Characterizations of Input-to-State Stability for Infinite-Dimensional Systems

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), and 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 noncoercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS (sISS) that is equivalent to ISS in the ODE case, but 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.

I. Introduction

For ordinary differential equations (ODEs), the concept of input-to-state stability (ISS) was introduced in [3]. The corresponding theory is now well developed and has a firm theoretical basis. Several powerful tools for the investigation of ISS are available and a multitude of applications have been developed in nonlinear control theory, in particular, to robust stabilization of nonlinear systems [4], design of nonlinear observers [5], analysis of large-scale networks [6]– [8], etc.