Penalty-Based Feedback Control and Finite Element Analysis for the Stabilization of Nonlinear Reaction-Diffusion Equations

TL;DR

This paper introduces a penalization approach combined with finite element analysis to stabilize nonlinear reaction-diffusion equations with Dirichlet boundary feedback, providing theoretical convergence and numerical validation.

Contribution

The work develops a penalization technique for boundary control of reaction-diffusion equations and analyzes its convergence and stability using finite element methods.

Findings

Penalized control solutions converge to Dirichlet boundary control as penalty parameter approaches zero.

Finite element semi-discrete scheme stabilizes the penalized problem.

Numerical experiments confirm theoretical convergence and stability results.

Abstract

In this work, first we employ a penalization technique to analyze a Dirichlet boundary feedback control problem pertaining to reaction-diffusion equation. We establish the stabilization result of the equivalent Robin problem in the \(H^{2}\)-norm with respect to the penalty parameter. Furthermore, we prove that the solution of the penalized control problem converges to the corresponding solution of the Dirichlet boundary feedback control problem as the penalty parameter \(\epsilon\) approaches zero. A \(C^{0}\)-conforming finite element method is applied to this problem for the spatial variable while keeping the time variable continuous. We discuss the stabilization of the semi-discrete scheme for the penalized control problem and present an error analysis of its solution. Finally, we validate our theoretical findings through numerical experiments including showing that penalized solution converges to the original solution.