# Global Stabilization of 2D Forced Viscous Burgers' Equation Around Nonconstant Steady State Solution by Nonlinear Neumann Boundary Feedback Control:Theory and Finite Element Analysis

## TL;DR

This paper develops a nonlinear boundary feedback control method to achieve global stabilization of the 2D forced viscous Burgers' equation around a nonconstant steady state, with finite element analysis and numerical validation.

## Contribution

It introduces a novel nonlinear Neumann boundary feedback control for stabilizing the 2D Burgers' equation around nonconstant steady states, including error estimates and convergence analysis.

## Key findings

- Global stabilization achieved under smallness condition.
- Optimal error estimates in $L^
abla(L^2)$ and $L^
abla(H^1)$ norms.
- Numerical experiments confirm theoretical results.

## Abstract

Global stabilization of viscous Burgers' equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers' equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under some smallness condition on that steady state solution. On discretizing in space using $C^0$ piecewise linear elements keeping time variable continuous, a semidiscrete scheme is obtained. Moreover, global stabilization results for the semidiscrete solution and optimal error estimates for the state variable in $L^\infty(L^2)$ and $L^\infty(H^1)$-norms are derived. Further, optimal convergence result is established for the boundary feedback control law. All our results in this paper preserve exponential stabilization property. Finally, some numerical experiments are documented to confirm our theoretical findings.