  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.
