Stabilization with finite dimensional controllers for a periodic parabolic system under perturbations in the system conductivity

TL;DR

This paper develops a control method to stabilize a periodic parabolic system affected by conductivity perturbations, ensuring the system can be controlled to exhibit periodic behavior despite disturbances.

Contribution

It introduces a finite dimensional control approach that guarantees stabilization of perturbed periodic parabolic systems, overcoming the lack of inherent periodic solutions.

Findings

Perturbed systems can be stabilized to periodic solutions using finite dimensional controls.

The control method is effective under general conductivity perturbations.

The approach ensures the perturbed system's behavior can be regulated to match desired periodic patterns.

Abstract

This work studies the stabilization for a periodic parabolic system under perturbations in the system conductivity. A perturbed system does not have any periodic solution in general. However, we will prove that the perturbed system can always be pulled back to a periodic system after imposing a control from a fixed finite dimensional subspace.