# Global Stabilization of BBM-Burgers' Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

## TL;DR

This paper develops nonlinear boundary feedback control laws to globally stabilize BBM-Burgers' equations, providing finite element error analysis, superconvergence results, and numerical validation of the stabilization and error estimates.

## Contribution

It introduces new nonlinear boundary feedback controls for BBM-Burgers' equations and establishes superconvergence results, advancing the theoretical and numerical understanding of stabilization.

## Key findings

- Global stabilization achieved using nonlinear boundary feedback laws.
- Optimal error estimates in multiple norms for the finite element solution.
- First-time establishment of superconvergence results for boundary feedback controls.

## Abstract

In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in $L^\infty(L^2)$, $L^\infty(H^1)$ and $L^\infty(L^\infty)$-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.