6.4.9, and 6.4.10 can be used to estimate the parameters 6 and p , and by symmetric arguments, 6’ and p‘.

6.4.13. Theorem. Let X , have Hilbert space structure. Assume either (i) H is inside the sector {a,b}, b > a, d = -(a+b)/2, q l = (b-a)/2, B is linear, I + Bd is one-to-one, and B‘ A ( I + Bd)-’B has gain g ( B ‘ ) 5 q 2 , or (ii) H is conic with constants (a,b), ab(b-a) < 0, q 1 = -2ab/(b-a), c = -(a+b)/(2ab), B‘ B + c I , and g ( B ‘ ) I q 2 .

If 6 = 1 - q 1 q2 > 0, then 6 is a margin of boundedness for (9’) and p is a gain factor for (9).

=

ql/6

The incremental form of Theorem 6.4.13 is also true. It follows directly from Theorem 6.4.10. In case (ii) of Theorem 6.4.13, the gain g ( B ’ ) can be estimated in several cases as follows. (a) If ab(b-a) < 0, then 6 = 1 - q l q2 > 0 provided that - B is inside the sector { -b-’ -6(b-a)/(2ab), 6 ’+6(b-a)/(2ab)}. Indeed, this statement is equivalent to saying that B is inside of the sector (c-((1 +S)/q,), c+((l -6)/v1)). (b) If a < 0 , b = + c o , and H I - a 1 is positive, then 6 = 1 - q 1 q 2 > 0 provided that - B is inside the sector { - 6 (+a- I), - a- + 6(+a- ’)}.

’

The parameters 6, 6‘ and p, p’ can also be estimated, using conicity conditions. We have the following result. 6.4.14. Theorem. Suppose that for some real constants r 2 0 and c we know that B is interior conic (c, (1 - S ) Y ) , and suppose that H satisfies one of the following conditions. (i) c2 > r 2 and H is exterior conic (- c/(c’ --r’), r / ( c 2- r’)), (ii) r’ > c2 and H is interior conic ( c / ( r 2- c’), r / ( r 2- c’)), or (iii) r 2 = c 2 and 2cH+Z is positive. Then 6 and 6‘ = 6/(1 + Icl/r) are margin of boundedness parameters for (9”)and p = (6r)-’, p’ = ( 1 3 ’ ) - ~ ( l c l +r - ’ ) are gain factor parameters for (9). Proof. Let H‘ = ( H - ’ + c I ) - ’ and B‘ = B + c I a s above. Then the method of proof used in Theorem 6.4.7 shows that H’ is interior conic (0,r - ’) and B‘ is interior conic (0, (1 - 6 ) ~ ) . Their respective gains are g ( H ’ ) < r - ’ and g ( B ’ ) < (1 - 6 ) ~ . Thus, I - g ( H ’ ) g ( B ’ ) = I -(1-6) = 6. The small gain theorem applied to (9’) shows that if u = u = 0, then jje’ll

< 6-’//x’+w’jj = 6-’/ / x + w / / .

VI

228

Since y

= y’,

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

x’ = x, and w

= w‘,

we have

g ( H ’ ) lle’ll 5

IIYII

Cll(r4l IIx+wII.

If x = w = 0, then by the small gain theorem, system (9) is stable with respect to X . Also, \if11 = IIu+ u s H” - c(u+ u ) By] 11

+

5

Ilu+4I +9(H’)Elcl IIu+uII +g(B’)llflll

= ((u+ull(I+r-’Icl)

Therefore,

llfll 5 8 - ’ ( 1 + r - l

and llzll = llz’-cfll

=

ICl)ll~+~il

Clcl+s(B’)I llfll 5 (lcl+r-l)llfll.

IlB’f-cfIl

This completes the proof.

+ r-’(1-h)rIlfll.

B

The sector conditions and the conicity conditions are the same conditions in different forms. To see this, take

c or equivalently,

=

--(-+A), 1 1 2 a

1 a=---

r-c

5(i-k)

r = 1 1

b

and

b

-1

.

=-

r+c

We now state and prove a boundedness result for a special case of ( Y ) using gain factors. Let us consider interconnected feedback systems of the type depicted in Fig. 6.3, which are described by functional equations of the form

ei

= ui

+ Biyi,

yi = H i e i ,

ui

=

xi

+ w i+

1

j= 1

C g y j , (6.4.15)

Y1

YL

Figure 5.3 Interconnected feedback system described by Eq. (6.4.15) where i = 1 ,

..., I.

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

229

..., 1, where w idenotes a reference signal, xiis an input, and y , and e, are output signals. Assume that xi E Xi,, wi E X i , ei E Xi,,and yi E Xi,. Here, Hi and B, are relations on X i , while C , is a relation on Xje x Xi,. System (6.4.15) may be viewed as an interconnection of I isolated subsystems described by the equations i = I,

e, = xi + wi + Biyi,

y , = H 1. eI.?

(6.4.16)

i = I,...,/.

6.4.17. Theorem. Suppose that each isolated subsystem described by Eq. (6.4.16) is bounded with respect to Xi and has gain factor p i . If the gains g(C& i,j= 1, ...,1, are finite and if the successive principal minors of the test

matrix

4

y..

=

1 -g(Cjj)pj,

i =j

-9 (C,) P j

i# j

7

are all positive, then the input-output relations E,, U i , and Yi associated with system (6.4.15) are all bounded with respect to X = X f Z l Xi.( U i denotes the relation associating a given input x with the variable u i . )

Proof. Since each subsystem has gain factor p i , we have for any time T, llyjllT

PjllujIIT

for a l l j = 1 , ..., 1, so that

~ ~IIuLllT)' ,, and define X , and W, similarly. Then the Define UT = ( ~ ~ u l..., above inequality can be written as

RUT 5 X T +

w~

where R is the test matrix given in the hypothesis. Since all off-diagonal terms of matrix R are nonpositive and since all successive principal minors of R are by assumption positive, R is an M-matrix. It follows from Theorem 2.5.2 that A = R-' exists and that R-' 2 0. Therefore,

U,

s

A ( X T + W,).

Letting T + 00 we see that U 5 A(X+ W ) < co. Thus, each ui E X i . For i = 1, ...,1 we now have IIyiII 5

Pi

IIuiII <

This concludes the proof.

and

lleill 5

I1uiII+ g(Bi) IIYiII < 00.

230

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Note that in proving Theorem 6.4.17 the method of Section 6.2 was used, rather than an application of any specificresult from that section. Furthermore, note that a continuity theorem corresponding to Theorem 6.4.17 is also true. In the subsequent specific examples we shall see that in applications the margin of boundedness parameter di of the isolated subsystems is actually of more importance than the gain factor pi. D. Examples We now consider two specific examples to indicate the types of elements that can be used as subsystems and interconnections and to illustrate application ofthe theory developed. In both examples the underlying space is X , = L,, while X = L, and boundedness and continuity are interpreted in the sense of the L,-norm. 6.4.18. Example. Consider the multiple-loop system depicted in Fig. 6.4. This system may be viewed as an interconnection of three linear timeinvariant systems (blocks H , , H , , H 3 ) , a time-varying gain (block H 5 ) , a piecewise nonlinearity (block If4), and a hysteresis nonlinearity (block I f s ) . This illustrates the variety of types of elements allowable. Elements with time delays can also be handled. H6

4

o(s+4)

-

0.25(1

s+4 (s+l) (s+2) (s+5)

-

cos t )

H5

Figure 6.4

Block diagram of Example 6.4.18.

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

231

Now let h , ( t ) be the inverse Laplace transform of (s+2O)/[(s+ l)(s+2)]. Then the block labeled H , represents a system having input x and output u(t) =

l

h , ( t - z ) x ( r ) dz

+ z(t)

for t 2 0, where z ( t ) accounts for initial conditions. Similar statements apply to the blocks labeled H , and H,. We first analyze the more general system specified in Fig. 6.5 and then return to the more special case of Fig. 6.4. The blocks labeled H I , H,, and H , are assumed to be linear and time-invariant operators belonging to class Yo.

SUBLOOP 1

SUBLOOP 2

Figure 6.5 A multi-loop system.

Zero input responses of these linear blocks are accounted for by the functions and z 3 which are assumed to belong to L , [0,03). The blocks labeled H4, H , , and H6 represent time-varying nonlinearities which are modeled by relations on L,,[O, a).The constant k is assumed to be nonnegative. We now use Theorem 6.4.7 to obtain boundedness conditions for the system of Fig. 6.5. Assume that the single loop system containing the two relations H , and - H4, identified as subloop 1 , is stable with margin of boundedness 6. This is ensured by assuming that H I is conic with constants (a,, b,) and by assuming that H4 is inside of the sector

zl, z,

{ - b; 1 - 6 (b, - a l ) (2a1b,) - 1, -a;

+ 6 (b, - a l ) (2a, b,) -

1

>,

where 6, < a, < 0 or a , < 0 < b,. Further, assume that H , is inside the sector { - b 2 , b 2 } , N, is inside the sector { - b 3 , b 3 ) , H , is inside the sector {O,b,}, and H , is inside the sector {0,b6},where b,, b,, b,, and b, are all positive.

232

v1

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

We now show that the system of Fig. 6.5 is bounded if

+ k b , ( b , + b , ) ( 2 b , ~ , / ( b i - a , ) )> 0.

d(l-bZb3bg)

(6.4.19)

First we see that each H i is conic with constants (a,,b,), where u4 = - b ; ' - 6 ( b , - ~ , ) / ( 2 ~ , b l ) , b, = - a ; ' + 6 ( b l - ~ , ) / ( 2 b l ~ l ) ,U, = -b2, a3 = - b 3 , and u5 = u6 = 0. In Theorem 6.4.7 select Do = (2,3,4,5,6} and C,, = (1). Then d2 = d , = 0, d , = -b5/2, d6 = -b,/2, q 1 = - ( 2 b l ~ l ) / ( b l - u ~ ) , q2 = b,, 11, = b,, q5 = b 4 2 , and q6 = b,/2. From the block diagram of Fig. 6.5 it is clear that matrix B assumes the form

B =

-1

-0

0

k

0

0

0-1

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

o

l

-1

o

0

o

o

-1

o

-

A

so that

and B'

= (I+

BD)-'( B + C ) is computed

as

Since subloop 1 has margin of boundedness 6 we have 1 - q 1 q 4

= 6.

Also

6.4

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

Figure 6.6 Nyquist diagrams for operators HI, H 2 ,and H 3 .

233

234

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Since d5 < 0 and d6 < 0, the last inequality implies the previous four. Using the expressions for yli and diand 1 - q l q4 = 6, the last inequality becomes inequality (6.4.19). We now show that in particular, the system of Fig. 6.4 satisfies inequality (6.4.19), and thus, this system is bounded. First we note that the linear operators H , , H , , and H , have impulse response functions belonging to L , [O, a). Their Nyquist diagrams are shown in Fig. 6.6. From these diagrams it is clear that H I is outside the sector { - 5.33, -0.5) and H , and H3 are inside the sector { -0.5, 0.5}. Note also that H , has input-output relation and that 0 5 dN,/ax 5 0.5. Thus, H , is inside of the sector {0,0.5}. Similarly, H , is inside of the sector (0.53,1.6} and H , is inside of the sector {0,0.5}. Recalling that a , = -0.5, b, = -5.33, b, = b, = b, = b, = 0.5, and k = 0.5 we compute 6 = 0.375,

arid see that inequality (6.4.19) is satisfied.

I

I

1

t coszt

4

G9

25 e-'4s (S+5)(S+lOJ

-

G3

I

LOOP 2

G41

1

'i-F-2 + cos t

G5

Figure 6.7 Block diagram of Example 6.4.20.

I

6.4

235

RESULTS INVOLVING SECTOR AND CONICITY CONDITIONS

6.4.20. Example. Consider now the interconnected system depicted in Fig. 6.7. We will use Theorem 6.4.17 to analyze this system. We make the identifications I = 3, H I = G I , B , = - G 2 , CZ1= k,, C3, = k,, H , = G,, B, = -C5G4, C32 = 0, Cl2 = Gg-G,, H3 = G,, B3 =-G7, C13 = GI,, C23 = G,, C , , = C,, = C3, = 0. The operators GI, G,, G,, and G, are in class 2 ' (see Section 6.1, Part D) and are characterized by their Laplace transforms. Also, G,, G,, G, belong to class Jlr (see Section 6.1, Part C) and have graphs which lie in the indicated shaded regions. Furthermore, G5 and G, are timevarying linear gains and GI, is an operator of class Af in cascade with a timevarying linear gain. The interconnections k, and k, are constant multipliers. For the sake of illustration we shall at first regard the linear elements G1, G,, G,, G, as being adjustable. At the end of our discussion we will make the additional assumption that these linear elements are those indicated in the figure. First we need to determine margins of boundedness and gain factors for each isolated subsystem (i.e., for Loops 1, 2, 3, when disconnected from the system).

a,

Loop I . G2 is interior conic (9,t). Setting c1 = - 2 and (1 - - 6 , ) r , = we note that c12> rlz, provided that > 6, > 0 and r I 2 > c I 2 if 1 > 6, > +. Therefore the isolated subsystem determined by Loop 1 has margin of boundedness 6, with 3 > 6 , > 0 if - G I lies outside of the sector {4(l-6J/(4-3d1), 4(1-6,)/(2-36,)} and 1 > 6, > $ if -GI lies in the samesector(seeTheorem6.4.14).Thegainfactorp1= r T ' 6 ; ' = 4(1 -61)6;1.

+

Loop 2. The operator G, has gain 0.75. If we require C, and G, to be interior conic with constants (0,b,) and (0,b4), respectively, then this loop has margin of boundedness 6, = 1 -0.75b3 b, if 6, is in the interval 0 < 6, < 1. This can be checked by using c2 = 0, (1 - 6 , ) r , = 0.75b4, and b, = r;' in part (ii) of Theorem 6.4.14. The corresponding gain factor is p2 = r;'6;' = 6, (1 - 0.75b3b4). Loop 3. The operator 2G,+Z is positive. If c3 = r , = 1 in part (iii) of Theorem 6.4.14 and if - G , is interior conic (- 1, 1 -6,), then this subsystem has margin of boundedness 6, and gain factor p3 = 6;'.

Next, we need to form the test matrix R , R

=

[

1

-(0.5+b4)p2

-0.256;'

-Ik,lP1

1

-0.56;'

- lkzl P1

0

1

1.

This matrix has positive successive principal minors if and only if

I - (0.5+b4)IklIp,p2 - ~ k 2 ~ p l { 0 . 5 ( 0 , 5 + b 4 ) p 2 6 ; ' + 0 . 2 5 6 ; ' > } 0.

236

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Recalling that p, = 4(1 -d1)6T1, 6, get the inequality 616263

=

1-0.75b3b,,

- (1-61){[b3 +j(1-62)1[21k1163

+Ik,(]

and p,

= b,6;',

we

+lk2I62) > 0. (6.4.21)

For each choice of 6,, 6,, S,, and b, there is a corresponding set of k, and k2 for which the stability condition (6.4.21) holds. In particular, this condition is satisfied for all k, and k, for which lkll and lk21 are sufficiently small. Next, consider the Laplace transforms indicated in Fig. 6.7. From the Nyquist plot of GI given in Fig. 6.8 it is seen that G, is inside of the sector { -0.5,2}. Therefore it is easy to calculate that 6, = 0.8. Similarly, G,(Jo) and C,(jo) lie inside circles in the complex plane centered at the origin with radii 0.5 and 1 , respectively (see Fig. 6.8). Therefore, b3 = 0.5, b, = 1, and 6, = 6. The Nyquist plot of G,, shown in Fig. 6.8, indicates that G , is inside the sector {0.5,1.5} so that 6, = 0.5. Substituting these numbers into inequality (6.4.21) we find that a sufficient condition for boundedness of the system given in Figure 6.7 is given by

0 < 1.21k,l+ 1.71k21 < 1 .

(6.4.22)

Examples 6.4.18 and 6.4.20 show that analysis of rather complicated interconnected systems by the present methods is indeed possible. These results

Figure 6.8 Nyquist plots for Example 6.4.20.

6.5

237

POPOV-LIKE CONDITIONS

may be conservative, but not excruciatingly so. In particular, these examples illustrate the variety of system elements that can be treated by the present approach. In addition, an indication is given of some of the methods which can be employed to verify sector or conicity conditions. We emphasize that many other types of relations may be analyzed by the present results. Note that in Example 6.4.20, condition (6.4.22) is sufficient for stability for any nonlinear memoryless elements with graphs in the indicated shaded regions. Terms with memory or delays can be treated just as easily. Thus, in Example 6.4.20, G, may be replaced by any relation which is conic (0.75,0.25). This is the case if, for example, ( G , x ) ( t ) = 0.75x(t)

+ u , x ( ~ - T , ) + a,x(t--z,)

+

where lull l a , l I 0.25,O < z1 < co,and 0 < T~ < co. This is also the case if, for example, ( G , x ) ( t ) = 0.75x(t)

+a

where b > 0 and lal/b < 0.25. 6.5 Stability of Large Scale Systems: Popov-Like Conditions

The systems considered in this section are special cases of composite system (6.4.15). In particular, we shall treat those systems which may be viewed as an interconnection of scalar input-scalar output subsystems of the Popov type. We will consider two types of system configurations. First we consider systems with configuration as shown in Fig. 6.9a, which are described by functional equations of the form t

ei = xi

+ wi - L i y i + 1 C , y j , j= 1

yi

=

N.e. I

I9

(6.5.1)

i = 1, ..., 1. Here x i , e i , y i E L , , ( R + , R ) = X , and wi E L , ( R + , R ) = X . The isolated subsystems of system (6.5.1) are given by

e1. = x .I + w .I - L . y . L

i = 1,

I

P

y . = N . e 1.9 I

(6.5.2)

..., 1.

6.5.3. Definition. Isolated subsystem (6.5.2) is said to have Property A if

(i) Ni E N and N i is interior conic with center bi/2 and radius bi/2 for some bi E (0, a); (ii) Li E Y with Fourier transform L , * ( j o ) ; (iii) Li = LizLil

238

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

x.+w. 1 1

.Yi

4-G x-tw.

(b)

Figure 6.9 (a) Interconnected feedback system (6.5.1) where i = 1 , ..., 1. (b) Transformed interconnected feedback system (6.5.1) where i = 1, ..., 1.

where ti, is a time-invariant linear operator with Fourier transform L r 2 ( j 0 )= l / ( l +jogi) for some positive constant gi;(iv) lLi*(jw)land IjoLi*(jo)Iare bounded for all o 2 0; and (v) the operator L i , (b,:' -Ai)Z is positive for some constant Ai > 0.

+

Let Xi be the range of Li2, i.e., Li,: L 2 ( R + ,R ) + Xi c L , ( R + , R ) and define X = Xi=, X i . Thus, Xi is the set of functions in L 2 ( R + ,R) which are absolutely continuous on finite subintervals of R' and whose derivatives are in L,. Define a norm 11. on Xi by

X i -+ L, Then X is a compatible subspace of X , = L,,(R+, R'). Now let Li,: be the right and left inverse of Li,. Then Li, involves differentiation. Note that in this case boundedness with respect to ( X , L,) will mean that whenever

6.5

239

POPOV-LIKE CONDITIONS

xi, dxi/dt, wi, dwi/dt E L,, then the outputs y i E L,. Since in our first stability result for composite system (6.5.1) (Theorem 6.5.7) the e , will be shown to be in Xi, we have e, deldt E L,(R+, R') so that e E C,. Moreover, y i ( t ) = N i e , ( t ) , so that y E C,, whenever the N i are all time invariant. This means that our proof (of Theorem 6.5.7) will not only show boundedness with respect to (A', L,) but also with respect to ( X , C,). 6.5.4. Definition. The interconnections C,, i,j = 1 , . . ., 1, in Eq. (6.5.1) are

said to possess Property B if they are time-invariant linear operators mapping L , ( R + , R ) + Xiwith finite gains. We note that if Property A and Property B are both true, then L,, = L,, L, maps L,, into itself, Li, is bounded with respect to L,, and has gain

g(Li,) = ess s~plL:~(jw)l< co. 020

Similarly, Li, C , is bounded with respect to L, and g(Li,

Cik) =

ess sup 1(1 + j o q i ) Cz(jo)I < co. W 2 0

Note that condition (v) of Definition 6.5.3 is equivalent to the requirement that Re((1 +joqi)L,*(jo)}+ b;' 2 Ai > 0

a.e. for o 2 0,

the familiar Popov condition. Here Aiis the minimum distance, parallel to the real axis, between the graph of the modified frequency response of Liand the Popov line with intercept -(!&I and slope (q,)-l. We note that the parameter Ai will play a role analagous to that of a,, the margin of boundedness, considered in Section 6.4. Before considering the main results of this section, we recall a result from the Popov theory. 6.5.5. Lemma. Let

I > 0 and define Kx by

Kx(t) = I

sd

for

e-'(('-')x(z) dz,

t 2 0,

where x E L2,. Suppose that g E .Af and satisfies Jg(x,t ) - 7x1 I y 1x1 for all 2 0, then

x E R and t E R'. If G ( x ) ( t ) g ( K x ( t ) ,t ) for all t IIG(x)-yxllT

5y

lIxIIT

for all x E L,, and T 2 0. Proof. The conclusion of this lemma is equivalent to or

(G(x)-yx, G(x)-w),.


2?<x, G ( x ) ) T 2 ( G ( x ) ,G ( x ) ) T '

VI

240

If y

= Kx,

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

then (letting g(y) = g(y( .), .)) this is equivalent to (9(Y>,9(Y)>T 2Y ( 4 g(Y))T

or since Ax

+ Ay,

= (dy/dt)

This concludes the proof.

W

We will also require the following preliminary result. E JV be interior conic (b/2,b/2)where 0 < b < co, let L be a linear time invariant operator on L Z e ,let A > 0 be a constant such that L (b- - A)Z is positive, and let e and y be given by

6.5.6. Lemma. Let N

+

'

y = Ne

e=x-Ly,

for some fixed x E L z e . Then Proof. Since x

=

lljjIlT

s p l1xllT for all T 2 0 where p

5 A-I.

e + Ly we have

<X,Y)T = < e , y ) T + (LY,Y)T 2 <e,y), + (A-b-')T

for any T > 0. Since N is inside the sector (0,b) it follows that N - ' - b - ' I is positive. Thus, <e,y),

=

(N-'Y>Y)T 2 b-'Y),

IIXIITIIYIIT

2

(X,Y)T

and

+

2 ( A - ~ - ' ) ( ~ , Y ) Tb-'(y,y)T

so that A ~ l y I ~ I ~/ x / T/ ~ for all T 2 0.

W

6.5

241

POPOV-LIKE CONDITIONS

We now state and prove the following stability result.

6.5.7. Theorem. Assume that all isolated subsystems (6.5.2) possess Property A and that all interconnections of composite system (6.5. I ) possess Property B. If the successive principal minors of the test matrix R = [ru] defined by

are all positive, then the input-output relations of composite system (6.5.1) are bounded with respect to (A', L2),i.e., for all x,e E A', the solutions y of Eq. (6.5.1) belong to L,(R+, R'). Proof. Operating on both sides of Eq. (6.5.1) by Li3 we obtain u, = L i 3 ( x i + w i )- L i l y i

I

+ C (Li3c,)yj,

y , = ~ , e ~ e,I. = L.12 u.I

j= 1

for i = 1, ...,1. This transformed system (see Fig. 6.9b) has isolated subsystems described by ui

= Li3( X i

+ Wi)

-

Li, yi,

(6.5.8)

yi = Ni Lizu i .

By Lemma 6.5.5, NiLi2 is interior conic (bi/2,bi/2). Thus, by Lemma 6.5.6, all isolated subsystems (6.5.8) have gain margins pi = A;' with respect to L,(R+, R). Noting that the test matrix R is an M-matrix, we conclude from Theorem 2.5.2 that R-' exists and that R-' 2 0. Let 6, denote the Kronecker delta and let

S

=

[6,-g(Li3C,)pj]

=

R diag[pl, ...,pl].

'

Then S - = diag [A1, ..., All R - 2 0. The conclusion of the theorem follows now by applying Theorem 6.4.17. The second configuration of interconnected systems which we consider is depicted in Fig. 6.1Oa and is described by functional equations of the form I

e,

=

x i + wi- Niyi+

1 C,yj,

yi = L I. e .

1 9

j= 1

(6.5.9)

i = 1, ...,1. In Eq. (6.5.9) all symbols have the same meaning as in Eq. (6.5.1). The isolated subsystems are in this case described by the set of equations e . = x. + w . 1 - N 1. y1.)

Yi

=

Liei,

(6.5.10)

i = 1, ..., 1. Isolated subsystem (6.5.10) is said to possess Property A if Li and Ni possess all the properties enumerated in Definition 6.5.3. We now prove the following result.

242

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

1

l Z&p - $

'it

zL

LU

(b)

Figure 6.10 (a) Interconnected feedback system (6.5.9) where i = 1 , formed interconnected feedback system (6.5.12) where i = 1 , ..., 1.

..., 1. (b) Trans-

6.5.11. Theorem. Assume that all isolated subsystems (6.5.10) possess Property A and that for all interconnections of composite system (6.5.9) we have g ( C U L j z< ) 03. Let

Di

=

Ai(g(Li1)CAi+g(Li1>1}-'.

I f the successive principal minors of the test matrix R y..

=

= Cry] defined

D i- g(CiiL j z ) ,

for

-g
for i # j

by

i =j

are all positive, then the input-output relations Yiof composite system (6.5.9) are bounded with respect to ( X , L 2 ) . Proof. System (6.5.9) is equivalent to the transformed system (see Fig. 6.10b) described by the equations ei = (xi+ivi) - ( N i L i 2 ) z + i

1

C1 C , L j 2 z j ,

j=

zi

=

L 11 . e.I

(6.5.12)

6.5

when yi = Lizzi. Let

c CYLj2Zj, I

ri

243

POPOV-LIKE CONDITIONS

=xi+wi+

N . L1. 2z r. = u.I I

j= 1

so that or

ei = ri - ui

zi = L i l r i - L i l u i .

Taking the inner product at time T with u i , we see that

+
(NiLi2Zi,Zi)T

=


As before, we have 2

(NiLiZzi,zi)T

and (ui, L i , ui)T

so that

2 (Ai-&; ' ) ( u i ,u i ) T 5

AiIIuill;

and

b,~'
5


=

=

bl"

IIuiII;

') ))ui))

(A;-b;

IIui I/T /ILilY iIIT

g(L,il)l l r i ! T -

Ai lluillT

Since zi = L i ,r i - Li, u i , we have I/ZiIIT

5

g(Lil)(/rillT+

IIuillT)

5 g(Lil)[l

+g(Lil)/Ail

//YilIT=

OF'

IIri11T.

Recalling the definition of ri and applying the triangle inequality, we have Oi IIZillT 5

IIxilIT

+ IIwiIIT

I

-k

j= 1

g('i,.

llZjIIT.

Letting 2, = (llzl ( I T , ..., \)zlllT)T and defining ,'A and W, similarly, the above inequality can be rewritten as R Z T

IXT+

WT.

Noting that R is an M-matrix, it follows that R-' exists and that R - ' 2 0 (see Theorem 2.5.2). We thus have ZT 5 R-'(&+WT)

for all T 2 0, where X + W = ( jlxl 11 the proof.

5 R-'(X+W) <

03

+ 11 w111, ..., llxl 1 + /

This concludes

Note that in Theorem 6.5.11, if Ci, has Property B, then g(CikLk2)

=

ess sup IIcifr(jo)(l +joqi)-'ll.

W 2 0

Let us next reconsider composite system (6.5.9), with somewhat different assumptions. Specifically, we assume that N i E N with Ni inside of the sector

244

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

{ a i , a,+b,} for some real constants ai < 0 and bi such that ai+bi > 0. As before L, = L,, L i , where Lr2(jw)= (1 +joq,)-'and - L i , is inside the sector {a,:', [a,+b,(l --bjhi)-']-'} for some di > 0. When these conditions are true, we say that isolated subsystem (6.5.10) possesses Property C. Now let us introduce the notation

where the indicated gains are with respect to the space L 2 ( R + ,R). 6.5.13. Theorem. Assume that all gains g ( L i , ) and g(C,iLj2) are finite and that all isolated subsystems (6.5.10) possess Property C . If the successive principal minors of the test matrix R = [rJ given by

are all positive, then the input-output relations Y, of composite system (6.5.9) are bounded with respect to ( X , L 2 ) . Proof. Let A = diag[a,, ..., ail.In the present case the transformation (T) of Section 6.4, Part A , reduces to e' = e

+ Az,

z'

=

z

so that Eq. (6.5.12) transforms to pi'

=

(xi+w,)- N , ' z ~+

I

C CGLj2zj,

j = 1

zi = L;,

pi',

where y i = Li2zi and N,' = N , Lj2-uiZ. The transformed operator Ni' is in is positive. Thus, Theorem 6.5.11 is the sector {O,b,} and L,'+b;'-h, applicable and the boundedness of zi follows. Sincey, = L i 2 z i ,it follows that Y E L,(R+,R'). Before considering a specific example, we point out that the method of the present section does not succeed in proving corresponding continuity conditions for interconnected systems of the Popov type considered herein. 6.5.14. Example. To demonstrate the applicability of the method of the present section, we consider the system shown in Fig. 6.1 1. In the present case we make the identifications I = 4, C,, = C,, = C,, = C,, = 0, C,, = C32 = L,, C,, = L,, C,I = C42 = L,, C43 = 0, C12 = C21 = 0, C,, = C23 = L,, C14 = C24 = L,.

Each N i , i = I , 2,3,4, is assumed to be a memoryless nonlinearity characterized by a function ni with the property 0 5 x n i ( x ) 5 b,x2 for all x E R and bi > 0 some constant. Also for each i, L, E 2' has Laplace transform Li*(s) with ILi*(jw)l and (joL,*(jo)l bounded for all o E R'.

6.5 POPOV-LIKE

CONDITIONS

245

Figure 6.11 Block diagram of Example 6.5.14.

We seek to determine restrictions on the nonlinearities which are sufficient to ensure boundedness of the system of Fig. 6.1 1. This means that for each isolated subsystem determined by the four indicated loops we must find a Popov line with intercept -b; and slope q; so that the plot of the modified frequency response, I i ( j o ) , of the corresponding linear element Li lies entirely to the right of this line. Furthermore, the distance from Zi(jw) to its Popov line, measured parallel to the real axis, is the parameter Ai for the ith isolated subsystem. Plots of the modified frequency responses of L i , i = 1,2,3,4, are given in Fig. 6.12, together with the Popov lines which are now discussed in the following. For the time being, let us assume that a suitable set of Popov lines has been found and that the corresponding margins of boundedness, Ai, have been identified. The next step is to form the test matrix R of Theorem 6.5.7. Boundedness of the system of Fig. 6.11 is then ascertained if the successive principal minors of R are all positive. Note however that the choice of the Popov slope parameters qi affects both Aiand the various gains which enter matrix R in somewhat subtle ways. Therefore, although the adjustable parameters qi provide a desirable degree of flexibility, there does not seem to be a straightforward way of selecting the qi's in an "optimal" manner. Returning to the example on hand, we must compute the various gains

246

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS w

Im

I

(b)

Figure 6.12 Modified frequency responses and Popov lines for Example 6.5.14. (a) A , = J ; q , = f $ . (b) A z = $ ; q 2 = $ .

6.5

POPOV-LIKE CONDITIONS

247

(d)

Figure 6.12 Modified frequency responses and Popov lines for Example 6.5.14. (c) A3=2.0; q 3 = & . (d) Ad= 1.7; q 4= f.

VI

248

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

g ( L i 3Cg)which occur in matrix R . As a sample calculation, we have S(L33c31) =

max

+jwq3)(6+jo)-II

020

=

max{&,43).

In this fashion we obtain

0

A1

R=[

A2

- max {b, q 3 )

-

- max (3, 44)

- max (3994)

-S(Ll3 cl3)

- max {I?41 }

-g(L23c23)

-max{1?q2)

max {b, q 3 )

A3

0

-S(L33

C34)

A4

The remaining gains in R cannot be specified in simple closed form until more is known about q l , q,, and q3. Roughly speaking, we want to choose the qi's so that the diagonal terms Ai are large and the off-diagonal terms are small in magnitude. As an illustration of the type of stability criteria which can be obtained, suppose that we allow b, and b, to be arbitrarily large, i.e., suppose we seek conditions on 6 , and b4 sufficient to ensure boundedness for any finite, fixed values of b, and b,. In an attempt to make A t and A, as large as possible without making the off-diagonal elements of R unnecessarily large, let us try q , = 14/15 and q2 = 4.Then, as indicated in Fig. 6.12, we have A1 2 4 and A2 2 for any finite b, and b,. For this q1 we have

+

s ( L , ~ C , , ) LA r

= maxl(l+joq1)(7+jo)(3+jw)-'(4+jw)-'I. 020

The maximum is attained at o2E 5.50, which yields r E 1.01. Similarly for this q2 we have g(L, , C, , )

=

maxI(1 +joq,)(7+jw)(3+j0)-~(4+jo)-'~A s

2.6525

020

with maximum attained at w 2 E 7.695. In order to minimize the remaining off-diagonal terms, we also require q3 I 6 and q4 I f . It is then easy to show that g(Lt3CI3= ) max)(l+joq3)(9+jw)(3+jw)-21= 1 W 2 0

with the maximum attained at w = 0. The off-diagonal elements of matrix R = [rV] are now completely determined, and we have A1 0 -r -1 R =

0

A2

-1

-1

-1

-1

6 3

-S

-+

A3

0

A4

6.6 L,-STABILITY AND

/,-STABILITY

OF LARGE SCALE SYSTEMS

249

It is now an easy matter to show that all of the successive principal minors of the test matrix R are positive if and only if 361 A2A3 A4 - ( s A ~+rAJ (I +3A4) - ($A1 +A,) A3 > 0. If we set AI and A, equal to their minimum possible values and substitute for r and s, we obtain the condition A3A4 > 0.833A3

+ 0.429A4 + 0.854

(6.5.15)

as a sufficient condition for boundedness. Therefore, the composite system of Fig. 6.11 will be bounded if we can find Popov lines for the isolated subsystems determined by Loop 3 and Loop 4 with q3 I & and q4 < and with margins of boundedness satisfying inequality (6.5.15). One suitable choice is q3 = A, q4 = +, A3 = 2.0, and A4 = 1.7, which leads to the Popov lines shown in Fig. 6.12. These choices are made most conveniently by graphical rather than analytic techniques. An approximate graphical analysis yields the bounds for the corresponding nonlinearities as

+

b3

E

0.47

and

b4 z 0.55.

6.6 L,-Stability and /,-Stability Systems

of Large Scale

In the present section we prove some results for L, boundedness (and I, boundedness) using the shifted L,-theory mentioned in Section 6.1. This approach is not the only method of proving boundedness with respect to this space. For example, it may be possible to use the theory developed in Section 6.2, together with some space such as L,, I , , BC, C , , Co, and the like. Also, recall from Section 6.5 that in Theorem 6.5.7 the solutions y i of system (6.5.1) are not only in L2(R+, R) but also in Co c L , (see the remark preceding Definition 6.5.4). Since shifted L2-spaces are used repeatedly in this section, we employ the notation X(a, m) = (L2 (R+R")), 9

where g is the weighting function g ( f ) = exp(at) and Xe(a,m)= (L!&), is the extended space. Also, we use the notation

and Ilfllu,T

=

We will require the following result.

:r',.

250

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

6.6.1. Lemma. If 0 > CT > a1 = a - b and if the relation H satisfies any one of the properties (i) H is inside the sector {a, b } , (ii) H is outside the sector {a,b}, (iii) H + al is positive, with respect to the space X(o,,m), then H satisfies the same properties with respect to X(a, m). Proof. If (i) is true, let cp(t) = ( H x ( t ) - b x ( t ) ) T ( H x ( t ) - a x ( t ) ) e - 2 " 1 ' and let

$(T)=

c

q ( z ) dz = ( H x - a x , H x - b x ) , , , , .

Then $ ( T ) 5 0 for all T > 0 by (i). Integrating by parts yields ( H x - ax, H x - bx),, =

=

s,'I

dtj ( t ) / d t ]exp ( - 2bt) dt

$ ( T )exp(-26T)

+ 26

rTr

Jo lJo rt

1

1 cp (z) dz exp (- 2bt) dt

Similar arguments can also be employed in the proofs of cases (ii) and (iii). Since the statements in the following definition can be restated in terms of sector and positivity conditions, these statements will hold for a E (al, 0) whenever they hold for any a1 < 0. In this definition we concern ourselves with multi input-multi output systems of the type considered in Sections 6.2-6.4 (see in particular system (9) in Section 6.4, Part A). 6.6.2. Definition. A multi input-multi output system of the form e=x+w+z,

y=He

f = ~ + ~ + y ,

z=Bf

viewed as a single-loop system, is said to possess Property D with weight a < 0 if for some real constants c, r 2 0 and 6 E (0,l) and for the space X(a, m), the relation B is interior conic (c, r(l -6)) and the relation H satisfies one of the following conditions:

6.6 L,-STABILITY AND

251

OF LARGE SCALE SYSTEMS

I,-STABILITY

6) c2 > r 2 and H i s exterior conic ( - C ( C ~ - ~ ~ r) (-c ~2 -,r 2 ) - l ) , or (ii) r 2 > c2 and H is interior conic ( c ( r 2 - c2)-’, r ( r 2 - c ’ ) - ’ ) , or (iii) r 2 = c2 and 2cH+ I is positive. We now prove the following preliminary result.

6.6.3. Lemma. If system (9) possesses Property D, then in either of the special cases x = w = 0 or u = u = 0, the input-output relation connecting u or x tof or e, respectively, has finite gain which cannot exceed (Icl+ lr1)(6r)-

’.

Proof. This is just Theorem 6.4.14 and its proof restated. If u = u = 0, then by the proof of that theorem the transformed entities e’ = e

satisfy

+ cf,

y‘ = y

IlYll I g(H’)lle’ll I r - l lle’ll.

IIe’II I 6 - l IIx+wII,

Thus,

llell 5 IIe’II + I C I

f’ = H ‘ e ’

=f

IlfI = lle’ll + I C I IIYII I 6 - l IIx+wII

+ lclr-16-1 IIx+wII

= (I~l+r)(6r)-~IIx+wII.

If x

=

w

= 0, then

llfll

by the proof of Theorem 6.4.14 we have

I 6-’(1+r-’Icl)llu+ull

= (r+Ic1)(6r)-l

IIu+uII.

This concludes the proof. We now consider interconnected systems described by the set of equations ei = ui

+ Biyi,

yi

=

Hiei,

ui = xi

+ w,+

I j= 1

C , y j , (6.6.4)

i = 1, ..., 1. In Eq. (6.6.4) we have e i ( t ) , u i ( t ) , yi(t), x i ( t ) , w i ( t ) E R“‘, and n= ni. This system may be viewed as an interconnection of isolated subsystems described by the set of equ at’ions

zfel

e, = xi

+ wi + B i y i ,

y,

=

H .I e I. )

(6.6.5)

i = 1, ...,1. The relations B i , Hi,and C , have the same form as corresponding ones in Eq. (6.4.15) while X i is the space X ( o i ,ni) where oi < 0 for i = 1 , ...,1. The structure of system (6.6.4) is depicted in Fig. 6.3. In the subsequent results we require the notion of L,-memory (see Definition 6.1.22 and Theorems 6.1.23 and 6.1.24). 6.6.6. Theorem. Assume that all isolated subsystems (6.6.5), defined on Xe(ai,ni), satisfy Property D for some constants ai < 0, ri 2 0, and ci, i = 1, ...,I. Let o = maxi{oi} < 0. If each Hi has decaying L,-memory, if

VI

252

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

each mapping (C, H j ): X ( B ,n j ) X ( o , ni) has finite gain and if the successive principal minors of the test matrix R = [rJ defined by --f

are all positive, then the output relations &, i = I , . . ., Z, associated with composite system (6.6.4) are all bounded with respect to (L:, LZ). Proof. By Lemma 6.6.1 we can assume that B = a1 = ... = oI.Furthermore, Lemma 6.6.3 and Eq. (6.6.4) imply that for each i,

IIui It u,T

IIei tIu, T 5 ( IciI + ri) ( S i ri) -

I

I (/cil+ri)(8iri)-1{

/IXi+WiIIu,T

+ j = 1 g(cGHj)/lejllu,7'

1

.

Let E, = [lie, / / a , T ,..., l/e,l/u,T]T and define ( X + W ) , similarly. Then the above inequalities can be written as

RE, I (X+W)T where R is the test matrix. Observing once more that R is an M-matrix, it follows that R-' exists and that R-' = D = [di] 2 0, so that ET

I D(X+W),

or

whenever x , w E X(a, n). If x, w E L>, then certainly x, w E X(a, n) for any have

B

< 0. Furthermore, we

5 IIxj+wjIIL,[(e-2uT- 1)/(-2o)I1/' 5 / / x j +~ ~ ( ( ~ , e - ~ ~ / ( - 2 o ) ' ~ ~ ,

so that I

1

euT IIeiIIu,, I (dG/(-20)'/~) IIxj+ j= 1

WjlIL,.

By Theorem 6.1.24 it now follows that llyll,,, < co. When C, and Hi are linear time-invariant operators belonging to 9, with C,eu E 9 and Hie" E 9 for some B < 0, then the gains g ( C G H j )can be

6.6 L,-STABILITY AND

I,-STABILITY

OF LARGE SCALE SYSTEMS

253

obtained as the gains of the relations e-"C,Hje" from L 2 ( R + R"j) , to L , ( R + , R"'), that is, g(Cikffk)

=

ess s u p I c ~ ( o + j m ffk*(a+jm)I. )

UJhO

In this case we may replace the gains g(CikHk) in R by (6.6.8) and check that the successive principal minors of the resulting matrix are all positive. For it then follows by continuity that the test matrix R will have the desired properties for I CI sufficiently small. Using Theorem 6.6.6 as a model, an incremental counterpart of this theorem can easily be stated and proved. Analogous discrete time systems can be treated in a similar manner with L,-spaces replaced by I,-spaces. In this case Definition 6.6.2 and Lemma 6.6.3 remain valid without change, while in the proof of Lemma 6.6.1 we need to replace integration by parts with summation by parts, given by N

C ui(ui+l-ui) i= 1

Letting I

=

=z

uN+luN+1-utul-

N

C (ui+l-ui)ui+l.

i= 1

{0, 1,2, ...}, we use in the present case the notation Z(0, m) =

(I2

(4 R")),

where for t E 1, g(t) = exp(ot). For the discrete time version of composite system (6.6.4) with corresponding isolated subsystems of the form (6.6.5) the following result is now easily established.

6.6.9. Theorem. Assume that all isolated subsystems (6.6.5) of composite system (6.6.4) possess Property D for some constants ai < 0, r i 2 0, and c i . Let a = max{a,, ..., a,}. If each Hi has decaying I,-memory, if each mapping C,Hj maps Z ( a , n j )into Z ( o , n i )with finite gain g ( C , H j ) and if all successive principal minors of the test matrix R = [rV]determined by

are positive, then the input-output relations associated with composite system (6.6.4) are bounded with respect to ( I : , 1:). We conclude this section with a specific example.

6.6.10. Example. Consider the continuous time interconnected system shown in Fig. 6.13. Here T I , T,, T3, and a are positive constants. The operators

254

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Figure 6.13 Block diagram of Example 6.6.10 where a, T1, T2, T3are positive constants.

L,, L,, L , characterized by the indicated Laplace transforms belong to class Yeand for some constants CT; < 0, eU1Li E 9, i = I , 2,3. The operators N , , N , , and N , are assumed to belong to class h’” and are assumed to have graphs in the indicated shaded regions. Figure 6.13 indicates the identification 1 = 3, C,,= C,, = C,, = C,, = 1, and all other C, = 0. In the sequel we take full advantage of the remark associated with (6.6.8). Thus, we consider the unshifted Nyquist plots of the linear elements Li, i = 1,2,3. When a conicity condition is referred to, no mention will be made of a particular weight. As indicated in the remark for (6.6.8), if the conditions which we will derive are satisfied, then a negative weight CT, with 101 sufficiently small, can be found so that La-boundedness is guaranteed by Theorem 6.6.6 (with all conicity conditions, positivity conditions, etc., being interpreted with respect to that weight). Specifically, in what follows we seek to establish a relationship between the two positive parameters a and R , in Fig. 6.13 to guarantee that this system be L,-bounded. We first analyze the isolated subsystems determined by the indicated loops. Loop 1. N , is interior conic (0, R l ) . Setting c1/(rlZ-clz) = 0 and r , / ( r I 2 - c l 2 ) = R , , we find c, = 0 and y1 = l / R , . This subsystem will have

margin of boundedness 6, provided that L, is interior conic with center 0 and radius (1 -ij,).r,. This will be the case if ~ e ~ ’ “ ‘ * ~ ( u + j o 5 ) - (1 ’ ~-ijl)r1 for all o E R ’ . This is true for all TI 2 0 if I / a I ( 1 -6,)/R,. The best value of 6, is therefore given by 61

1 - (Rl/a).

(6.6.1 1)

is interior conic (0,p). Setting ~ ~ / ( r ~ ~ = - c0 , and ~ ) find c2 = 0 and r , = 2. This subsystem will have margin of boundedness 6 , provided that L , is interior conic (0,2( 1 - d 2 ) ) . This will be the case if I e - ’ ” T 2 ( 2 + j o ) ~ 1 ( 3 + ~ o ) - 1 ~-6,) ~ 2 ( for l all W E R’. This is Loop 2. N ,

r 2 / ( r 2 ’ - c,’)

= +, we

6.7 true for all T, 2 0 if

255

ANALYSIS AND DESIGN PROCEDURE

6 5 2(1 -8,).

The best value of 6, is therefore given by

6 =I-'="

12

12'

Loop 3. N3 is interior conic (+, 1). Setting ~ ~ / ( r ~= ~3 -a nd c ~ ~ ) r 3 / ( r 3 2- c),' = 1, we obtain c3 = 3 and r3 = 4. This subsystem will have margin of boundedness 6, if L , is interior conic (3, (1 - S3)+). This will be the case if /e-joT3[3(1 + j o ) ] - ' - 3 ] 5 ( I -8,)Q for all o E R'. This is true for all T3 2 0 if 1 I (1 -a3)+. The best value of 6, is therefore given by 83

-1-9-1

-

4

-

4'

Using the above calculations together with the gainsg(L,) g ( L 3 )= 3, we obtain the test matrix

R=

[

6,

-]/a

0

0

-1/3

11/12

-1/3

-1/6

1/6

]

=

I/a, g(L,)

= 4,

.

The successive principal minors of R are all positive if and only if

6, > 4/(7a).

(6.6.12)

Combining (6.6.1 1) and (6.6.12) it follows from Theorem 6.6.6 that the system of Fig. 6.13 is &-bounded if a

> R, + 417,

i.e., if the single pole of L , is at least a distance R , +4/7 into the left-half plane. Note that this condition is independent of T , 2 0, T, 2 0, T3 2 0. As expected, the condition on L , becomes more restrictive as the condition on N , becomes less restrictive, and conversely. 6.7 Analysis and Design Procedure

The results of Sections 6.2-6.6 can readily be used in analysis and design procedures of large scale systems. For purposes of discussion, we consider Theorem 6.4.17 in the following procedure. Similar statements apply to a great portion of the other results. Step 1. Impose the constraints that each isolated subsystem (described by Eq. (6.4.16)) have margins of boundedness ai, 0 < hi < 1. Calculate the corresponding gain factors pi.

Step 2. Form the test matrix R (of Theorem 6.4.17). Boundedness conditions are obtained by requiring that each of the successive principal minors of matrix R be positive.

256

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

Step 3. If the boundedness conditions are not satisfied, modify some or all of the isolated subsystems so as to reduce p i . Repeat Step 2. Although the present results may not represent the “best” stability conditions, they have the desirable property that they single out a class of modifications that can be made to each isolated subsystem separately to enhance overall system stability for the types of systems considered herein. This is possible in part because we frequently view large scale systems (i.e., multi input-multi output systems) as interconnected systems and in part because we often focus attention on large scale systems with subsystems which are single-loop feedback systems. Such a viewpoint is often possible when there is nontrivial feedback in a large scale system, and is a natural one when one attempts to stabilize such systems using local feedback. To be more specific, consider a system consisting of I “forward-loop’’ relations Hiinterconnected by a number of feedback relations C,, as shown in Fig. 6.14. If the conditions of Theorem 6.2.3 do not already yield boundedness, place a compensating feedback relation Bi around each forward loop relation H i , each Bi being chosen so that the resulting single-loop is bounded with margin of boundedness di. This procedure presupposes only that each Hisatisfies one of the conicity conditions of Theorem 6.4.14 for some choices of parameters r , c, and 6, which is not a very restrictive requirement. The compensated system then has the structure of Eq. (6.4.15). If the boundedness condition of Theorem 6.4.17 is not satisfied, then alter each of the feedback relations Bi,and thus the corresponding 6, and p i ,until it is. This can be accomplished by making the quantity pi sufficiently small. (In this connection, note that the test matrix R = [r,] of Theorem 6.4.17 is required to be an M-matrix. Furthermore, recall that a sufficient condition for R to be an M-matrixis the set ofinequalities r i i - C f = l , i +( r ji j I > 0, i = I , ..., 1.) At this point we hasten to emphasize that in some cases, application of the preceding compensation procedure may not be practical nor desirable. For example, certain “forward loop” relations may not be physically accessible to compensation, the cost may be prohibitive, etc. The similarity between the above procedure and those suggested in earlier

Figure 6.14 Typical interconnected feedback system where i = 1 , ...,I.

6.8

NOTES AND REFERENCES

257

chapters is of course not coincidental. After all, we have pursued throughout this book the same basic method of analysis. Only the technical details, frequently not trivial, have changed under vastly different situations.

6.8 Notes and References

Good references on systems theory (dealing with input-output properties of feedback systems) include the books by Willems [I], Holtzman [l], and Desoer and Vidyasagar [I]. A great deal of the fundamental work dealing with this subject is due to Sandberg (see, e.g., Sandberg [l-3, 7-91), Zames C1-41, and Popov [I]. For additional details concerning the variety of spaces considered in Section 6.1, refer to the books by Corduneanu [2, Chapter I] and Miller [ I , Section 5 of Chapter 111 and Chapter V]. Almost periodic functions are discussed in detail by Besicovitch [I] and Fink [l]. For applications of ultimately periodic or almost periodic spaces, see BeneS [3,4], BeneS and Sandberg [I], and Miller [2]. Weighted spaces are dicussed in Corduneanu [2], Massera and Schaeffer [a], Miller [l], and Gollwitzer [l], as well as in Willems [I], Desoer and Vidyasagar [l] and the work by Sandberg and Zames for the special case of shifted L,-spaces. In Definition 6.1.4 the term “compatible space” is used while in Corduneanu [2j, Massera and Schaeffer [2j, Gollwitzer [l], and Dotseth [l J the term “ X is a subspace of X , with stronger topology than that inherited from X,” is used to express the same idea. Results on boundedness and continuity of operators H E Lfe defined on X = L,” can be found in the work of Sandberg and Zames and in Desoer and Vidyasagar [l], Holtzman [l], and Willems [I]. Boundedness results of operators H E Sfe defined on various spaces are also given in Corduneanu [2], Massera and Schaeffer [2], Miller [l], Gollwitzer [I], and Dotseth [l]. Shifted L,-spaces and decaying L,-memory are discussed in detail in Zames [2]. Finally, for a connection between Lyapunov stability and input-output stability, refer to Willems [2]. The results in Section 6.2 are due to Lasley and Michel [I-61 and Lasley [l j. As presented here, these results are stated in sufficiently general form to include the interesting results of Araki [3], Cook [2], Tokumaru, Adachi, and Amemiya [l], and part of the work of Porter and Michel [3-51 and Porter [I]. The instability results in Section 6.3 are an adaptation of the interesting work of Takeda and Bergen [l]. For related work which generalizes or partially generalizes the results of Takeda and Bergen, refer to Vidyasagar [1) and Miller and Michel [l]. An integral equation example illustrating instability is given in the next chapter.

258

VI

INPUT-OUTPUT STABILITY OF LARGE SCALE SYSTEMS

The results in Section 6.4 are an adaptation of the work of Porter and Michel 13-51, Porter [I], Lasley and Michel [1,4,6], and Lasley [l]. For related results see Araki [3j, Cook [ 2 ] , and Sundareshan and Vidyasagar [I]. The Popov-like results of Section 6.5 are due to Lasley and Michel [2, 4, 61 and Lasley [l]. The material presented in Section 6.6 dealing with L,- and /,-stability is based on results by Lasley and Michel [3,5,6] and Lasley [l]. For additional related work refer to the papers by Cook [l], Miller and Michel [2], Rosenbrock [3], and Callier, Chan, and Desoer [1,2]. Also, refer to Kevorkian [l] and Ozguner and Perkins [I], where structural aspects of interconnected systems are considered.

CHAPTER VII

htegrodzfferential Systems

In the present chapter we apply the general stability and instability results of Chapter VI to interconnected systems described by Volterra integrodifferential equations. Throughout this chapter we employ the notation and conventions of the previous chapter. In the first section some preliminary background material is provided. In the second section L,-stability and instability results for interconnected systems described by Volterra integrodifferential equations are established. The emphasis in the third section is on linear systems and nearly linear systems. In the fourth section results of this chapter are applied to the point kinetics model of a coupled nuclear reactor with several cores. This chapter is concluded with a discussion of references in the fifth section.

7.1

Preliminary Results

A linear integrodifferential equation with kernel H the form B(t> = HY(t) + f @ ) 259

E

2ZIe is an equation of

(L)

VII

260

INTEGRODIFFERENTIAL SYSTEMS

for t 2 0 and y ( 0 ) = y o and inputfE X,. The resolvent of (L) is the function

R E Po,such that

R*(s) = (sZ- H*(s))-'

where R*(s) and H *(s) denote the Laplace transforms of the operators R and H , respectively. We say that H possesses Property F if det[jwZ-H*(jw)]

-a < w < co.

# 0,

Also, we say that H possesses Property Q if

det(s1-H*(s)) # 0 for all complex numbers s = o+jw such that 0 = Res 2 0. If R is the resolvent of H , then a simple argument involving Laplace transforms shows that the solution of (L) is given by Y ( t >= W ) y o + J ) ( t - r ) f ( 4

&.

(7.1.1)

Moreover, the following results are proved in Miller [4] and Grossman and Miller [I]. 7.1.2. Theorem. If H E 2,then R E Toif and only if H has Property Q. In this case the derivative R is also in To. 7.1.3. Theorem. If H E Tl and H has Property F then there is a finite (possibly empty) set P = {sl,. .., sN} of complex numbers such that Resj > 0, there is a set { M j k of } matrices, there are integers m j 2 1, and there is an operator S such that IS1 and belong to L , ( R + ) for all p E [ l , 001 such that

Is1

R*(s) = S*(S)+

N

C

2 mi

j=l k = l

The operator S in Theorem 7. I .3 is called the residual resolvent of Eq. (L). When P = @ then R E Toand S = R, by Theorem 7.1.2. All nonlinear integrodifferential equations considered in this chapter are of the form i ( t ) = Hx(t) B(x,,t) + f ( t ) (N)

+

for t 2 0 with x ( 0 ) = xo given. Here H E T l e , f € X,, B is a continuous mapping on X , x R + into R", and xt denotes the truncated function. These assumptions ensure that Eq. (N) has solutions which are unique ifBsatisfies a Lipschitz condition in x (see Miller [l, Chapter 111). If one thinks of B ( x , , t ) + f ( t ) as known, then from Eqs. (7.1.1) and (N) the variation of constants formula x(t)

follows.

=

R(t)xo + RCf(t)+B(x,,t)I

(V)

7.2 L,-STABILITY, INSTABILITY

OF INTERCONNECTED SYSTEMS

261

7.2 &-Stability and Instability of Interconnected Systems

Now in particular, let X , = Lz, and let X = L,". In the following, we apply some of the results from Sections 6.2 and 6.3 to equation (N). Let us assume that Eq. (N) can be written in the form ii(t)

Hizi(t)+

1

1

C1 B,(zjt, t > + A ( t ) ,

j=

(9)

i = 1, ..., I, for t 2 0 with zi(0) = zio given. Here Hi E PIC is n i x n, matrix valued, B,: L2,(R+, R"j) x R + + L,,(R+, R"'), E Xi, = L,,(R+, R"[), and n= ni. Following our earlier viewpoint, we consider composite system (9') as an interconnection of 1 isolated subsystems ($) described by equations of the form

x:f=

Yi(0) = zio, with interconnectincg structure specified by B, , i,j = 1 , ..., 1. Jji(t)

=

ffiyi(t) +f,(t),

($1

7.2.1. Theorem. Assume that HiE Y1 possesses Property Q and that fi E Xi for i = 1, ..., 1. Assume that all B, are bounded with respect to X and assume that the gains g(BJ < co for i,j = 1, ... ,1. If the successive principal minors of the test matrix A = [a,] defined by

are all positive, then the input-output relations Zi for composite system (9) are stable with respect to X . (Here thef, are inputs and the zi are outputs.) ProoJ By Theorem 7.1.2 the operator H = diag [ H , , .. ., I f l ] has resolvent R = diag[R,, ..., RJ E 9,By . the results in Section 6.1, Part D, R maps X , into X , with finite gain. Thus, Eq. (9') is equivalent to the variation of constants equation (V) and Theorem 6.2.3 can be applied to (V). The conclusion of the theorem follows. H

7.2.2. Remark. Note that the above theorem remains valid if ,'A is any extended space such that R maps X , into itself and X is the completely compatible subspace. For example, we may have X , = Lie and X = L," for 1 5p I 00, or we may have X , = C and X = BC. Of course the gains must always be computed with respect to the given space X . The conclusion of Theorem 7.2.1 can be strengthened as follows. 7.2.3. Theorem. Assume that all hypotheses of Theorem 7.2.1 are true. Then the solution x ( t ) of composite system (9) is in C , n L,".

262

VII

INTEGRODIFFERENTIAL SYSTEMS

Proof. By Theorem 7.2.1, f E L," = X implies that x E X . Hence, F ( t ) 4 f ( t ) + B ( x , ,t ) E X . By Theorems 7.1.2 and 7.1.3 we also have R E L , ( R f , R"') for p = 1,2 and R E L , ( R f , RRz)for p = 1,2. Thus, the convoIution

( R F ) ( t )=

sd

R(t-T)F(z) dz

is in BC (by the Schwarz inequality) and IRF(t)l i IIRIILzlIF\lL2 for t 2 0. Furthermore, the derivative

is also in L,". Thus, ( R F ) ( t )+ 0 as t + co and x(t) =

R ( t ) x , + ( R F ) ( t ) E C,.

It is also possible to establish reasonable conditions for composite system

(9) such that instability of one of the isolated subsystems (q)implies instability of (9). 7.2.4. Theorem. Assume that HiE 9, andJ;: E X for i = 1, ...,1. Assume that all B, are stable with respect to X with finite gains g(B& for i,.j = I , ..., 1. Suppose that each H i possesses either Property F or Property Q and suppose that for at least one value i = i,, Hi satisfies Property F only. Define M [mJ = [ g ( B g ) g , ( R j ) ]and let r ( M )be its spectral radius, i.e.,

=

r ( M ) = max { / A i l : liis an eigenvalue of M ) . If either r ( M ) < 1 or if r ( M )= 1 and mg > 0 for all i a n d j , then the inputoutput relations for composite system (9) are not bounded with respect to X . Proof. We shall apply Theorem 6.3.1. Pick zio Eq. (V) becomes zi = R , ( J ; . + E ~ =B,zj), , or

ei =f,+ 2. =

=0

for i = 1, . ..,I so that

1

j= 1

Bezj

(7.2.5)

R 1. e1.

for i = 1, ..., 1. HereA is in the orthogonal complement of the stable manifold of R , , i.e.,j; E S(R,)'. In order to apply Theorem 6.3.1 it is necessary to show that g , ( R i ) < co for all i and that S(R,)' # @ when Ri 4 9,. Since each Ri satisfies Property F it follows that R,*(ju) is bounded, say 1 R,*( j w ) I 5 M i on 0 5 u < co. If y = Ri cp and if cp E X i , then by the Parseval equation and the bound for IR,*(jo)l we have

Thus, g , ( R i ) I M ifor all i .

7.3

263

LINEAR EQUATIONS AND LINEARIZED EQUATIONS

For i = i, we assume that Ri possesses Property F but not Property Q. By Theorem 7. I .3 there is a nonempty set P = {sl,..., sN} in the right half of the complex plane such that

Since d" ds"

1

cu

-cp*(sj)

=

djm(cp)

(-z)"e-"Jrcp(7)dz

then IR,cpl can be in L, if and only if djm(cp)= 0 for 0 I m I mi- 1 and ..., N . Thus, when i = i, we have

j = 1,

S(Ri)' = {cp

E

Xi: dj,(cp) # 0 for at least o n e j and m }

#

0.

The conclusion of the theorem now follows from Theorem 6.3.1. Note that by Lemma 6.2.8 a sufficient condition for r ( M ) < 1 is that all successive principal minors of the test matrix A = [aij]defined by a.. =

1 - g(Bii)gc(Ri),

if

i

-g('ij)

if

i#,j

gc(Rj),

=j

are all positive. 7.3 Linear Equations and Linearized Equations

We now consider some properties of linear equations (L) and nearly linear equations (N). In the present section we define boundedness of system (N) as follows.

7.3.1. Definition. Given a space X we say that system (N) is bounded with respect to X if for eachfE X and each initial condition x, E R" the solution of (N) is in X . Note that system (L) is included as a special case of (N). We will require the following result. 7.3.2. Theorem. Consider system (L) where H E Y1is defined as Hcp(t) = h,cp(t)

+

6

h(t-z)cp(z) dz

and t'lh(t)l E L , ( R + ) for some r > 1 . Then the following conditions are equivalent.

VII

264

INTEGRODIFFERENTIAL SYSTEMS

(i) The resolvent R E Y o ; (ii) det(s1- h, - h*(s)) # 0 for all s such that Re s 2 0; (iii) system (L) is bounded with respect to X = L,". Proof. The equivalence of (i) and (ii) is just a restatement of Theorem 7. I .2. To show that (ii) is equivalent to (iii), assume that (ii) is true. Then by Theorems 7.1.2 and 7.1.3 it follows that R = S and IS1 E L, for p = 1,2. Since IRI E L2 we have R ( t ) y oE X . Since IRI E L , then by Theorem 6.1.13, Rf E X = L," i f f E X. Applying these facts to Eq. (7.1.1) we see that the solution y E X . Conversely, to prove that (iii) implies (ii), assume that (ii) is not true. Then there is a nonzero vector y o and a complex number so with Re so 2 0 such that [so

I - ho - Fi*(s,)] yo

=

0.

Define y ( t ) = exp(sot)yo and f(t)

=

1'

h ( t - z ) y ( z ) dz =

-a,

Then

So y solves Eq. (L) and y is not in X . But the Schwarz inequality shows that

ThusfE X but the solution y $ X. Combining Theorems 7.2.1 and 7.3.2 we obtain the following result for composite system (9').

7.3

265

LINEAR EQUATIONS AND LINEARIZED EQUATIONS

7.3.3. Theorem. For composite system (9') (described in Section 7.2) assume that X = L," and that all Hiand B, are linear and are specified by the formulas

(7.3.4)

where hi E L , ( R + ,R"'), b, E L , ( R + ,R"'), and t' /hi/ E L,, t' lb,l E L, for some r > 1 and for i , j = 1 , ..., 1. If the successive principal minors of the test matrix A = [a,] defined by

are all positive, then the resolvent of the linear composite system (9'is )in Yo. Of course the gains of R, and B, in the above theorem can be computed as g(Rk) = max IRk*(jo)l W 2 0

and (BiA

=

max I Bi"k( j o )I. U l t O

Note that in the scalar case these gains can effectively be estimated by graphical means. Theorem 7.3.3 can be combined with Theorems 6.1.13 and 6.1.14 to prove the following result.

7.3.5. Theorem. If the hypotheses of Theorem 7.3.3 are satisfied and if X is any of the spaces BC, C , , Co, A , , A P , or L," for some p in the interval 1 < p < co, then (linear) composite system (9') is bounded with respect to X . Theorems 7.2.4 and 7.3.2 can also be combined, to yield the following result.

7.3.6. Theorem. Suppose that X = L,", Hiand B, satisfy Eq. (7.3.4) with /Ail E L , , lb,l E L , , and trlhil E L,, t'lb,l E L , for some r > 1 and for i , j = 1, ..., 1. Suppose that each Hisatisfies either Property Q or Property F and for at least one i = i, it satisfies Property F only. Let M = [mi]= [g(B,)g,(Rj)] and let r ( M ) be the spectral radius of M . If either r ( M ) < 1 or r ( M ) = 1 and m, > 0 for all i andj, then the resolvent of linear composite system (9') is not in Y o Thus, . (9') is also not bounded with respect to L," or with respect to BC. An alternate form of the above instability theorem can be established.

7.3.7. Theorem. Consider composite system (9) with X

= L,"

and with Eq.

266

VII

INTEGRODIFFERENTIAL SYSTEMS

(7.3.4) true and [hi]E L , and lbijlE L , for all i,j = 1, ..., I. Suppose that for each i, H i satisfies either Property Q or Property F and that for at least one i = i, there is a complex number so with Res, > 0 such that (soZ- H;(s)) = 0. Let M = [g(Bc)g,(Rj)] have spectral radius r ( M ) < 1. Then the resolvent of linear composite system (9) is not in Zo.Indeed, there is a complex number s1 with Res, > 0 such that det[~l-H,*(s)6~-Bt(s)]

=

0

when

s = s1

(7.3.8)

where ii, denotes the Kronecker delta. Proof. Pick y > 0 and in Eq. ( Y ) replace z i ( t ) by zi(t)ePY‘,hi(t) by h i ( t ) e - Y ‘ ,etc. The stability equations for the modified subsystems are now of the form

det[(s+y)Z-

Hi*(s+y)]

=

0,

R e s 2 0.

If y is sufficiently small, then Property Q or F for the original subsystems will be preserved for the modified subsystems and r ( M ’ ) < 1 for the modified is unbounded matrix of gains M ’ . By Theorem 7.3.6 the modified system (9) with respect to X = L,” and thus also with respect to BC. By Theorem 7.1.2 it follows that det[(s+y)Z-

Hi*(s+y)4-B,j*(s+y)] = 0

has a root s2 with Res, 2 0. Thus, Eq. (7.3.8) is true with s,

= s,+y.

A major reason for the importance of determining stability of a linear system, i.e., for determining whether or not its resolvent is in Po,is that local stability or instability results for nonlinear systems can be determined by linearization. In order to illustrate this, let us consider systems described by equations of the form j(t)

(7.3.9)

=

with y ( 0 ) = y o E R” given. Suppose that n and m are continuously differentiable. For y o andf“small” we hope that y ( t ) will be ‘‘small” so that we may replace Eq. (7.3.9) by

j ( t ) = n‘(O)y(t) where nT = (nIT, ..., n:), matrices determined by n’(0)

=

+

b ( t - z ) m ‘ ( O ) y ( z ) dz + f ( t ) ,

(7.3.10)

mT = (mIT, ..., m:), and n’(0) and m’(0) are the

[30)],

m’(0) = p 3 0 ; l

7.4

267

AN EXAMPLE

Once the stability or instability of Eq. (7.3.10) is analyzed, we can return to the stability problem for the nonlinear equation (7.3.9) in the following fashion. 7.3.11. Theorem. If the resolvent of the operator H defined by

H&)

=

n’(O)q(t)

+

sd

b(t-z)rn’(O)cp(z) dz

(7.3.12)

, Eq. (7.3.9) is locally stable in the sense that given & > 0 there is is in 9,then I l f l l L , < 6, then y E BC

a 6 > 0 such that whenever lyol< 6, f E BC, and and Iy(t)l‘< d for all t 2 0.

For the proof of Theorem 7.3.1 1 refer to Grossman and Miller [I]. Instability of Eq. (7.3.10) corresponds to having a solution so to the equation det [sl- n’(0)- b*(s)rn’(O)]

=

0

(7.3.13)

with R e s 2 0. In the critical case where all roots satisfy Res, = 0 it is not possible to conclude anything about stability or instability of Eq. (7.3.9) from the stability properties of Eq. (7.3.10). However, in certain noncritical cases Res, > 0, the following is known. 7.3.14. Theorem. If H is given by Eq. (7.3.12), if Eq. (7.3.13) has at least one solution so with Res, > 0 and no critical solutions s with Res = 0, then there is a constant 6 > 0 and points y , inside the ball B(6) with arbitrarily close to the origin such that the solution of Eq. (7.3.9) withf(t) = 0 and y ( 0 ) = must leave the ball B(6) in finite time. For a proof of Theorem 7.3.14 see Miller and Nohel [I]. 7.4 An Example

In order to demonstrate the applicability and usefulness of the ideas and methods discussed above, we reconsider once more the nuclear reactor problem treated in Example 5.7.18. Throughout this section we use the same notation as was used in that example. Specifically, consider the point kinetics model of a coupled nuclear reactor with 1 cores (see Akcasu, Lillouche, and Shotkin [l] and Plaza and Kohler [I]) given by

268

VII

for i = I , ..., 6 and j given by

=

INTEGRODIFFERENTIAL SYSTEMS

I , ..., 1. Assume that the expression for reactivity p i , pj(t> =

J0 T t ; . ( t - z ) P j ( z )dz

is correct at least to linear terms and that the feedback function W , E L , . Solving for the c,(t) in terms of p j we see that c i j ( t )= c y ( 0 ) e - ' j t

+

*pj)

&(eAiJt

where * denotes the convolution integral. Substituting this expression into the equation for p j ( t ) and linearizing, we obtain

(7.4.1) , j = 1, ..., 1. This system has the form of

(9) (see Section 7.2) with

z j = p j , all

(7.4.2) (7.4.3) and

f,(t)

6

1 c,(O)e-"J'.

i= I

The resolvent R, for an isolated subsystem (one core) has Laplace transform Rk*(s)= l/D,(s) and is in Yoif

when R e s 2 0 . This condition can be checked graphically from a plot of 0 < a3 in the complex plane. The gain g ( & ) can also be determined since l / g ( R k )is the minimum distance from the graph of &(ja) on 0 I w < a3 to the origin. Moreover, D k ( j w ) on 0 I

I f the successive principal minors of the test matrix A , A

1

1- Cg(Ri)g(Bi)l

7.5

NOTES AND REFERENCES

269

are all positive, then by Theorem 7.2.3 the solutions of Eq. (7.4.1) are in C , n L,. Moreover, by Theorem 7.3.11, the original nonlinear point kinetics model is locally stable in this case. An alternate method of analysis would be to put Bjk = 0 in Eq. (7.4.3) when j = k and add a term A y 1 ( h j j* z j ) ( t ) to Eq. (7.4.2). The left-hand side of Eq. (7.4.4) will have the added term A; 'h;(s) while g ( B j j ) = 0 for all j . The rest of the stability analysis proceeds as before. If D k ( j o )# 0 for --GO < o < co and k = 1, ..., I, and &(s) = 0 for some k = k , and s = so with Res, > 0, then the instability results can be applied. As above, l/gc(Rk)is the minimum distance from the graph of D k ( j o )on -co < w < co to the origin. If the spectral radius of the matrix M = [g(B&g,(Rj)] is less than one, then system (7.4.1) is L,-unstable (see Theorem 7.3.7). Finally, from Theorem 7.3.14 we see that the original nonlinear point kinetics model is in this case also unstable in the sense of that theorem. 7.5

Notes and References

Basic background material on Volterra integral and integrodifferential equations can be found in the books by Bellman and Cooke [I, Chapter 71, Volterra [l], Miller [l], and Corduneanu [2]. The stability aspects of Property Q and local stability via linearization of integrodifferential equations are studied in Grossman and Miller [I] and Corduneanu [I]. Unstable linear systems satisfying Property F are investigated in Miller [4] while local instability via linearization is studied in Miller and Nohel [I]. In addition, related results for integral equations can be found in several other papers. In particular, the significant work of Sandberg [l-91 develops the systems theoretic viewpoint used in studying both L, and L , stability. The work of BeneS 11-41 and BeneS and Sandberg [l] exploits fixed point theorems while Wu and Desoer [l] use operator techniques. Hale [3] develops C,-semigroup theory for a class of linear problems while Levin [I] and Driver [l] use Lyapunov functions to study Volterra integrodifferential equations. For additional references, see also the bibliography in Miller [l]. The results in Sections 7.2 and 7.4 are based on Miller and Michel [l]. Related results for integral equations can also be found in Miller and Michel [2]. Theorem 7.3.2 is an adaptation and expansion of a result given in Grossman and Miller [l]. Theorems 7.3.3 and 7.3.5-7.3.7 are new. An introduction to reactor dynamics and to the related engineering literature can be found in Akcasu, Lillouche, and Shotkin [l]. For a more mathematical treatment of the stability of point kinetics models of nuclear reactors refer to Bronikovski, Hall, and Nohel 111, Bronikovski [l], Levin and Nohel [l], Ergen, Lipkin, and Nohel [l], Hsu [I], and the bibliographies in these references.

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References

Aizerman, M. A. and Gantmacher, F. R. [I] Absolute Stability of Regulator Systems, Holden-Day, Inc. San Francisco, California, 1964. Akcasu, Z., Lillouche, G. S., and Shotkin, L. M. [l] Mathematical Methods in Nuclear Reactor Dynamics, Academic Press, New York, 1971. Apostol, T. M. [l] Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1957. Araki, M. [I] Stability of composite control systems, Ph.D. Dissertation, Dept. of Electronics, Kyoto Univ., Kyoto, Japan (1971). [2] M-matrices, Dept. of Computing and Control, Imperial College of Science and Technol., London, England, Rep. No. 74/19 (March 1974). [3] Input-output stability of composite feedback systems, Dept. of Computing and Control, Imperial College of Science and Technol., London, England, Rep. No. 75/1 (January 1975). [4] Application of M-matrices to the stability problems of composite dynamical systems, J. Math. Anal. Appl. 52 (1975). 309-321. Araki, M. and Kondo, B. [I] Stability and transient behavior of composite nonlinear systems, 1EEE Trans. Automatic Control 17 (1972), 537-541.

2 71

272

REFERENCES

Araki, M., Ando, K., and Kondo, B. [I] Stability of sampled-data composite systems with many nonlinearities, ZEEE Trans. Automatic Control 16 (1971), 22-27. Arnold, L. [I] Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974. Athans, M., Sandell, N., and Varaiya, P. [I] Stability of interconnected systems, Proceedings of the 1975 ZEEE Conference on Decision and Control, Houston, Texas, IEEE, December 1975, pp. 456-462. Bailey, F. N. [I] Stability of interconnected systems, Ph.D. Dissertation, Univ. of Michigan, Ann Arbor, Michigan (1964). [2] The application of Lyapunov’s second method to interconnected systems, SZAM J. Controf 3 (1966), 443462. Barbu, V. and Grossman, S. I. [l] Asymptotic behavior of linear integrodifferential systems, Trans. Amer. Math. SOC. 171 (1972), 277-288. Bellman, R. [ I ] The boundedness of solutions of infinite systems of linear differential equations, Duke Math. J. 14 (1947), 695-706. [2] Vector Lyapunov functions, S l A M J. Control 1 (1962), 32-34. [3] Introduction to Matrix Analysis, McGraw-Hill, New York, 1970. [4] Large systems, IEEE Trans. Automatic Control 19 (1974), 465. Bellman, R. and Cooke, K. L. [ I ] Differential Difference Equations, Academic Press, New York, 1963. BeneS, V. [ I ] A fixed point method for studying the stability of a class of integrodifferential equations, J. Math. and Physics 40 (1961), 55-67. [2] A nonlinear integral equation from the theory of servomechanisms, Bell System Tech. J . 40 (1961), 1309-1321. [3] Ultimately periodic solutions to a nonlinear integrodifferential equation, Bell System Tech. J . 41 (1962), 257-268. [4] Ultimately periodic behavior in a class of nonlinear servomechanisms, Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963. BeneS, V. and Sandberg, I. W. [ I ] On the response of a time variable nonlinear system to almost periodic signals, J. Math. Anal. Appf. 10 (1965), 245-268. Bers, L., John, F., and Schechter, M. [ I ] Partial Differential Equations, Amer. Math. SOC.,Providence, Rhode Island, 1964. Bertram, J. E. and Sarachik, P. E. [I] Stability of circuits with randomly time-varying parameters, IRE Trans. Circuit Theory 6 (1959), 260-270. Besicovitch, A. S. [ I ] Almost Periodic Functions, Dover, New York, 1954. Bhatia, N. P. and Szego, G. P. [ I ] Stability Theory of’ Dynamical Systems, Springer-Verlag, Berlin and New York, 1970. Blight, J. D. and McClamroch, N. H. [ I ] Graphical stability criteria for large-scale nonlinear multiloop systems, Preprints, Sixth ZFAC World Congress, Boston, Massachusetts, August 1975, IFAC, 1975 (Paper No. 44.5).

REFERENCES

273

Bose, A. [I] Stability and compensation of systems with multiple nonlinearities, Ph.D. Dissertation, Dept. of Electrical Engineering, Iowa State Univ., Ames, Iowa (1974). Bose, A. and Michel, A. N. [I] Qualitative analysis of large-scale systems, Proceedings of the 17th Midwesr Symposium on Circuit Theory, Univ. of Kansas, Lawrence, Kansas, September 1974, Western Periodicals, North Hollywood, California, 1974, pp. 1-8. [2] Qualitative analysis of large-scale systems: stability, instability and boundedness, 2. Angew. Math. Mech. 56 (1976), 13-20. Brezis, H. [I] Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed., Academic Press, New York, 1971, pp. 101-156. [2] Operateurs Maximaux Monotones, North-Holland Publ., Amsterdam, 1973. Bronikovski, T. A. [l] An integrodifferential system which occurs in reactor dynamics, Arch. Rational Mech. Anal. 37 (1970), 363-380. Bronikovski, T. A,, Hall, J. E., and Nohel, J. A. [I] Quantitative estimates for a nonlinear system of integrodifferential equations arising in reactor dynamics, SIAM J. Math. Anal. 3 (1972), 567-588. Buck, R. C. [I] Advanced Calculus, McGraw-Hill, New York, 1956. Burns, J. A. and Herdman, T. L. [I] An application of adjoint semigroup theory for a functional differential equation with infinite delays to a Volterra integrodifferential system, J. Math. Anal. Appl. (to appear). Callier, F. M., Chan, W. S.,and Desoer, C. A. [l] Stability theory of interconnected systems-Part I : Arbitrary interconnections, Electronics Research Laboratory, Univ. of California, Berkeley, California, Rep. NO. ERL-M565 (August 1975). [2] Stability theory of interconnected systems-Part I1 : Strongly connected subsystems, Electronics Research Laboratory, Univ, of California, Berkeley, California, Rep. No. ERL-M565 (November 1975). Chaffee, N. [I] A stability analysis for a semilinear parabolic partial differential equation, J. Differential Equations 15 (1 974), 522-540. Chaffee, N. and Infante, E. F. [l] A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974), 17-37. Coddington, E. A. and Levinson, N. [l] Theory of Ordinary Diferential Equations, McGraw-Hill, New York, 1955. Coleman, B. D. and Mizel, V. J. [l] Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87-123. 123 On the stability of solutions of functional differential equations, Arch. Rational Mech. Anal. 30 (1968), 173-196. Cook, P. A. [I] Modified multivariable circle theorems, in Recent Mathematical Developments in Control, D. J. Bell, Ed., Academic Press, London, 1973, pp. 367-372. [2] On the stability of interconnected systems, Internat. J. Conrrol20 (1974), 407-415.

274

REFERENCES

Corduneanu, C. [l] Some differential equations with delay, Proceedings of EQUADIFF 3, Brno, Czechoslovakia, f972 (Czechoslovak Conference on Differential Equations and Their Applications), pp. 105-1 14. [2] Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. Crandall, M. G. [I] Semigroups of nonlinear transformations in Banach spaces, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed., Academic Press, New York, 1971. Crandall, M. G. and Liggett, T. M. [I] Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. Dantzig, G. B. [l] Linear Programming and Extensions, Princeton Univ. Press, Princeton, New Jersey, 1963. Defermos, C. M. and Slemrod, M. [ I ] Asymptotic behavior of nonlinear contraction semigroups, J . Functional Analysis 13 (1973), 97-106. Desoer, C. A. and Vidyasagar, M. [ I ] Fkedback Systems: Input-Output Properties, Academic Press, New York, 1975. Dieudonne, J. [ I ] Foundations of Modern Analysis, Academic Press, New York, 1960. Doetsch, G. [ I ] Theorie und Anwendung a'er Laplace-Transformarion, Springer-Verlag, Berlin and New York, 1937. Doob, J. L. [ l ] Stochastic Processes, Wiley, New York, 1953. Dotseth, G. M. [ I ] Admissibility results on subspaces of C(R,R") and LLP(R,R"),Math. Systems Theory 9 (1975), 9-17. Driver, R. D. [ I ] Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal. 10 (1962), 401426. Dunford, N. and Schwarz, J. T. [ I ] Linear Operufors, Part 1, Wiley (Interscience), New York, 1958. Ergen, W. K., Lipkin, H . J., and Nohel, J. A. [ I ] Application of Lyapunov's second method in reactor dynamics, J. Math. and Phys. 36 (1957), 3 6 4 8 . Fiedler, M. and Ptak, V. [l] On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J. 12 (1962), 382400. Fink, A. M. [ I ] Altnost Periodic Differential Equations (Lecture Notes in Mathematics), SpringerVerlag, Berlin and New York, 1974. Gantmacher, F. R. [ I ] The Theory ofMutrices, Vol. I , Chelsea, Bronx, New York, 1959. [2] The Theory ofMutrices, Vol. 11, Chelsea, Bronx, New York, 1959. Godunov, A. N. [I] Peano's theorem in Banach spaces, Functional Anal. Appl. 9 (1975), 53-56.

REFERENCES

275

Gollwitzer, H. E. [I] Admissibility theory and the global behavior of solutions of functional equations, in Delay and Functional Differential Equations, K. Schmitt, Ed., Academic Press, New York, 1972. Grimmer, R. [I] Stability of a scalar differential equation, Proc. Amer. Math. SOC.32 (1972), 452-456. Grossman, S. I. and Miller, R. K. [I] Nonlinear Volterra integrodifferential systems with L’-kernels, J. Differential Equations 13 (1973), 551-566. Grujid, Lj. T. and Siljak, D. D. [I] On stability of discrete composite systems, ZEEE Trans. Automatic Control 18 (1973), 522-524. [2] Asymptotic stability and instability of large-scale systems, ZEEE Trans. Automatic Control 18 (1973), 636645. [3] Exponential stability of large-scale discrete systems, Internat. J. Control 19 (1974). 48 1-49 1. Hahn, W. [11 Theorie und Anwendung der Direkten Methode von Ljapunov, Springer-Verlag, Berlin and New York, 1959. [2] Stability of Mofion, Springer-Verlag, Berlin and New York, 1967. Halanay, A. [I] Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 1966. Hale, J. K. [I] Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39-59. [2] Functional Differential Equations, Springer-Verlag, Berlin and New York, 197 I. [3] Functional differential equations with infinite delays, J. Math. Anal. Appl. 48 (1974), 276283. [4] The solution operator with infinite delays, International Conference on Differential Equations, Brown Univ., Providence, Rhode Island, August 1974, Academic Press, New York, 1975, pp. 330-336. Hille, E. and Phillips, R. S. [I] Functional Analysis and Semi-Groups (Amer. Math. SOC. Colloquium Pub]., Vol. 33), Amer. Math. SOC.,Providence, Rhode Island, 1957. Holtzman, J. M. [I] Nonlinear System Theory: A Functional Analysis Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1970. Hsu, C. [I] Stability of reactor systems via Lyapunov’s second method, Trans. ASME J . Basic Eng. 89 (1967), 307-310. Infante, E. F. and Walker, J. A. [I] On the stability properties of an equation arising in reactor dynamics, Lefschetz Center for Dynamical Systems, Lecture Notes 74-10, Brown Univ., Providence, Rhode Island (1974). Kalman, R. E. [ I ] Lyapunov functions for the problem of Lurk, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 201-205. Kalman, R. E. and Bertram, J. E. [ I ] Control system analysis and design via the “second method” of Lyapunov-

2 76

REFERENCES

Part 1: Continuous-time systems, Trans. A S M E J . Basic Eng. 82 (1960), 371-393. [2] Control system analysis and design via the “second method” of LyapunovPart 11: Discrete-time systems, Trans. A S M E J . Basic Eng. 82 (1960), 394400. Kalman, R. E., Ho, Y. C., and Narendra, K. S. [ I ] Controllability of linear dynamical systems, Contributions to Differential Equations 1 (1963), 189-213. Kamke, E. [ I ] Zur Theorie der Systeme gewohnlicher Differentialgleichungen, 11, Acta Math. 58 (1932), 57-85. Kats, I . la. and Krasovskii, N. N. [ I ] On the stability of systems with random parameters, PMM 24 (1960), 809-823. Kevorkian, A. K. [ I ] Structural aspects of large dynamic systems, Preprints, Sixth IFAC World Congress, Boston, Massachusetts, Angust 1975, IFAC, 1975 (Paper No. 19.3). Kozin, F. [ I ] A survey of stability of stochastic systems, Automatika 5 (1969), 95-1 12. Krasovskii, N. N. [ I ] Stability of Motion, Stanford Univ. Press, Stanford, California, 1963. Krein, S. G . [ I ] Linear D[flerential Equations in Banach Space (Translations of Mathematical Monographs, Vol. 29), Amer. Math. Soc., Providence, Rhode Island, 1970. Kron, G . [I] Diakoptics, Macdonald, London, 1963. K u o , B. C . [ I ] Discrete-Data Control Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1970. Kurtz, T. [ I ] Convergence of sequences of semigroups of nonlinear equations with applications to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259-272. Kushner, H . [I] Stochastic Stability and Control, Academic Press, New York, 1967. 121 Converse theorems for stochastic Lyapunov functions, SIAM J. Control 5 (1967), 228-233. [3] Introduction to Stochastic Control, Holt, Rinehart and Winston, New York, 1971. Ladas, G . E. and Lakshmikantham, V. [I] D(Jferentia1Equations in Abstract Spaces, Academic Press, New York, 1972. Lakshmikantham, V. [I] Stability and asymptotic behavior of solutions of differential equations in a Banach space, Technical Rep., Dept. of Mathematics, Univ. of Texas, Arlington, Texas. Lakshniikantham, V. and Leela, S. 111 Diff2rential and Integral Inequalities, Vol. I , Academic Press, New York, 1969. 121 Difitwntial and Integral Inequalities, Vol. 11, Academic Press, New York, 1969. LnSalle, J. P. [ I ] Vector Lyapunov functions, Bulletin of the Institute of Mathematics, Academia Siriica 3 (1975), 139-150. LaSalle, J . P. and Lefschetz, S. [ I ] Stability by L-vapunou’s Direct Method with Applications, Academic Press, New York, 1961. Lasdon, L. S. s New York, 1970. [ I ] Optimization Theory of Large S ~ ~ t e n iMacmillan,

REFERENCES

277

Lasley, E. L. [l] The qualitative analysis of composite systems, Ph.D. Dissertation, Dept. of Electrical Engineering, Iowa State Univ., Ames, Iowa, (1975). Lasley, E. L. and Michel, A. N. [I] Input-output stability of composite systems, Proceedings of the 8th Asilomar Conference on Circuits, Systems, and Computers, PaciJic Grove, California, December 1974, Western Periodicals, North Hollywood, California, 1974, pp. 472-482. [2] Input-output stability of interconnected systems, Proceedings of the IEEE International Symposium on Circuits and Systems, Boston, Massachusetts, April 1975, IEEE, 1975, pp. 131-134. [3] L,- and /,-stability of interconnected systems, Proceedings of the IEEE Conference on Decision and Control, Houston, Texas, December 1975, IEEE, 1975, pp. 375-382. [4] Input-output stability of interconnected systems, IEEE Trans. Automatic Control 21 (1976), 8 4 8 9 . [ 5 ] L,- and I,- stability of interconnected systems, IEEE Trans. Circuits and Systems 23 (1976), 261-270. [6] Input-output stability of large-scale systems, in Large-Scale Dynamical Systems, R. Saeks, Ed., Point Lobos Press, Los Angeles, California, 1976, pp. 195-220. Lasota, A. and Yorke, J. A. [I] The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13 (1973), 1-12. Lefschetz, S. [ I ] Stability of Nonlinear Control Systems, Academic Press, New York, 1965. Levin, J . J. [l] The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc. 14 (1963), 534547. Levin, J. J. and Nohel, J. A. [I] On a system of integro-differential equations occurring in reactor dynamics, J . Math. Mech. 9 (1960), 347-368. Massera, J. L. [l] Contributions to stability theory, Annals of Math. 64 (1956), 182-206. Massera, J . L. and Schaeffer, J. J. [I] Linear differential equations and functional analysis-Part I, Annals of Math. 67 (1958), 517-573. [2] Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. Matrosov, V. M. [ 11 Method of Lyapunov-vector functions in feedback systems, Automat. Remote Control 33 (1972), 1458-1469. [2] The method of vector Lyapunov functions in analysis of composite systems with distributed parameters, Automat. Remote Control 34 (1973), 1-16. Matzer, E. [I] Zur Stabilitat gekoppelter Systeme, D. Sc. Dissertation, Technical Univ. of Graz, Graz, Austria (1973). McClamroch, N. H. [I] A representation for multivariable feedback systems and its use in stability analysis, Dept. of Engineering, Univ. of Cambridge, England, Report No. CUED/B-Control TR/98 (1 975). McClamroch, N. H. and Ianculescu, G. D. [l] Global stability of two linearly interconnected nonlinear systems, IEEE Trans. Automatic Control 20 (1979, 678-682.

278

REFERENCES

McKenzie, L. W. [I] The matrix with dominant diagonal and economic theory, Proceedings of a Symposium on Mathematical Methods in the Social Sciences, Stanford Univ. Press, Palo Alto, California, 1960, pp. 277-292. Mesarovid, M. D. and Takahara, Y. [I] General Systems Theory: Mathematical Foundations, Academic Press, New York, 1975. MesaroviC, M. D., Macko, D., and Takahara, Y. [ I ] Theory of Hierarchical Multilevel Systems, Academic Press, New York, 1970. Metzler, L. [I] Stability of multiple markets: The Hicks Conditions, Econometrica 13 (1945), 277-292. Michel, A. N. On the bounds of the trajectories of differential systems, Internat. J . Control 10 (1969), 593-600. Quantitative analysis of systems: stability, boundedness and trajectory behavior, Arch. Rational Mech. Anal. 38 (1970), 107-122. Stability, transient behavior and trajectory bounds of interconnected systems, Internat. J. Control 11 (1 970), 703-7 15. Quantitative analysis of simple and interconnected systems: stability, boundedness, and trajectory behavior, IEEE Trans. Circuit Theory 17 (1970), 292-301. Stability analysis of interconnected systems, Berichte der MathematischStatistischen Sektion im Forschungszentrum Graz, Bericht Nr. 4, Technical Univ. of Graz, Graz, Austria (1973). Stability analysis and trajectory behavior of composite systems, Proceedings of the 1974 IEEE International Symposium on Circuits and Systems, San Francisco, California, IEEE, April 1974, pp. 240-244. Stability analysis of interconnected systems, S I A M J . Control 12 (1974), 554-579. Stability analysis of stochastic large-scale systems, Proceedings of the 8th Annual Princeton Conference on Information Sciences and Systems, Princeton Univ., Princeton, New Jersey, March 1974, pp. 285-288. Stability analysis and trajectory behavior of composite systems, IEEE Trans. Circuits and Systems 22 (1979, 305-312. Stability analysis of stochastic large-scale systems, Z. Angew. Math. Mech. 55 (1979, 93-105. Stability analysis of stochastic composite systems, IEEE Trans. Automatic Control 20 (1 975), 246-250. Michel, A. N. and Heinen, J. A. [l] Quantitative stability of dynamical systems, Internat. J. Systems Sci. 1 (1971), 303-306. [2] Quantitative Mengenstabilitat von Systemen, Regelungstech. Prozep-Datenverarbeit. 20 (1972), 113-121. [3] Comparison theorems for set stability of differential equations, Internat. J. Systems Sci. 12 (1972), 317-324. [4] Quantitative and practical stability of systems, Automatic Control Theory Appl. 1 (1972), 9-15. Michel, A. N. and Porter, D. W. [I] Analysis of discontinuous large-scale systems : Stability, transient behavior, and trajectory bounds, Proceedings of the 1970 IEEE International Symposium on Circuit Theory, Atlanta, Georgia, IEEE, December 1970, pp. 92-94.

REFERENCES

279

Analysis of discontinuous large-scale systems: stability, transient behavior, and trajectory bounds, Int. J. Systems Sci. 2 (1971), 77-95. [3] Stability analysis of composite systems, IEEE Trans. Automatic Control 17 (1972), 222-226. [4] Practical stability and finite-time stability of discontinuous systems, IEEE Trans. Circuit Theory 19 (1972), 123-129. Michel, A. N. and Rasmussen, R. D. [I] Stability analysis of stochastic interconnected systems, Proceedings of the 12th Annual Allerton Conference on Circuit and System Theory, Univ. of Illinois, Urbana, Illinois, October 1974, pp. 77-86. [2] Stability of stochastic composite systems, IEEE Trans. Automatic Conirol 21 (1976), 89-94. [3] Stability anlysis of stochastic interconnected systems, in Advances in Control and Dynamical Systems: Theory and Applicaiions, Vol. 13, C. T. Leondes, Ed., Academic Press, New York, 1976. Miller, R. K. [l] Nonlinear Volterra Zntegral Equations, Benjamin, Menlo Park, California, 1971. [2] Almost-periodic behavior of solutions of a nonlinear Volterra system, Quart. Appl. Math. 28 (1971), 553-570. [3] Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac. 17 (1974), 39-55. [4] Structure of solutions of unstable linear Volterra integrodifferential equations, J. Diff. Equations 15 (1974), 129-157. Miller, R. K. and Michel, A. N. [l] L,-stability and instability of large-scale systems described by integrodifferential equations, SIAM J . Math. Anal. (to appear). [2] Stability of multivariable feedback systems containing elements which are openloop unstable, Proceedings of the 13th Annual Allerton Conference on Circuit and System Theory, Univ. of Illinois, Urbana, Illinois, October 1975, pp. 580-589. Miller, R. K. and Nohel, J. A. [ I ] A stable manifold theorem for a system of Volterra integrodifferential equations, SIAM J. Math. Anal. 6 (1975), 506-522. Mitra, D. and So, H. C. [I] Existence conditions for L , Lyapunov functions for a class of nonautonomous systems, IEEE Trans. Circuit Theory 19 ( I 972), 594-598. Miiller, M. [I] Uber das Fundamentaltheorem in der Theorie der gewohnlichen Differentialgleichungen, Math. Z. 26 (1926), 619-645. Narendra, K. S. and Neuman, C. P. [l] Stability of contipuous time systems with n-feedback nonlinearities, AIAA 1. 11 (1967), 2021-2027. Narendra, K. S. and Taylor, H. J. [11 Frequency Domain Criteria for Absolute Sfabilify,Academic Press, New York, 1973. Ostrowski, A. [l] Determinanten mit iiberwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen, Comment. Math. Helv. 30 (1956), 175-210. Ozgiiner, U. and Perkins, W. R. [I] On the multilevel structure of large scale composite systems, IEEE Trans. Circuit.v and SyJtems 22 (1979,618-622. [2]

280

REFERENCES

Pai, M. A. and Narayan, C. L. [I] Stability of large scale power systems, Preprints, Sixth IFAC World Congress, Boston, Massachusetts, August 1975, IFAC, 1975 (Paper No. 31.6). Pao, C. V. [ I ] Semigroups and asymptotic stability of nonlinear differential equations, SIAM J . Control3 (1972), 371-379. Pazy, A. [ I ] Semigroups of linear operators and applications to partial differential equations, Univ. of Maryland, Dept. of Mathematics, Lecture Notes No. 10 (1974). Piontkovskii, A. A. and Rutkovskaya, L. D . [I] Investigation of stability theory problems by the vector Lyapunov function method, Automat. Remote Control 10 (1967), 1422-1429. Plant, R. H. and Infante, E. F. [ I ] Bounds on motions of some lumped and continuous dynamical systems, Trans. A S M E Ser. E. J. Appl. Mech. 38 (1972), 251-256. Plaza, H . and Kohler, W. H. 111 Coupled-reactor kinetics equations, Nuclear Science and Engineering, 22 (1966), 4 19422. Popov, V. M. [ I ] Absolute stability of nonlinear systems of automatic control, Automat. Remote Control 22 (1961), 857-875. Porter, D. W. [ I ] Stability of multiple-loop nonlinear time-varying systems, Ph.D. Dissertation, Iowa State Univ., Ames, Iowa (1972). Porter, D. W. and Michel, A. N. [ I ] Stability of composite systems, Proceedings of the 4th Asilomar Conference on Circuitsand Systems, Pacific Grove, California, Nooember 1970, Western Periodicals, North Hollywood, California, 1970, pp. 634-638. 121 Stability analysis of composite systems with nonlinear interconnections, Proceedings of the 14th Midwest Symposium on Circuit Theory, Univ. of Denver, Dencer, Colorodo, May 1971, Western Periodicals, North Hollywood, California, 1971, pp. 6.6.1-6.6.10. [3] Stability of multiple-loop nonlinear time-varying systems, Rept. No. ISU-ERIAMES-73167, Iowa State Univ., Ames, Iowa (1973). [4] Input-output stability of multiloop feedback systems, Proceedings of the 1974 Joint Automatic Control Conference, Austin, Textis, IEEE, June 1974, pp. 75-82. [5] Input-output stability of time varying nonlinear multiloop feedback systems, IEEE Trans. Automatic Control 19 (1974), 422427. Quirk, J. and Saposnik, R. [ I ] Introchrciion to General Equilibrium Theory and Welfare-Economics, McGraw-Hill, New York, 1968. Rasmussen, R. D. [ I ] Lyapunov stability of large-scale dynamical systems, Ph.D. Dissertation, Dept. of Electrical Engineering, Iowa State Univ., Ames, Iowa (1976). Rasmussen, R. D . and Michel, A. N. [ 11 On vector Lyapunov functions for stochastic dynamical systems, Proceedings of ihe 3rd Milwaukee Symposium on Automatic Computation and Control, Milwaukee, Wisconsin, April 1975, Western Periodicals, North Hollywood, California, 1975, pp. 161-166. 121 Stability analysis of large scale dynamical systems, Proceedings of the 18th Midwest

REFERENCES

281

Symposium on Circuit Theory, Montreal, Canada, August 1975, Western Periodicals, North Hollywood, California, 1975, pp. 144-148. [3] On vector Lyapunov functions for stochastic dynamical systems, IEEE Trans. Automatic Control 21 (1976), 250-254. [4] Stability of interconnected dynamical systems described on Banach spaces, IEEE Trans. Automatic Control 21 (1976), 464471. Rosenbrock, H. H. [I] A Lyapunov function with applications to some nonlinear physical systems, Autornatika 1 (1963), 31-53. [2] A Lyapunov function for some naturally occurring linear homogenous time dependent equations, Automatika 1 (1963), 97-109. [3] Multivariable circle theorems, in Recent Mathematical Developments in Control, D, J. Bell, Ed., Academic Press, London, 1973, pp. 345-364. Sandberg, I. W. On the response of nonlinear control systems to periodic input signals, Bell System Tech. J . 43 (1964), 911-926. On the L,-boundedness of solutions of nonlinear functional equations, Bell System Tech. J. 43 (1964), 1581-1599. A frequency domain condition for the stability of feedback systems containing a single time-varying nonlinear element, Bell System Tech. J . 43 (1964), 1601-1608. A condition for the L,-stability of feedback systems containing a single timevarying nonlinear element, Bell System Tech. J . 43 (19641, 1815-1817. On truncation techniques in approximate analysis of periodically time varying nonlinear networks, IEEE Trans. Circuir Theory 11 ( 1964), 195-201. A stability criterion for linear networks containing time varying capacitors, IEEE Trans. Circuit Theory 12 (1965), 2-1 I. On the boundedness of solutions of nonlinear integral equations, Bell Sysfem Tech. J. 44 (1965), 439453. Some results on the theory of physical systems governed by nonlinear functional equations, Bell System Tech. J . 44 (1965), 871-898. Some stability results related to those of V. M. Popov, Bell System Tech. J . 44 (1965), 2133-2148. Some theorems on dynamic response of nonlinear transistor networks, Bell System Tech. J . 48 (1969), 35-54. Shaw, L. [1] Existence and approximation of solutions to an infinite set of linear time-invariant differential equations, SIAM J . Appl. Math. 22 (1972), 266-279. Sirazetdinov, T. K. [I] On the theory of stability of processes with distributed parameters, PMM 31 (1967), 3748. Slemrod, M. [l] Asymptotic behavior of a class of abstract dynamical systems, J. Dijerential Equations 7 (1970), 584-600. [2] Asymptotic behavior of C,-semigroups as determined by the spectrum of the generator, Indiana (lniu. J . Math. (to appear). Suhadloc, A. [I] On a system of integrodifferential equations, SIAM J. Appl. Math. 21 (1971), 195-206. Sundareshan, M. K. and Vidyasagar, M. [ I ] L,-stability of large-scale dynamical systems: criteria via positive operator theory (to appear).

282

REFERENCES

Szarski, J. [ 11 Dijerential Inequalities, Polish Scientific Publ., Warzawa, 1965. Takeda, S. and Bergen, A. R. [I] Instability of feedback systems by orthogonal decomposition of L2,IEEE Trans. Automatic Control 18 (1973), 631-636. Tdussky, 0. [I] A recurring theorem on determinants, Amer. Math. Monthly 56 (1949), 672-676. Tewarson, R. P. [ I ] Sparse Matrices, Academic Press, New York, 1973. Thompson, W. E. [ I ] Stability of interconnected systems, Ph.D. Dissertation, Michigan State Univ., East Lansing, Michigan (1968). 121 Exponential stability of interconnected systems, IEEE Trans. Automatic Control 15 (1970), 5044506. Thompson, W. E. and Kocnig, H. E. [ I ] Stability of a class of interconnected systems, Internat. J. Control 15 (1972), 751-763. Tokumaru, H., Adachi, N., and Amcmiya, T. [ I ] On th.2 input-output stability of interconnected systems, Systems and Control (J. Japan Association of Automatic Control Engineers) 17 (1973), 121-125. [2] Macroscopic stability of interconnected systems, Preprints, Sixth IFAC World Congress, Boston, Massachusetts, August 1975, IFAC, I975 (Paper No. 44.4). Vidyasagar, M. [ I ] L2-instability criteria for interconnected feedback systems, Proceedings of the 1976 IEEE lnternational Symposium on Circuits and Systems, Munich, Germany, April 1976, IEEE, 1976, pp. 386-389. Volterra, V. [ 11 Theory of Functionals and of IntcJgral and Integrodiflrrentia1 Equations, Dover, New York, 1959. Walker, J. A. [ I ] On the application of Lyapunov’s direct method to linear dynamical systems, Lefschetz Center for Dynamical Systems, Lecture Notes TR 74-7, Brown Univ., Providence, Rhode Island (1974). Walker, J. A. and Infante, E. F. [ I ] Some results on the precompactness of orbits of dynamical systems, Lefschetz Center for Dynamical Systems, Technical Rep. 74-2, Brown Univ., Providence, Rhode Island (1974). Walter, W. [ I ] Dijirential and Integral Inequalities, Springer-Verlag, Berlin and New York, 1970. Wang, P. K. C. [ I ] Asymptotic stability of distributed parameter systems with feedback controls, IEEE Trans. Automatic Control 11 (1 966), 46-54. [2] Stability analysis of elastic and aeroelastic systems via Lyapunov’s direct method, J . Franklin Inst, 281 (1966), 51-72. [3] On the stability of equilibrium of a mixed distributed and lumped parameter control system, Internat. J. Control 3 (1966), 130-147.

REFERENCES

283

Stability for a mixed distributed and lumped parameter control system, Internat. J . Control 6 (1967), 65-73. [5] Theory of stability and control for distributed parameter systems (a bibliography), Internat. J. Control 7 (1968), 101-116. Wazewski, T. [ I ] Systemes des equations et des inegalitts differentielles ordinaires aux deuxiemes membres monotones et leurs applications, Ann. Soc. Poln. Mar. 23 (1950), 112-166. Webb, G. F. [I] Autonomous nonlinear functional differential equations and nonlinear semigroups, J . Mafh. Anal. Appl. 46 (1974), 1-12. Weiss, L. and lnfante, E. F. [I] Finite time stability under perturbing forces and on product spaces, IEEE Trans. Automatic Control 12 (1967), 54-59, Weissenberger, S. [l] Stability regions of large-scale systems, Automatika 9 (1973), 653-663. Willems, J. C. [I] The Analysisof Feedback Systems, M. I. T. Press, Cambridge, Massachusetts, 1971. [2] The generation of Lyapunov functions for input-output stable systems, SIAM J. Control 9 (1971), 105-134. Wong, E. [ I ] Stochastic Processes in Information and Dynarnical Systems, McGraw-Hill, New York, 1971. Wu, M. Y. and Desoer, C. A. [I] LP-stability (1 s p I co) of nonlinear time-varying feedback systems, SIAM J . Control 7 (1969), 356-364. Yacubovich, V. A. [ I ] Solution of certain matrix inequalities occurring in the theory of automatic control, Dokl. Acad. Nauk SSSR 143 (1962), 1304-1307. Yoshizawa, T. [ I ] Stability Theory by Lyapunov’s Second Method, Math. SOC.Japan, Tokyo, Japan, 1966. Zames, G. [I] Functional analysis applied to nonlinear feedback systems, IEEE Trans. Circuit Theory 10 (1963), 393404. [2] Nonlinear time-varying feedback systems-conditions for L,-boundedness derived using conic operators on exponentially weighted spaces, Proceedings of the 3rd Annual Allerton Conference on Circuit and System Theory, Univ. of Illinois, Urbana, Illinois, October 1965, pp. 46W71. [3] On the input-output stability of time-varying nonlinear feedback systemsPart 1: Conditions derived using the concept of loop, gain, conicity and positivity, IEEE Trans. Automatic Confrol 11 (1966), 228-238. [4] On the input-output stability of time-varying nonlinear feedback systemsPart 11: Conditions involving circles in the frequency plane and sector nonlinearities, IEEE Trans. Automatic Control 11 (1966), 465476. Zemanian, A. H. [I] Distribution Theory and Transform Analysis, McGraw-Hill, New York, 1965. Zubov, V. I. [ I ] Methods of A. M . Lyapunov and Their Application, P. Noordhoff Ltd., The Netherlands, 1964. [4]

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A

Absolute stability, 70, 72 Abstract initial value problem, 136 Accretive, 139 Admissible nonlinearity, 70, 72 Admissible pair, 143 Aircraft (longitudinal motion), 69 Asymptotic stability differential equation, 15, 30, 141 in the large, 15, 141 with probability one, 1 1 I , 116, 126 Autonomous differential equation, 19, I19 B Backward diffusion equation, 154

W ) ,13

Biases, 207

Boundary, 13 Bounded, 15,30 Bounded linear map, 135 Bounded relation, 198, 199 Bounded system, 263 C Capacitor (circuit), 79 Causal, 198 Center (interior conic relation), 199 Circle criterion, 230-237 Closed linear map, I35 Closure, 13 Column dominance, 49 Comparison principle, 56, 125, 187 Compatible subspace, 197, 198 Compensated system, 88 Complete instability, 19, 38, 53, 96

285

286

INDEX

Completely compatible space, 197 Composite system, 22,93, 1 1 3, 156, 213,261 Conditional gain, I98 Conditional probability, 110 Conic (relation), 199, 200, 223 Conicity conditions, 220 Contraction semigroup, 137, 138 Control problem direct, 84, 85 indirect, 72, 74, 87, 128

D Decaying L,-memory, 204, 206 Decentralized system, 22 Decrescent, 17 Degree of stability, 45, 53

v,,

110

Diagonal dominance condition, 45 Diagonal matrix, I4 Diakoptics, 1 Difference equation, 92 Diffusion equation, 154 Discrete time systems, 91 Dissipative (linear operator), 138 Distribution function, 154 Domain, 13, 139 of attraction, 15, 141 Dominance conditions, 49 Dynamical system linear, 137 nonlinear, 138

E Economic system, 77-79 Eigenvalue, 13 Elliptic partial differential equation, 152, 188 Expectation operator, 110 Expected value, 110 Exponential stability in Banach space, 141, 163 definition, 15 discrete case, 99 in the large, 16, 33, 51, 141, 165 with probability one, 1 1 I , 119, 126 in quadratic mean, f 12 Equilibrium point, 14, 141

Euclidean space, 13 Extended space, 195, 197 Exterior conic (relation), 199, 200

F Fading memory, 147, 181 Finite time stability, 64 Fourier transform, 203 Free subsystem, 21, 23 Frequency response, 203 Functional differential equation, 145, 183

G Gain, 199, 207, 208 Gain factor, 206, 226, 255-256 Gaussian random process, 131 Generalized solution, 137 Generator (of a semigroup), 139

H Hicks condition, 77-79 Hilbert space structure, 196 Hybrid system, 155, 173 Hypervector, 23

I Incremental gain, 199 Incrementally conic relation, 199 Incrementally positive relation, 200 Independent increment Markov process, 11 1 Indirect control problem (see control) Infinitesimal generator, 137, 139 weak, I12 Inner product, 136 Input, 21 Input-output, 21 Input-output stability, 208-21 5 Inside (a sector), 199 Instability, 38, 53, 168, 216 Integral operators, 201 lntegrodifferential equation, 259

INDEX

Interconnected system, 22,93, 113, 156,213, 261 Interior conic (relation), 199, 200 Invariant, 19 Inverse relation, 139 Isolated equilibrium, 14 Isolated subsystem, 21,23, 113, 156,213,261 Ito calculus, 131 Ito differential equation, 154

K K (class K ) , 16 same order of magnitude, 16, 32 K' (class K i ) , 110 K R (class KR), 16 L A(A), 13

Laplace transform, 203,260 Largescalesystem,22,93, 113, 156,213,261 Linear map bounded, 135 closed, 135 Local negative feedback, 45 &-space, 136, 194 Lurk type Lyapunov function, 87 Lyapunov function, 20, 92, 1 12, 142 Lyapunov matrix equation, 73 Lyapunov theorems, 20, 142 M Margin of boundedness, 226 of continuity, 227 of stability, 45, 53 Markov process, 110, 111 Matrix diagonal, 14 Lyapunov equation, 73 Minkowski, 46 norm, 14, 110 Maximal solution, 57 Maximum principle for parabolic equations, 187

287

Memoryless relation, 198 Minimal solution, 57 Minkowski matrix, 46 M-matrix, 46, 98, 166 Modified frequency response, 203 Multi-loop system, 22 Multiple feedback systems, 84, 87 Multiple input-output system, 209 Multivalued function, 138, 198

N Negative definite function, 17 Negative semidefinite function, 17 Nonautonomous differential equation, 19 Nonlinear transistor, 81 IIDII., 110 Norm (of a bounded linear map), 136 Nuclear reactor, 179, 267 Nyquist plot, 203 0 output, 21 Outside (of a sector), 199 P P ( A ) , 110 P ( A I B ) , 110 Partial differential equations, 149, 187 elliptic, 152, 188 parabolic, 152, 188 Passive operator, 218 properly, 2 I 8 strictly, 218 Point kinetics model, 179, 267 Poisson step process, 1 11, 1 12 Popov conditions, 239 Positive definite function, 17 Positive relation, 199, 200 Positive semidefinite function, 17 Power system, 88 Practical stability, 64 Probability measure, 110 Probability space, 110 Pseudonorm, 195

288

INDEX

Q Quasicontractive semigroup, 138 Quasimonotone function, 59 R Radially unbounded function, 17 Radius (of a relation), 199 Random variable, 110 Range, 13, 139 Reference signals, 207 Relation, 138, 198 Residual resolvent, 260 Resolvent kernel, 260 map, 135 set, 135 R o w dominance. 49 S Same order of magnitude, 16 Sample continuous (function), 1 II Sample space, 110 Sampled data system, 104 Sampling time, 104 Sector theorems, 207, 222 Semigroup C,, 137 of contractions, 137 differentiable, 144 nonlinear, 138 Seminorm, 195 Shifted Fourier Transform, 204 Shifted L,-theory, 196 Shifted L,-theory, 249 Shifted Nyquist plot, 204 0-algebra, I10 Singular point, 14, 141 Small gain theorem, 207 Space C, BC, Co, etc., 195 Spectral radius. 216 Spectral set, 135 Spectrum, 135 Stability, 14, 17, 30 absolute, 70 finite time, 64

of first approximation, 61, 78, 266-267 input-output, 208-21 5 L,, 249 practical, 64 with probability one, 1 1 I with respect t o sets, 64-69 for semigroups, 141, 142 total, 66 uniform total, 66 Stabilization, 88 Stable manifold, 198 Stochastic differential equation, 110, 153 Stochastic process, 110-1 1 1 Strongly elliptic operator, 152 Strongly parabolic (partial differential equation), 152 Strong solution, 136, 140

T Tearing problem, 25 Time invariant differential equation, 19 Time invariant relation, 198 Time-varying capacitor, 79 TrA, I10 Trajectory, 136 Trajectory behavior, 63-69 Trajectory bounds, 63-69 Transfer system, 2 I Transistor network, 8 I Transition function, 154 Trival solution, 14, 141 Truncation, 194

U Uniform boundedness, 15, 18, 40, 54, 143 Uniform ultimate boundedness, 15, 18, 40, 54, 142, 169 Uniform asymptotic stability Banach space, I 4 I,159, 1 89 differential equations, 15, 17, 26, 28, 48 discrete systems, 94, 99 functional differential equations, 185 in the large, 16,29,48, 141 sampled data systems, 107 Uniformly convex space, 140 Uniform stability, 15, 31

289

INDEX

Unstable (also see Instability) Banach space, 141, 168 differential equations, 15, 18, 38,53 discrete systems, 95-100 input-output, 216 integral equations, 261 with probability one, 111 with respect to sets, 64-69 V Variation of constants, 260 Vector comparison equation, 58

A 8 7

c a

0 9

E F 6 H

O 1 2 3

1 4 1 5

Vector Lyapunov function, 44, 56, 122, 187 Volterra equation, 147, 259 W W-accretive, 139 Weak coupling conditions, 46, 54 Weighted spaces, 195 Weighted sum (of Lyapunov functions), 44 Well posedness, 136, 140, 157 White noise, 13 I Wiener process, 1 II , 1 12

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