Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces

TL;DR

This paper establishes a fundamental equivalence between input-to-state stability (ISS) and the existence of ISS Lyapunov functions in Banach space systems, providing new insights and constructions for both nonlinear and linear infinite-dimensional systems.

Contribution

It proves the equivalence between ISS and Lyapunov functions in Banach spaces and offers new methods to construct these functions for linear systems.

Findings

ISS is equivalent to coercive Lyapunov functions in nonlinear systems.

Linear systems admit non-coercive Lyapunov functions for ISS.

Constructive methods for Lyapunov functions are provided for linear systems.

Abstract

We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.