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The second edition of this textbook provides a single source for the analysis of system models represented by continuous-time and discrete-time, finite-dimensional and infinite-dimensional, and continuous and discontinuous dynamical systems. For these system models, it presents results which comprise the classical Lyapunov stability theory involving monotonic Lyapunov functions, as well as corresponding contemporary stability results involving non-monotonic Lyapunov functions. Specific examples from several diverse areas are given to demonstrate the applicability of the developed theory to many important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, and artificial neural networks.

The authors cover the following four general topics:

- Representation and modeling of dynamical systems of the types described above

- Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces involving monotonic and non-monotonic Lyapunov functions

- Specialization of this stability theory to finite-dimensional dynamical systems

- Specialization of this stability theory to infinite-dimensional dynamical systems

Replete with examples and requiring only a basic knowledge of linear algebra, analysis, and differential equations, this book can be used as a textbook for graduate courses in stability theory of dynamical systems. It may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, economics, and the physical and life sciences.

Review of the First Edition:

“The authors have done an excellent job maintaining the rigor of the presentation, and in providing standalone statements for diverse types of systems. [This] is a very interesting book which complements the existing literature. [It] is clearly written, and difficult concepts are illustrated by means of good examples.”

- Alessandro Astolfi, IEEE Control Systems Magazine, February 2009

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
We summarize the aims and scope of the book and we give an outline of its contents. We also present a brief perspective on the development of stability theory.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 2. Dynamical Systems

Abstract
We give the definition of dynamical system and a classification of such systems: finite-dimensional and infinite-dimensional systems; continuous-time and discrete-time systems; continuous and discontinuous systems; autonomous and non-autonomous systems; and composite systems. Classes of finite-dimensional dynamical systems that we address include systems determined by ordinary differential equations, ordinary differential inequalities, ordinary difference equations, and ordinary difference inequalities. General classes of infinite-dimensional dynamical systems that we address include systems determined by differential equations and inclusions defined on Banach spaces and systems determined by linear and nonlinear semigroups. Specific classes of infinite-dimensional dynamical systems that we address include systems determined by functional differential equations, Volterra integrodifferential equations, and certain classes of partial differential equations. For all cases, we present specific examples.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 3. Fundamental Theory: The Principal Stability and Boundedness Results on Metric Spaces

Abstract
We present the Principal Lyapunov and Lagrange Stability Results, including Converse Theorems for continuous dynamical systems, discrete-time dynamical systems and discontinuous dynamical systems defined on metric spaces. All results presented involve the existence of either monotonic Lyapunov functions or non-monotonic Lyapunov functions. We show that the results involving monotonic Lyapunov functions reduce to corresponding results involving non-monotonic Lyapunov functions. Furthermore, in most cases, the results involving monotonic Lyapunov functions are in general more conservative than the corresponding results involving non-monotonic Lyapunov functions.We present stability results (sufficient conditions) for uniform stability, local and global uniform asymptotic stability, local and global exponential stability, and instability of invariant sets. We also present Converse Theorems (necessary conditions) for most of the enumerated stability types. Furthermore, we present Lagrange stability results (sufficient conditions) for the uniform boundedness and the uniform ultimate boundedness of motions of dynamical systems, as well as corresponding Converse Theorems (necessary conditions).The results of this chapter constitute the fundamental theory for the entire book because most of the general results that we develop in the subsequent chapters concerning finite-dimensional systems and infinite-dimensional systems can be deduced as consequences of the results of the present chapter.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 4. Fundamental Theory: Specialized Stability and Boundedness Results on Metric Spaces

Abstract
We present several important specialized stability and boundedness results for dynamical systems defined on metric spaces. It turns out that a number of the results that we will develop in the subsequent chapters concerning finite-dimensional and infinite-dimensional systems can be deduced as consequences of corresponding results of the present chapter.For autonomous and periodic dynamical systems we show that when an invariant set is stable (asymptotically stable) then it is also uniformly stable (uniformly asymptotically stable). For autonomous dynamical systems we also present necessary and sufficient conditions for the stability and the asymptotic stability of invariant sets.For dynamical systems determined by continuous-time and discrete-time semigroups defined on metric spaces we establish stability and boundedness results which comprise the LaSalle-Krasovskii invariance theory.We present for both continuous-time and discrete-time dynamical systems a comparison theory for the various Lyapunov and Lagrange stability types. This comparison theory enables us to deduce the qualitative properties of a complex dynamical system (the object of inquiry) from the qualitative properties of a simpler dynamical system (the comparison system).For general continuous-time dynamical systems we establish a Lyapunov-type result which ensures the uniqueness of motions of a dynamical system.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 5. Applications to a Class of Discrete-Event Systems

Abstract
We apply the stability theory of dynamical systems presented in Chapters 3 and 4 in the analysis of an important class of discrete-event systems. We show that these discrete-event systems determine dynamical systems. We establish necessary and sufficient conditions for the uniform stability and the uniform asymptotic stability of invariant sets with respect to the class of discrete-event systems considered. We apply these results in the analysis of two specific examples, a manufacturing system and a load balancing problem in a computer network.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 6. Finite-Dimensional Dynamical Systems

Abstract
We present the principal stability and boundedness results for continuous dynamical systems, discrete-time dynamical systems, and discontinuous dynamical systems involving monotonic and non-monotonic Lyapunov functions. We apply the results of Chapter 3 to arrive at these results. When considering various stability types, our focus is on invariant sets that are equilibria. Our results constitute sufficient conditions (the Principal Stability and Boundedness Results) and necessary conditions (Converse Theorems). We demonstrate the applicability of all results by means of numerous examples.
We also present results for uniform stability and for uniform asymptotic stability in the large involving multiple non-monotonic Lyapunov functions. The applicability of these results is demonstrated by means of a specific example.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 7. Finite-Dimensional Dynamical Systems: Specialized Results

Abstract
For autonomous and periodic continuous-time dynamical systems we show that stability and asymptotic stability imply uniform stability and uniform asymptotic stability, respectively. For such systems we also present specialized Converse Theorems. For continuous-time and discrete-time dynamical systems determined by semigroups, we present the LaSalle-Krasovskii invariance theory (involving monotonic Lyapunov functions). These results constitute sufficient conditions. For the special case of dynamical systems determined by linear autonomous homogeneous systems of differential equations and difference equations, we present invariance results which constitute necessary and sufficient conditions (involving monotonic Lyapunov functions). For general continuous-time and discrete-time dynamical systems we present invariance stability and boundedness results involving non-monotonic Lyapunov functions. We present results which make it possible to estimate the domain of attraction of an asymptotically stable equilibrium for dynamical systems determined by differential equations. We present stability results for dynamical systems determined by linear homogeneous differential equations and difference equations. Some of these results require knowledge of the state transition matrix while other results involve Lyapunov matrix equations. We present stability results for dynamical systems determined by linear periodic differential equations (the Floquet Theory). Also, we study in detail the stability properties of dynamical systems determined by second-order differential equations. We investigate various aspects of the qualitative properties of perturbed linear systems, including Lyapunov’s First Method (also called Lyapunov’s Indirect Method) for continuous-time and discrete-time dynamical systems; existence of stable and unstable manifolds in continuous-time linear perturbed systems; and stability properties of periodic solutions in continuous-time perturbed systems. We present a stability and boundedness comparison theory for finite-dimensional continuous-time and discrete-time dynamical systems.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 8. Applications to Finite-Dimensional Dynamical Systems

Abstract
We apply the results developed in Chapters 6 and 7 in the qualitative analysis of several important classes of continuous dynamical systems, discrete-time dynamical systems, and discontinuous dynamical systems. We address six specific classes of systems: nonlinear regulator systems; analog Hopfield neural networks and synchronous discrete-time Hopfield neural networks; digital control systems; pulse-width modulated feedback control systems; systems with saturation nonlinearities with an application to digital filters; and linear and nonlinear Hamiltonian systems subjected to persistent and intermittent dissipation.
Anthony N. Michel, Ling Hou, Derong Liu

Chapter 9. Infinite-Dimensional Dynamical Systems

Abstract
We address the Lyapunov stability and the boundedness of motions (Lagrange stability) of infinite-dimensional dynamical systems determined by differential equations defined on Banach spaces and by semigroups with an emphasis on the qualitative properties of equilibria. We consider continuous as well as discontinuous dynamical systems (DDS). Most of the results involve monotonic Lyapunov functions. However, some of the stability results for DDS involve non-monotonic Lyapunov functions as well.
We present the Principal Stability and Boundedness Results (sufficient conditions) and some Converse Theorems (necessary conditions) for dynamical systems determined by general differential equations defined on Banach spaces. Most of these results are consequences of corresponding results established in Chapter 3 for dynamical systems defined on metric spaces. We demonstrate the applicability of these results in the analysis of several specific classes of differential equations defined on different Banach spaces. For autonomous differential equations defined on Bansch spaces we present invariance results and we apply these results in the analysis of specific classes of systems. We develop a comparison theory for general differential equations defined on Banach spaces and we apply these results in the stability analysis of a point kinetics model of a multicore nuclear reactor described by Volterra integrodifferential equations. Finally, we present stability results for composite systems defined on Banach spaces described by a mixture of different differential equations and we apply these results in the analysis of a specific class of systems.
Special important differential equations in Banach spaces include retarded functional differential equations. For dynamical systems determined by such equations, some of the preceding results can be improved. We present stability and boundedness results for dynamical systems determined by retarded functional differential equations, including Razumikhin-type theorems, and invariance results for dynamical systems determined by retarded functional differential equations. We apply some of these results in the qualitative analysis of the Cohen–Grossberg neural network model endowed with multiple time delays.
Finally, we present stability and boundedness results for discontinuous dynamical systems determined by differential equations in Banach spaces (involving non-monotonic Lyapunov functions) and by linear and nonlinear semigroups defined on Banach spaces. We demonstrate the applicability of these results by means of several classes of infinite-dimensional dynamical systems.
Anthony N. Michel, Ling Hou, Derong Liu

Backmatter

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