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.

翻译：本文第一部分研究利用局部内部控制器对抛物型耦合系统进行反馈镇定。该系统对于任意$\omega>0$，均可实现指数衰减率$-\omega<0$的反馈镇定。通过求解合适的代数Riccati方程，可获得反馈形式的镇定控制律。第二部分采用协调有限元方法将连续系统近似为有限维离散系统。对于任意$\epsilon>0$，该近似系统也能实现指数衰减率$-\omega+\epsilon$的（一致）反馈镇定，反馈控制律通过求解离散代数Riccati方程获取。文中给出了镇定解与镇定反馈控制的误差估计，并通过数值算例验证了理论结果。