Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

TL;DR

This paper establishes that the existence of a non-coercive Lyapunov function is both necessary and sufficient for integral uniform global asymptotic stability in a broad class of nonlinear dynamical systems, including finite- and infinite-dimensional cases.

Contribution

It proves the equivalence between non-coercive Lyapunov functions and integral UGAS, extending stability analysis to less regular and more general systems.

Findings

Non-coercive Lyapunov functions imply integral UGAS.

The converse holds without regularity assumptions.

Characterization of UGAS via integral stability properties.

Abstract

In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system. Furthermore, we give a characterization of uniform global asymptotic stability in terms of the integral stability properties and analyze which stability properties can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.