Finite Difference Method for Global Stabilization of the Viscous Burgers' Equation with Nonlinear Neumann Boundary Feedback Control

TL;DR

This work addresses global stabilization of the viscous Burgers' equation using nonlinear Neumann boundary feedback. It introduces a theta-based finite difference scheme that unifies explicit and implicit time stepping and analyzes stability and convergence via a discrete energy framework. The authors prove exponential stability for $\theta\in[\tfrac12,1]$ (with conditional stability for smaller $\theta$), establish existence of the discrete solution, and derive error estimates showing first-order convergence for the state and boundary controls. Numerical experiments corroborate the theoretical results, demonstrating effective stabilization and expected convergence behavior of the proposed scheme.

Abstract

This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the $θ$-scheme with $θ\in [0,1]$, unifies explicit and implicit time discretizations and is suitable for handling the nonlinear boundary feedback control problem. Using the discrete energy method, we prove that the proposed difference scheme is conditionally stable for $0 \leq θ< \frac{1}{2}$ and unconditionally stable for $θ\geq \frac{1}{2}$. In addition, we establish the exponential stability of the fully discrete solution. The error analysis shows a first-order convergence rate of the state variable in the discrete $L^{2}$-, $H^{1}$-, and $L^{\infty}$-norms for $θ\geq \frac{1}{2}$, while preserving the exponential stability property. A first-order convergence rate for the boundary control inputs is also obtained. Numerical experiments are conducted to validate the theoretical findings and to demonstrate the effectiveness of the method for the inhomogeneous nonlinear Neumann boundary feedback control of the viscous Burgers' equation.

Figures (2)

• Figure 1: Example 5.1: (a) Both Controlled and Uncontrolled solution in discrete $L^{2}-$norm. (b) Feedback controller at $x=0$ with $\theta=1$, (c) Feedback controller at $x=1$ with $\theta=1.$
• Figure 2: Example 5.2: (a) Both Uncontrolled and Controlled solution in the discrete $L^{2}-$norm with different values of $c_{0}$ and $c_{1},$ (b) Controlled solution in the discrete $L^{2}-$norm with various values of $\theta$, (c) Absolute value of the feedback controllers with different values of $\theta$ at $x=0$, (d) Absolute value of the feedback controller with different values of $\theta$ at $x=1$, (e) Feedback controller with different values of $\nu$.

Theorems & Definitions (11)

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