Observability and null-controllability for parabolic equations in $ L_p $-spaces

Abstract

<p style='text-indent:20px;'>We study (cost-uniform approximate) null-controllability of parabolic equations in <inline-formula><tex-math id="M2">\begin{document}$ L_p( \mathbb{R}^d) $\end{document}</tex-math></inline-formula> and provide explicit bounds on the control cost. In particular, we consider systems of the form <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = -A_px(t) + {\bf{1}}_ E u(t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ x(0) = x_0\in L_p ( \mathbb{R}^d) $\end{document}</tex-math></inline-formula>, with interior control on a so-called thick set <inline-formula><tex-math id="M5">\begin{document}$ E \subset \mathbb{R}^d $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ p\in [1,\infty) $\end{document}</tex-math></inline-formula>, and where <inline-formula><tex-math id="M7">\begin{document}$ A $\end{document}</tex-math></inline-formula> is an elliptic operator of order <inline-formula><tex-math id="M8">\begin{document}$ m \in \mathbb{N} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M9">\begin{document}$ L_p(\mathbb{R}^d) $\end{document}</tex-math></inline-formula>. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case <inline-formula><tex-math id="M10">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula>.</p>