Local null controllability of the penalized Boussinesq system with a reduced number of controls

Jon Asier Bárcena-Petisco (1, 2) , Kévin Le Balc'H (3, 4, 5)

1 UAM - Universidad Autónoma de Madrid
2 Fundación Deusto
3 IMB - Institut de Mathématiques de Bordeaux
4 CaGE - Control And GEometry
5 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions and defined in a regular domain Ω ⊂ R^N for N = 2 and N = 3. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter ε > 0. We prove that our system is locally null controllable using a control with a restricted number of components, defined in an open set ω contained in Ω and whose cost is bounded uniformly when ε → 0. The proof is based on a linearization argument and the null-controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This observability inequality is obtained thanks to the coercivity of some second order differential operator involving crossed derivatives.

Mots clés: Carleman inequality; Controllability; Nonlinear system; Penalized Stokes system

Domaines: Equations aux dérivées partielles [math.AP]

Jon Asier Bárcena-Petisco, Kévin Le Balc'H. Local null controllability of the penalized Boussinesq system with a reduced number of controls. Mathematical Control and Related Fields, 2022, 12 (3), pp.641. ⟨10.3934/mcrf.2021038⟩. ⟨hal-02913358v2⟩