Rapid stabilization in a semigroup framework

TL;DR

This paper establishes rapid stabilization of linear systems within a semigroup framework, demonstrating well-posedness, exponential decay, and explicit feedback design without relying on optimal control theory.

Contribution

It introduces a direct method for proving rapid stabilization of linear systems using semigroup theory, avoiding optimal control techniques.

Findings

Proves well-posedness of the closed-loop system.

Shows the perturbed operator generates a strongly continuous group.

Justifies exponential decay of solutions.

Abstract

We prove the well-posedness of a linear closed-loop system with an explicit (already known) feedback leading to arbitrarily large decay rates. We define a mild solution of the closed-loop problem using a dual equation and we prove that the original operator perturbed by the feedback is (up to the use of an extension) the infinitesimal generator of a strongly continuous group. We also give a justification to the exponential decay of the solutions. Our method is direct and avoids the use of optimal control theory.