In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay $-\omega<0$ for any $\omega>0$. The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay $-\omega+\epsilon$ for any $\epsilon>0$. The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.

翻译：本文探讨了使用局部内部控制实现粘性Burgers方程围绕非恒定稳态的反馈镇定，并利用有限元方法对镇定系统进行误差估计。该系统不仅可实现反馈镇定，且对任意$\omega>0$均呈现指数衰减率$-\omega<0$。通过求解线性化系统对应的适当代数Riccati方程，我们得到了反馈形式的镇定控制律。在文章第二部分，我们采用协调有限元方法对连续系统进行离散化，得到有限维离散系统。该近似系统同样被证明可实现（一致）反馈镇定，且对任意$\epsilon>0$具有$-\omega+\epsilon$的指数衰减率。离散系统的反馈控制通过求解离散代数Riccati方程获得。为验证方法的有效性，我们给出了镇定解与镇定反馈控制律的误差估计。数值实验进一步支持并验证了理论结果。