Approximation of feedback gains for the Oseen system

Abstract

We consider the Oseen system with a Dirichlet boundary control, and its semidiscrete approximation by a finite element method (FEM). We show that these two systems fit with the abstract setting recently introduced in [5]. We obtain convergence rates for Riccati based feedback laws and their discrete approximation, in terms of the discretization parameter h (the mesh size of the FEM). We also prove convergence rates between the solution of the closed-loop Oseen system and the solution of the semidiscrete closed-loop Oseen system. These results are based on new error estimates, previously known for the Stokes system in polyhedral or polygonal convex domains, that we have recently extended to the Oseen system in polyhedral (or polygonal) convex or non-convex domains. We also prove a uniform bound for the discrete control operator $B_h$, which seems to be totally new in the context of the numerical approximation of feedback laws.