Global stabilization and finite element analysis of the viscous Burgers' equation with memory subject to Neumann boundary feedback control

TL;DR

This paper develops a boundary feedback control strategy to globally stabilize the viscous Burgers' equation with memory, using finite element methods to analyze the semi-discrete scheme and provide error estimates.

Contribution

It introduces a novel boundary feedback control approach for the viscous Burgers' equation with memory, including finite element analysis and optimal error estimates.

Findings

Global stabilization achieved in multiple norms

Finite element semi-discrete scheme is stable and accurate

Error estimates for control laws are established

Abstract

This paper presents a global stabilization result of the viscous Burgers' equation with the memory term by applying Neumann boundary feedback control laws. We construct suitable feedback control inputs using the control Lyapunov functional and establish stabilization in the \(L^{2}, H^{1},\) and \(H^{2}\)-norms. The existence and uniqueness of the solution are established through the Faedo-Galerkin method. Moreover, we show the global stabilization where the diffusion coefficient $\nu$ is unknown. Then, we apply a \(C^{0}\)-conforming finite element method to the spatial variable while keeping the time variable continuous. Furthermore, we obtain global stabilization of the semi-discrete scheme and optimal error estimates for the state variable in the \(L^{\infty}\), \(L^{2}\), and \(H^{1}\)-norms, using the Ritz-Volterra projection. Additionally, error estimates for the feedback control laws are established. Lastly, we present some numerical simulations to demonstrate the theoretical findings.