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

Abstract

We analyze the interior controllability problem for a nonlocal Schrödinger equation involving the fractional Laplace operator $(-\Delta {)}^s$, $s\in (0,1)$, on a bounded $C^{1,1}$ domain $\Omega \subset {\mathbb{R}}^n$. The controllability from a neighborhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. The results follow applying the multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian, and from an explicit computation of the spectrum of the operator in the one-dimensional case.