Characterizations of input-to-state stability for infinite-dimensional systems

TL;DR

This paper provides new characterizations of input-to-state stability (ISS) for a broad class of infinite-dimensional control systems, extending known criteria and introducing the concept of strong ISS with specific criteria and counterexamples.

Contribution

It generalizes ISS criteria to infinite-dimensional systems, introduces strong ISS, and explores the implications of ISS Lyapunov functions in this context.

Findings

Broader ISS criteria for Banach space differential equations

Introduction of the strong ISS (sISS) concept and criteria

Counterexamples showing limitations of ODE-based characterizations

Abstract

We prove characterizations of input-to-state stability (ISS) for a large class of infinite-dimensional control systems, including some classes of evolution equations over Banach spaces, time-delay systems, ordinary differential equations (ODE), switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang for ODE systems. For the special case of differential equations in Banach spaces we prove even broader criteria for ISS and apply these results to show that (under some mild restrictions) the existence of a non-coercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS which is equivalent to ISS in the ODE case, but which is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for the sISS property. At the same time, we show by means of counterexamples, that many characterizations, which are valid in the ODE case, are not true for general infinite-dimensional systems.