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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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Last updated on March 17, 2022

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