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

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain \begin{document}$ \Omega\subset\mathbb R^N $\end{document} for \begin{document}$ N = 2 $\end{document} and \begin{document}$ N = 3 $\end{document} . The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter \begin{document}$ \varepsilon > 0 $\end{document} . We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set \begin{document}$ \omega $\end{document} contained in \begin{document}$ \Omega $\end{document} . We also show that the control cost is bounded uniformly with respect to \begin{document}$ \varepsilon \rightarrow 0 $\end{document} . The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.