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**Impulsive differential equations. Transl. from the Russian by Yury Chapovsky.**
*(English)*
Zbl 0837.34003

World Scientific Series on Nonlinear Science. Series A. 14. Singapore: World Scientific. ix, 462 p. (1995).

The book under review is an extended translation of the original monograph ‘Differential equations with impulse action’. Kiev: Vyshcha Shkola (1987) written by the well-known specialists in impulsive differential equations (IDE). It presents a systematic exposition of basic results on the subject which have laid the foundations of the modern theory of IDE, and significant part of these results can be already found in the literature (see, for example, the recent monographs V. Lakshmikantham, D. Bainov and P. S. Simeonov ‘Theory of impulsive differential equations’. Singapore: World Scientific (1989; Zbl 0719.34002); D. Bainov and P. Simeonov ‘Impulsive differential equations: periodic solutions and applications’, John Wiley & Sons, New York (1993; Zbl 0815.34001) or D. Bainov and V. Covachev ‘Impulsive differential equations with a small parameter’. Singapore: World Scientific (1994; Zbl 0828.34001).

Besides of the first six chapters (General description of impulsive differential systems, Linear systems, Stability of solutions, Periodic and almost periodic impulsive systems, Integral sets of impulsive systems and Optimal control in impulsive systems) corresponding completely to those of the Russian edition (unfortunately, a number of inaccuracies and misprints of the original text has been preserved) the book contains also a new chapter Asymptotic study of oscillations in impulsive systems devoted to the applications of the asymptotic method of Krylov-Bogolyubov and of the averaging method for the study of second order IDE. Further, in the Supplement A written by S. I. Trofimchuk the recent results on the existence and uniqueness of the Carathéodory solutions and on the existence of periodic/almost periodic solutions of the IDE extending some theorems presented in Chapters 1, 3 and 4 are given. Possible applications of the results are illustrated by a substantial number of nontrivial examples. Finally, an extensive bibliography on the subject containing 186 items, the most part in Russian (some of the references with small inaccuracies) is given.

Written in a lucid and rigorous manner, the book whose translation into English was expected since its publication in 1987 will benefit all who are interested in IDE, though it does not reflect further significant progress in IDE during the last decade except the results outlined in Supplement A.

Besides of the first six chapters (General description of impulsive differential systems, Linear systems, Stability of solutions, Periodic and almost periodic impulsive systems, Integral sets of impulsive systems and Optimal control in impulsive systems) corresponding completely to those of the Russian edition (unfortunately, a number of inaccuracies and misprints of the original text has been preserved) the book contains also a new chapter Asymptotic study of oscillations in impulsive systems devoted to the applications of the asymptotic method of Krylov-Bogolyubov and of the averaging method for the study of second order IDE. Further, in the Supplement A written by S. I. Trofimchuk the recent results on the existence and uniqueness of the Carathéodory solutions and on the existence of periodic/almost periodic solutions of the IDE extending some theorems presented in Chapters 1, 3 and 4 are given. Possible applications of the results are illustrated by a substantial number of nontrivial examples. Finally, an extensive bibliography on the subject containing 186 items, the most part in Russian (some of the references with small inaccuracies) is given.

Written in a lucid and rigorous manner, the book whose translation into English was expected since its publication in 1987 will benefit all who are interested in IDE, though it does not reflect further significant progress in IDE during the last decade except the results outlined in Supplement A.

Reviewer: Yu.V.Rogovchenko (Firenze)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34A37 | Ordinary differential equations with impulses |