Numerical Study of Semidiscrete Penalty Approach for Stabilizing Boussinesq System with Localized Feedback Control

We study the numerical approximation of the stabilization of the semidiscrete linearized Boussinesq system around an unstable stationary state.The stabilization is achieved by internal feedback controls applied on the velocity and the temperature equations, localized in an arbitrary open subset.This article follows the framework of [11], considering the continuous linearized Boussinesq system.The goal is to study the approximation by penalization of the free divergence condition in the semidiscrete case.More precisely, considering infinite time horizon LQR optimal control problem, we establish convergence results for the optimal controls, optimal solutions and Riccati operators when the penalization parameter goes to zero.We then propose a numerical validation of these results in a two-dimensional setting.