Perturbing Lyapunov functions and global results

Abstract

THE PROOFS of many results in stability theory depend on dividing the vicinity of some kind of invariant set (or other convenient set) into suitable subsets and then showing that the solutions cannot leave such sets. This allows us to obtain global results in terms of arbitrary sets [2] and Lyapunov functions that satisfy some weaker assumptions. It is also known [3,4] that by perturbing Lyapunov functions and employing a family of Lyapunov functions, nonuniform stability results can be proved under weaker hypothesis. The main idea of perturbing the Lyapunov function is to make use of a Lyapunov function that may not be satisfying all the desired conditions and to perturb it in such a way, one gets a Lyapunov function that is useful in the investigation of stability of different systems. In this paper, we shall present two global theorems of a general character using Lyapunov functions and certain arbitrary sets. These results are then applied to obtain equi-stability and equi-asymptotic stability, showing the nonuniform behavior proved with less requirements. Our approach enlarges the class of useful Lyapunov functions and offers more flexibility. These global results can be utilized to prove several known results in stability and boundedness.