Approximation of feedback gains for abstract parabolic systems

Institut de Mathematiques de Toulouse, Universite Paul Sabatier & CNRS, 31062 Toulouse Cedex, France

^* Corresponding author: mehdi.badra@math.univ-toulouse.fr

Received: 21 November 2023
Accepted: 23 November 2024

Abstract

We consider parabolic controlled systems represented by a pair (A, B), where (A, D(A)) is the infinitesimal generator of an analytic semigroup on a Hilbert space Z and B is an unbounded control operator from a control space U into Z. We consider approximate controlled systems (A_ε, B_ε), for ε > 0, where (A_ε, D(A_ε)) is the infinitesimal generator of an analytic semigroup on a Hilbert space Z_ε and B_ε is an unbounded control operator from the control space U into Z_ε. Since Z_ε is not included in Z, we are in the case of nonconforming approximations. We assume that both Z and Z_ε are Hilbert subspaces of another Hilbert space H, and that there exist projectors P ∈ ℒ(H) and P_ε ∈ ℒ(H) such that Z = PH and Z_ε = P_εH, and for which (A, B, P) and (A_ε, B_ε, P_ε) satisfy suitable approximation assumptions. When the pair (A, B) is exponentially feedback stabilizable in Z, we first prove that the pair (A_ε, B_ε) is exponentially feedback stabilizable in Z_ε, uniformly with respect to ε ∈ (0, ε₀), for some ε0 > 0. We next prove that Riccati-based feedback laws stabilizing (A, B) in Z can be approximated by feedback laws stabilizing (A_ε, B_ε) in Z_ε. This type of results has been established in the eighties and the nineties in the case of conforming approximation, that is when Z_ε ⊂ Z. To the best of our knowledge nothing is known in the case of nonconforming approximations. We also extend, to the case of nonconforming approximations, convergence rates obtained in the case of conforming approximations. Nonconforming approximations play a central role in fluid mechanics. In M. Badra and J.-P. Raymond, Approximation of feedback gains for the Oseen system. Preprint (2025). https://hal.science/hal-04880955, we have shown that the results proved in the present paper apply to the Oseen system (the Navier–Stokes equations linearized around a steady state) and its semidiscrete approximation by a Finite Element Method.