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

Abstract

<p style='text-indent:20px;'>In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon > 0 $\end{document}</tex-math></inline-formula>. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in <inline-formula><tex-math id="M6">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. We also show that the control cost is bounded uniformly with respect to <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. 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.</p>