Non-coercive Lyapunov functions for infinite-dimensional systems

TL;DR

This paper demonstrates that non-coercive Lyapunov functions can guarantee uniform global asymptotic stability in infinite-dimensional systems with disturbances, under mild conditions, extending recent linear system results.

Contribution

It introduces new sufficient conditions for stability using non-coercive Lyapunov functions in infinite-dimensional systems, including simple assumptions for Banach spaces and new converse theorems.

Findings

Non-coercive Lyapunov functions imply UGAS under mild conditions.

Results extend to linear switched systems on Banach spaces.

New converse Lyapunov theorems are established.

Abstract

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.