Feedback stabilization of a parabolic coupled system and its numerical study

In the first part of this article, we study feedback stabilization a parabolic coupled system by using localized interior controls. The is stabilizable with exponential decay $ -\omega<0 for any \omega>0 $. A stabilizing control found in form solving suitable algebraic Riccati equation. second part, conforming finite element method employed to approximate continuous dimensional discrete system. approximated also (uniformly) -\omega+\epsilon $, \epsilon>0 and obtained error estimate stabilized solutions as well controls are obtained. We validate theoretical results numerical implementations.