Exponential stabilization of infinite-dimensional systems by finite-dimensional controllers

TL;DR

This paper investigates how to stabilize infinite-dimensional systems using finite-dimensional controllers, providing new conditions and a novel input-output gain concept, with applications to parabolic and hyperbolic equations.

Contribution

It introduces necessary and sufficient conditions for feedback stabilizability and a new input-output gain for infinite-dimensional systems.

Findings

Necessary and sufficient conditions for stabilization are established.

A novel input-output gain is introduced for stability analysis.

Quasi-finiteness is verified for certain classes of PDEs.

Abstract

This paper studies the feedback stabilization of abstract Cauchy problems with unbounded output operators by finite-dimensional controllers. Both necessary conditions and sufficient conditions for feedback stabilizability are presented. The proof of closed-loop stability is based on a novel input-output gain introduced in this paper. For systems satisfying a property we call quasi-finite, an equivalent characterization of feedback stabilizability is obtained. Quasi-finiteness is verified for classes of parabolic and hyperbolic equations.