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Stability analysis of impulsive functional differential equations. (English) Zbl 1189.34001
de Gruyter Expositions in Mathematics 52. Berlin: Walter de Gruyter (ISBN 978-3-11-022181-7/hbk). viii, 230 p. EUR 99.95; $ 155.00 (2009).

Impulsive functional differential equations are a natural generalization of impulsive ordinary differential equations (without delay) and functional differential equations (without impulses). At the present time the qualitative theory of such equations undergoes a rapid development, and this book aims to provide a systematic treatment of those developments. It presents the main results on stability and boundedness theory for impulsive functional differential equations, and in addition it provides a unified general structure applicable to study the dynamics of mathematical models based on such equations. This monograph is the first book dedicated to a systematic development of stability theory for impulsive functional differential equations.

The book consists of four chapters.

Chapter 1 is an introductory one. In Chapter 2 the main definitions on Lyapunov stability and boundedness of solutions of impulsive functional differential equations are given. Using the Lyapunov-Razumikhin technique and comparison technique theorems on Lyapunov stability, boundedness and global stability are proved. A lot of examples are considered to illustrate the feasibility of the results. Chapter 3 is dedicated to some extensions of Lyapunov stability and boundedness. Theorems on stability and boundedness of sets, conditional stability, parametric stability, eventual stability and boundedness, practical stability, Lipschitz stability, stability and boundedness in terms of two measures are presented. A lot of interesting results are considered in which assumptions allowing the derivatives of Lyapunov function to be positive are used to impulsively stabilize functional differential equations.

Finally, in Chapter 4, applications of stability and boundedness theory to Lotka-Volterra models, neural networks and economic models are presented. The impulses are considered either as perturbations or as control.

Each chapter is supplied with notes and comments.

The book is addressed to a wide audience of professionals such as mathematicians, applied researches and practitioners.

MSC:
34-02Research monographs (ordinary differential equations)
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
34K60Qualitative investigation and simulation of models
34K12Growth, boundedness, comparison of solutions of functional-differential equations
92B20General theory of neural networks (mathematical biology)
92D25Population dynamics (general)