Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

Abstract

<p style='text-indent:20px;'>We analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator <inline-formula><tex-math id="M1">\begin{document}$ (-\Delta)^{\, {s}}{} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s\in(0, 1) $\end{document}</tex-math></inline-formula>, on a bounded <inline-formula><tex-math id="M3">\begin{document}$ C^{1, 1} $\end{document}</tex-math></inline-formula> domain <inline-formula><tex-math id="M4">\begin{document}$ \Omega\subset{\mathbb{R}}^N $\end{document}</tex-math></inline-formula>. We first consider the problem in one space dimension and employ spectral techniques to prove that, for <inline-formula><tex-math id="M5">\begin{document}$ s\in[1/2, 1) $\end{document}</tex-math></inline-formula>, null-controllability is achieved through an <inline-formula><tex-math id="M6">\begin{document}$ L^2(\omega\times(0, T)) $\end{document}</tex-math></inline-formula> function acting in a subset <inline-formula><tex-math id="M7">\begin{document}$ \omega\subset\Omega $\end{document}</tex-math></inline-formula> of the domain. This result is then extended to the multi-dimensional case by applying the classical multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian.</p>