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 for and . The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter . We prove that our system is locally null controllable using a control with a restricted number of components, defined in an open set contained in and whose cost is bounded uniformly when . 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.