We analyze the interior controllability problem for a non-local 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 $. We first consider the problem in one space dimension and employ spectral techniques to prove that, for $ s\in[1/2, 1) $, null-controllability is achieved through an $ L^2(\omega\times(0, T)) $ function acting in a subset $ \omega\subset\Omega $ 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.
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The domain
First 15 eigenvalues of the Dirichlet fractional Laplacian
Gap between the first 15 eigenvalues of the Dirichlet fractional Laplacian
Example of the domain