Stability analysis in terms of two measures. (English) Zbl 0797.34056

Singapore: World Scientific. xi, 402 p. (1993).
One hundred years ago A. M. Lyapunov introduced the concept of stability of a rest point of a differential system we have been using since that time. It says that the rest point is called stable if the deviation of the state variable remains arbitrarily small along the motions provided that it was small enough at the initial time. During the last hundred years many other stability concepts have been used. One of them was the stability in terms of two measures introduced by A. A. Movchan in 1960. It says that the rest point is \((h_ 0,h)\)-stable if the deviation of the state variable measured by measure \(h\) remains arbitrarily small provided that the deviation measured by \(h_ 0\) at the initial time is small enough. This concept unifies a variety of known concepts of stability and boundedness; therefore, it serves as a good framework to develop a general stability theory. Such a theory was needed from another important point of view. If the state space is of infinite dimension (e.g., FDE, PDE), then there are several non-equivalent norms of the state variable, so the different measures appear very naturally.
This book gives an abstract theory of the stability in terms of two measures. The first chapter establishes the basic theory with such concepts as boundedness, partial stability, practical stability, invariance principle, comparison method and converse theorems. The second chapter generalizes such refinements of stability theory to the new concept as several Lyapunov functions, perturbations of Lyapunov functions, integral stability, cone-valued Lyapunov functions.
The third chapter is devoted to the extensions for FDE’s, control systems, random differential systems, impulsive integro-differential system. The book is concluded by important and interesting applications to mechanics, aircraft engineering, economics, population dynamics. Many results of the monograph are new, and so far most of the material was available only in articles.
Reviewer: L.Hatvani (Szeged)


34D20 Stability of solutions to ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations