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

TL;DR

The paper addresses global stabilization of the viscous Burgers' equation with memory under Neumann boundary feedback by designing a control Lyapunov functional that yields exponential decay in $L^{2}$, $H^{1}$, and $H^{2}$ norms. It develops a Faedo–Galerkin framework to prove existence and uniqueness and employs a $C^{0}$-conforming finite element method in space with a Ritz-Volterra projection to obtain optimal error estimates for the state and the boundary controls, including an adaptive strategy for unknown diffusion $\nu$. The analysis shows that the semi-discrete scheme inherits the stabilization properties with an optimal convergence rate of $O(h^{2})$ in relevant norms. Numerical experiments corroborate the theoretical findings, illustrate the influence of memory terms, and verify the predicted convergence orders and controller behavior.

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 $ν$ 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.

Figures (3)

• Figure 1: Example \ref{['ex1']}: (i) Controlled and uncontrolled solution in the $L^{2}$-norm for various values of $c_{0}$ and $c_{1}$. (ii) Control input for different values of $c_{0}$ and $c_{1}$ at the left boundary $x=0$. (iii) Control input for various values of $c_{0}$ and $c_{1}$ at the right boundary $x=1$.
• Figure 2: Example \ref{['ex2']}: (i) Controlled solution in the $L^{2}$-norm for various values of $\nu$ with $\rho=1$. (ii) Control input for different values of $\nu$ with $\rho=1$ at the left boundary $x=0$. (iii) Control input for various values of $\nu$ with $\rho=1$ at the right boundary $x=1$.
• Figure 3: Example \ref{['ex2']}: (i) In semi-log, controlled solution in the $L^{2}$-norm for different values of $\rho$ with fixed $\nu=0.1$. (ii) Control input for numerous values of $\rho$ with fixed $\nu=0.1$ at the left boundary $x=0$. (iii) Control input for various values of $\rho$ with fixed $\nu=0.1$ at the right boundary $x=1$.

Theorems & Definitions (12)

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