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Systems with impulse effect stability, theory and applications. (English) Zbl 0683.34032
Ellis Horwood Series in Mathematics and its Applications. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press. 255 p. £39.95 (1989).
The book consists of three parts. In Part A a description of the systems with impulse effect is given. Here are considered systems of the form: $$dx/dt=f(t,x),$$ $$t\neq \tau_ k(x)$$, $$\Delta x=I_ k(x)$$, $$t=\tau_ k(x)$$ where f: $${\mathbb{R}}_+x\Omega \to {\mathbb{R}}^ n$$; $$\tau_ k: \Omega \to {\mathbb{R}}_+$$; $$I_ k: \Omega \to {\mathbb{R}}^ n$$; $$\Omega$$ is a domain in $${\mathbb{R}}^ n$$ and $$0<\tau_ 1(x)<\tau_ 2(x)<...$$, $$\lim_{k\to \infty}\tau_ k(x)=\infty$$ for $$x\in \Omega$$ and are presented theorems of existence, uniqueness and continuability of the solutions; theorems of continuity and differentiability of the solutions with respect to initial data and a parameter. In Part B, Lyapunov’s first method for systems with impulse effect is presented, that is: stability of linear systems, characteristic exponents, stability by linear approximation, perturbation theorems. In Part C, Lyapunov’s second method for systems with impulse effect is developed. The authors prove direct and converse theorems of stability and theorems of comparison using piecewise continuous functions for comparison specially introduced here. The theory is developed in full analytic rigour, and is illustrated by many examples and applications.
Reviewer: S.Biranas

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations