Input-to-state stability of infinite-dimensional control systems

TL;DR

This paper develops new tools for analyzing input-to-state stability in infinite-dimensional control systems, including methods for constructing ISS-Lyapunov functions and a generalized small-gain theorem, with applications to reaction-diffusion equations.

Contribution

It introduces methods for constructing ISS-Lyapunov functions in Banach spaces and generalizes the small-gain theorem for infinite-dimensional systems.

Findings

Existence of ISS-Lyapunov functions implies ISS for certain classes of inputs.

Two methods for constructing local and global ISS-Lyapunov functions are developed.

The generalized small-gain theorem applies to interconnected infinite-dimensional systems.

Abstract

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.