Vector Lyapunov functions method: nonlinear analysis of dynamical properties. (Q2747614)

This superb monograp,written by the renowned specialist in stability theory, reflects valuable contributions to the qualitative theory of nonlinear differential equations made by the author over forty years. The book can be logically divided into two principal parts related to the two main streams in the development of the classical direct Lyapunov method. The first part (Chapters 1 to 3) is dedicated to the extension of the method of Lyapunov functions to equations with discontinuous right-hand sides, employment of several Lyapunov-type functions and vector Lyapunov functions. Applications to various stability problems in mechanical systems with dissipation and gyroscopic systems are discussed. The second part of the monograph (Chapters 4 and 5) is concerned with the vector Lyapunov functions method.NEWLINENEWLINENEWLINEChapter 1 introduces the reader to the fundamental theory of differential equations with discontinuous right-hand sides based on the concept of the right-handed solutions. Differential inequalities and equations of motion of several important mechanical systems are also discussed. In the second chapter, classical Lyapunov theorems and their various extensions to systems with discontinuous right-hand sides are presented. The stability of a gyroscopic system on a moving base is investigated. The third chapter deals with extensions of classical Lyapunov theorems obtained by replacing the standard assumption that the derivative of a Lyapunov function is sign definite with a less restrictive one requiring that the derivative has constant sign and the set where it vanishes satisfies certain additional conditions. This refinement of the classical method extends significantly its applications and is achieved by considering several scalar Lyapunov-type functions. Several examples of gyroscopic systems with full dissipation and mechanical systems with dry friction are examined. In Chapter 4, differential equations in Banach space with unbounded operators on the right-hand sides are studied. For finite-dimensional systems with discontinuous right-hand sides, differential inequalities that can be used for comparison with upper solutions, quasi-solutions and various types of generalized solutions are introduced. The main result of this chapter is a comparison principle obtained by the author that determines algorithms for stating and proving comparison theorems for different classes of dynamical systems based on their definitions. In the final chapter, the vector Lyapunov functions method is developed and used for deducing specific comparison theorems and theorems on dynamic properties of solutions to nonlinear differential equations in Banach space. NEWLINENEWLINENEWLINEThe monograph is written very carefully, the exposition is self-contained, rigorous and transparent. Numerous examples greatly contribute to a better understanding of the theoretical material. An extensive list of references, mostly to papers and books written in Russian, is provided. There is no doubt that everyone interested in stability theory will benefit a lot from reading this excellent contribution to the qualitative theory of nonlinear differential equations, and a translation of this work into English is a must.