<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>\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math>\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math>\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter <inline-formula><tex-math>\begin{document}$ \varepsilon &gt; 0 $\end{document}</tex-math></inline-formula>. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set <inline-formula><tex-math>\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in <inline-formula><tex-math>\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. We also show that the control cost is bounded uniformly with respect to <inline-formula><tex-math>\begin{document}$ \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>

中文翻译：

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减少控制数量的惩罚 Boussinesq 系统的局部零可控性

<p style='text-indent:20px;'>在本文中，我们考虑具有齐次 Dirichlet 边界条件的 Boussinesq 系统，定义在正则域 <inline-formula><tex-math>\begin{document}$ \Omega \subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math>\begin{document}$ N = 2 $\end{document }</tex-math></inline-formula> 和 <inline-formula><tex-math>\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula >。流体的不可压缩性条件被它的近似值所取代，该近似值通过一个小参数 <inline-formula><tex-math>\begin{document}$ \varepsilon > 进行惩罚。0 $\end{document}</tex-math></inline-formula>。我们证明我们的系统是局部空可控的，使用具有有限数量组件的控件，本地化在一个开放集 <inline-formula><tex-math>\begin{document}$ \omega $\end{document}</ tex-math></inline-formula> 包含在 <inline-formula><tex-math>\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> 中。我们还证明了控制成本对于 <inline-formula><tex-math>\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline- 是一致有界的公式>。证明基于线性化论证。线性化系统的零点可控性是通过证明伴随系统的新卡尔曼估计来获得的。