Feedback stabilization of a parabolic coupled system and its numerical study

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