Classical Converse Theorems in Lyapunov's Second Method

TL;DR

This paper surveys classical converse theorems related to Lyapunov's second method, exploring conditions under which stability properties imply the existence of suitable Lyapunov functions.

Contribution

It provides a comprehensive overview of existing results on the converse theorems in Lyapunov's stability theory.

Findings

Summarizes key classical converse theorems.

Highlights conditions for the existence of Lyapunov functions.

Discusses open problems and future directions.

Abstract

Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.