Penalty-Based Feedback Control and Finite Element Analysis for the Stabilization of Nonlinear Reaction-Diffusion Equations
This work addresses boundary control stabilization for reaction-diffusion equations, which is an incremental theoretical and numerical contribution to PDE control theory.
The authors tackled the stabilization of nonlinear reaction-diffusion equations using a penalty-based feedback control approach, proving convergence of penalized solutions to Dirichlet boundary control solutions as the penalty parameter approaches zero and validating results with numerical experiments.
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 \(ε\) 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.