Feedback stabilization of parabolic coupled system and its numerical study

03/20/2023

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by   Wasim Akram, et al.

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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 -ω<0 for any ω>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 -ω+ϵ, for any ϵ>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.