# Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

## TL;DR

This paper develops a nonlinear boundary feedback control for the 2D viscous Burgers' equation to achieve global stabilization, and analyzes its finite element approximation with confirmed exponential decay and numerical validation.

## Contribution

It introduces a novel nonlinear Neumann boundary feedback control law for global stabilization of the 2D viscous Burgers' equation and provides finite element error analysis.

## Key findings

- Exponential stabilization of the 2D viscous Burgers' equation achieved.
- Finite element method yields optimal error estimates in relevant norms.
- Numerical experiments confirm theoretical stability and convergence results.

## Abstract

In this article, global stabilization results for the two dimensional (2D) viscous Burgers' equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $C^0$-conforming finite element method in spatial direction, optimal error estimates in $L^\infty(L^2)$ and in $L^\infty(H^1)$- norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.