V. Lakshmikantham

Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems

1. Why several Lyapunov functions?.- 2. Refinements.- 3. Extensions.- 4. Applications.- References.

Stability of moving invariant sets and uncertain dynamic systems on time scales

Abstract In this paper, we study the stability of moving invariant sets and uncertain dynamic systems on time scale. A control application is considered.

Stability In The Models of Real World Phenomena

In this chapter we offer several examples of real world models to illustrate the general methods of stability analysis developed in the previous chapters.

Stability of Motion in Terms of Two Measures

This chapter is devoted essentially to the investigation of stability theory in terms of two measures. It also provides a converse theorem for uniform asymptotic stability in this set up and stresses the importance of employing families of Lyapunov-like functions and the theory of inequalities.

Stability of Perturbed Motion

In this chapter, stability considerations are extended to a variety of nonlinear systems utilizing the same versatile tools, namely, Lyapunov-like functions, theory of appropriate inequalities and different measures, that were developed in the previous chapters. In order to avoid monotony, we have restricted ourselves to present only typical extensions which demonstrate the essential unity achieved and pave the way for further work.

Variation of Parameters and Monotone Technique

In this chapter we stress the importance of nonlinear variation of parameters formulae for a variety of nonlinear problems as well as monotone iterative technique that offers constructive methods for the existence of solution in a sector.

Why several Lyapunov functions

As is well known, Lyapunov’s second method forms the core of what he himself called his second method for dealing with questions of stability. The main characteristic of the method is the introduction of a function, namely Lyapunov function which defines a generalized distance from the origin of the motion space. The concept of Lyapunov function together with the theory of inequalities furnishes a very general comparison principle under much less restrictive assumptions. It is widely recognized today, as an indispensable tool not only in the theory of stability but also in the investigation of various other properties of solutions of differential equations.