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Journal Articles Mathematical Control and Related Fields Year : 2022

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

Abstract

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.
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Dates and versions

hal-02913358 , version 1 (08-08-2020)
hal-02913358 , version 2 (07-06-2021)

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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⟩
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