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

Abstract

We analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator (−∆) s , s ∈ (0, 1), on a bounded C 1,1 domain Ω ⊂ R N . We first consider the problem in one space dimension and employ spectral techniques to prove that, for s ∈ [1/2, 1), null-controllability is achieved through an L 2 (ω × (0, T )) function acting in a subset ω ⊂ Ω 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.

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

“…The null controllability from the interior (that is, the case where the control function is localized in a nonempty subset ω of (−1, 1)) of the one-dimensional fractional heat equation has been recently investigated in [6] where the authors have shown that the system is null controllable at any time T > 0 if and only if 1 2 < s < 1. The interior null controllability of the Schrödinger and wave equations have been studied in [5]. The approximate controllability from the exterior of the super-diffusive system (that is, the case where u tt is replaced by the Caputo time fractional derivative D α t u of order 1 < α < 2, has been very recently considered in [27].…”

“…Since the system is not exact or null controllable (if δ > 0), it is the best possible result that can be obtained regarding the controllability of such systems. The null/exact controllabilty from the interior of the pure (without damping) wave equation (with strong zero Dirichlet exterior condition) associated with the bi-fractional Laplace operator has been investigated in [3] by using a Pohozaev identity for the fractional Laplacian established in [28]. More precisely, the author in [3] has considered the following problem:…”

Control and numerical approximation of fractional diffusion equations

“…Nevertheless, in recent years, several results have been obtained on the controllability properties of hyperbolic (wave) and dispersive (Schrödinger) models involving the fractional Laplacian. The interested reader may refer, for instance, to [12,18,92]. As for the numerical approximation of wave-type models, this issue is known to be quite delicate.…”

“…In addition to that, control problems in the non-local setting have been largely considered in recent years. An incomplete bibliography at this respect includes [5,12,14,15,18,21,33,43,66,69,70,91,92,93].…”

Multiplicity of solutions for fractional $q(.)$-Laplacian equations

“…The research on fractional Laplace operators and their applications is very attractive and extended. In the last decade, many authors from different fields of the pure and applied mathematics have considered PDE models involving the fractional Laplacian and addressed many relevant questions such as existence, uniqueness and regularity of solutions [5,10,13,14,15,17,31,32,34,39,40,41,47,48,49,50], spectral properties [22,23,28], or even more applied issues, for example control problems [8,9,11,12,42,43,44,45] or the description of several phenomena arising in finance and quantum mechanics [4,14,27].…”