Feedback Stabilization and Finite Element Error Analysis of Viscous Burgers Equation around Non-Constant Steady State

TL;DR

This paper develops a feedback stabilization method for the viscous Burgers equation around a non-constant steady state, using algebraic Riccati equations and finite element discretization, with proven exponential decay and error estimates.

Contribution

It introduces a novel feedback stabilization approach for the viscous Burgers equation around non-constant steady states, including finite element error analysis and numerical validation.

Findings

System is feedback stabilizable with exponential decay.

Finite element discretization preserves stabilization properties.

Error estimates confirm the accuracy of stabilized solutions.

Abstract

In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay $-\omega<0$ for any $\omega>0$. The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay $-\omega+\epsilon$ for any $\epsilon>0$. The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.