Boundary Stabilizability of Generalized Burgers-Huxley Equation with Memory

TL;DR

This paper develops a boundary feedback control method to stabilize solutions of a generalized Burgers-Huxley equation with memory, using eigenfunction-based controllers and fixed point theory.

Contribution

It introduces a linear boundary feedback controller for stabilization of a nonlinear PDE with memory, extending previous methods to this complex system.

Findings

The controller stabilizes the stationary solution under homogeneous boundary conditions.

The same controller also stabilizes the nonlinear system via fixed point arguments.

Remarks on stabilization around the zero solution under nonhomogeneous conditions.

Abstract

In this paper, we study a generalized Burgers-Huxley equation with memory, subject to nonhomogeneous Dirichlet boundary conditions. We construct a linear, finite-dimensional Dirichlet boundary feedback controller aimed at stabilizing the stationary solution corresponding to the homogeneous boundary condition. This controller is designed using eigenfunctions of the Laplace operator. We begin by analyzing the stabilization of a linear system under the proposed feedback law. Subsequently, we demonstrate that the same controller also stabilizes the full nonlinear system by applying the Banach fixed point theorem. Finally, we provide a remark on the stabilization of the generalized Burgers-Huxley equation with memory around the zero solution under nonhomogeneous Dirichlet boundary conditions.