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

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 \(ε\) 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.

Figures (2)

• Figure 1: Example \ref{['ex1']}: (i) Approximate solution at $x=0.5$. (ii) Approximate solution in the $L^{2}$-norm for various values of $\epsilon$. (iii) Controller in the $L^{2}$-norm for different values of $\epsilon$. (iv) Approximate solution in the $L^{2}$-norm.
• Figure 2: Example \ref{['ex2']}: (i) The state variable in the $L^{2}-$norm for different values of $\nu$. (ii) Control input in the $L^{2}-$norm for different values of $\nu$. (iii) The state variable in the $L^{2}-$norm with various values of $\delta$ with $\nu=0.1$. (iv) Control input in the $L^{2}-$norm for various values of $\delta$.

Theorems & Definitions (18)

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