Nonlinear systems. (English) Zbl 0969.34001

Khalil, Hassan K.

New York, NY: Macmillan Publishing Company. xii, 564 p. (1992).

As the author states in the first sentence of his preface, this beautiful book is intended as a text for a first-year graduate-level course on nonlinear systems, which may also be used for self-study or reference by engineers and applied mathematicians; based on the author’s teaching experience, there is in the preface a proposed schedule for a course consisting of 40 lectures of 50 minutes each.
In fact, the book is much more than a text, as can be seen from the contents, consisting of an introduction and seven chapters: Fundamental properties, Lyapunov stability, Advanced stability theory, Applications of Lyapunov stability, Periodic orbits, Perturbation theory and averaging, and Singular perturbations. The level of mathematical sophistication grows from chapter to chapter; more involved calculations and proofs are given in the two appendices. There are 113 references, a list of symbols, and an index.
While most of the material may be found in different sources, the author succeeds in making a proper selection of the most relevant topics; also included are such topics as the center manifold theorem and Poincare-Bendixson theorem, as well as topics reflecting the author’s preferences, based on his research experience.
One of the best features of the book is the very large number of examples and exercises. Most are based on research papers or special monographs mentioned in the references.

Reviewer: A.Halanay (MR 93k:34001)

MSC:

34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34H05	Control problems involving ordinary differential equations
34Cxx	Qualitative theory for ordinary differential equations
34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
93-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
93C10	Nonlinear systems in control theory

Keywords:

nonlinear systems; Lyapunov stability; periodic orbits; perturbation theory; averaging; singular perturbations