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

U. Biccari Internal control for a non-local Schrödinger equation involving the fractional Laplace operator (2021)

Abstract: We analyze the interior controllability problem for a nonlocal Schr\”odinger 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. As a consequence of that, we obtain the controllability for a nonlocal wave equation involving the higher order fractional Laplace operator (-\Delta)^{2s}:=(-\Delta)^s(-\Delta)^s , s\in[1/2,1) . 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.

Read Full Paper