Boundary feedback stabilization of Fisher's equation

TL;DR

This paper develops an explicit boundary feedback controller to stabilize Fisher's equation in various function spaces, extending previous results to handle any instability level, with demonstrated numerical effectiveness.

Contribution

It introduces a novel explicit finite-dimensional boundary feedback stabilization method for Fisher's equation applicable to any instability level.

Findings

Successfully stabilizes Fisher's equation in $L^2(0,1)$ and $H^1(0,1)$.

Extends stabilization techniques to arbitrary instability levels.

Numerical simulations confirm the effectiveness of the proposed controller.

Abstract

The aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to Fisher's equation in both $L^2(0,1)$ and $H^1(0,1)$. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any level of instability, which extends the result of \cite{02} for nonlinear parabolic equations. The effectiveness of the approach is illustrated by a numerical simulation.