##
**Liapunov functions and stability in control theory.**
*(English)*
Zbl 0968.93004

Lecture Notes in Control and Information Sciences. 267. London: Springer. xi, 208 p. (2001).

The monograph consisting of a preface, six chapters and a bibliography deals with stability and stabilization of continuous-time deterministic state (input) nonlinear systems described by finite dimensional ODEs by using the method of Lyapunov functions. The class of admissible inputs is constituted by all measurable essentially bounded functions throughout the book.

Chapter 1 presents preliminaries concerning the existence of solutions for ODEs and differential inclusions. Chapter 2 is focused on time-invariant continuous (smooth) systems. Main known results concerning linear systems, stability and stabilizability of nonlinear systems are surveyed. Chapter 3 re-interprets the problems from the previous chapter in the more general context of time-varying systems. The emphasis is laid on possible notions of internal stability and their relations. The topics include converse Lyapunov theorems and stabilizing feedback. Chapter 4 deals with nonsmooth systems. Direct and converse theorems on stability and asymptotic stability including their applications to external stabilization are presented. A new approach to prove converse Lyapunov theorems, which is shorter and easier than previous proofs, is included. Chapter 6 reviews selected tools from nonsmooth analysis which may be useful for the analysis of nondifferential systems with discontinuous Lyapunov functions. Motivated by applications in automatic control, the book presents a variety of converse Lyapunov theorems stating the existence of Lyapunov functions with suitable properties (regularity). The rigorous mathematical presentation is supplied with illustrative examples. The readers get an overview about the state of the art, which is presented in a self-contained systematic way. The book will be useful to researchers, professionals and graduate students in the subject areas.

Chapter 1 presents preliminaries concerning the existence of solutions for ODEs and differential inclusions. Chapter 2 is focused on time-invariant continuous (smooth) systems. Main known results concerning linear systems, stability and stabilizability of nonlinear systems are surveyed. Chapter 3 re-interprets the problems from the previous chapter in the more general context of time-varying systems. The emphasis is laid on possible notions of internal stability and their relations. The topics include converse Lyapunov theorems and stabilizing feedback. Chapter 4 deals with nonsmooth systems. Direct and converse theorems on stability and asymptotic stability including their applications to external stabilization are presented. A new approach to prove converse Lyapunov theorems, which is shorter and easier than previous proofs, is included. Chapter 6 reviews selected tools from nonsmooth analysis which may be useful for the analysis of nondifferential systems with discontinuous Lyapunov functions. Motivated by applications in automatic control, the book presents a variety of converse Lyapunov theorems stating the existence of Lyapunov functions with suitable properties (regularity). The rigorous mathematical presentation is supplied with illustrative examples. The readers get an overview about the state of the art, which is presented in a self-contained systematic way. The book will be useful to researchers, professionals and graduate students in the subject areas.

Reviewer: LubomĂr Bakule (Barcelona)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93D30 | Lyapunov and storage functions |

93D15 | Stabilization of systems by feedback |

93D20 | Asymptotic stability in control theory |

34A60 | Ordinary differential inclusions |