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 (2022), Vol. 11, No. 1: 301-324. doi: 10.3934/eect.2021014

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. 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.

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