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

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon). This work was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), by the Elkartek grant KK-2020/00091 CONVADP of the Basque government and by the Grant FA9550-18-1-0242 of AFOSR

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  • 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.

    Mathematics Subject Classification: 35R11, 35S05, 35S11, 93B05, 93B07.

    Citation:

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  • Figure 1.  The domain $ \Omega $ with the partition $ (\Gamma_0, \Gamma_1) $ of its boundary and the neighborhood $ \omega $ of $ \Gamma_0 $

    Figure 2.  First 15 eigenvalues of the Dirichlet fractional Laplacian $ (-d_x^{\, 2})^{{s}}{} $ on $ (-1, 1) $ for $ s\in (0, 1/2] $ (left) and $ s\in(1/2, 1) $ (right)

    Figure 3.  Gap between the first 15 eigenvalues of the Dirichlet fractional Laplacian $ (-d_x^{\, 2})^{{s}}{} $ on $ (-1, 1) $ for $ s\in(0, 1/2] $ (left) and $ s\in(1/2, 1) $ (right)

    Figure 4.  Example of the domain $ \Omega $ with the partition of the boundary $ (\Gamma_0, \Gamma_1) $ and the two neighborhood of the boundary $ \widehat{\omega} $ and $ \omega $

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