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Nonlinear systems analysis. 2nd ed. (English) Zbl 0759.93001

Englewood Cliffs, NJ: Prentice Hall. xi, 498 p. (1992).
The first edition of the book Nonlinear Systems Analysis by M. Vidyasagar appeared in 1978. This second edition comes after more than a decade. It is not a simple revision, but many parts have been updated and rewritten. Moreover, some new recent developments have been included. The book provides a very broad and comprehensive insight into the modern mathematical theory of control and systems, with emphasis on the analysis rather than the synthesis procedures. The author deals mainly with nonlinear systems, but some necessary material about linear theory is recalled and often reviewed under new interesting points of view. In fact, certain problems are often discussed both for the linear and nonlinear case. The different approaches are deeply compared and contrasted. After a brief introduction, devoted to the basic terminology, the reader finds a presentation of some general theorems about ordinary differential equations (existence theory). Then the author focuses on autonomous planar systems: the treatment includes local behavior near equilibria and periodic solutions (Bendixson’s theory, index theory). Most of the book is devoted to stability theory. In particular, Lyapunov theory is developed in Chapter 5. Many aspects are considered in a very complete and extensive way (nonautonomous systems, uniform and asymptotic stability, local and global stability, instability, converse theorems). Moreover, there is a clear treatment of the so called Lure’s problem (absolute stability). The notion of input-output stability, more suited to control systems, is developed in Chapter 6 in a very general way. The treatment includes linear and nonlinear, as well as time varying and time invariant feedback systems. Finally, a section is devoted to discrete time systems. The last chapter is an introduction to the differential geometric approach. Same classical problems such as feedback linearization are reviewed in this framework. The book is concluded by three appendices and a long list of references.

MSC:

93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93C10 Nonlinear systems in control theory
34H05 Control problems involving ordinary differential equations
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