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

J.A. Bárcena-Petisco, Kevin Le Balc’H. Local null controllability of the penalized Boussinesq system with a reduced number of controls (2021)

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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