Approximate Controllability from the Exterior of Space-Time Fractional Wave Equations

In this paper we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. For the fractional elliptic PDEs this has been recently investigated by the authors in [5]. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the source/control either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem. We present an approach on how to approximate the parabolic Dirichlet solutions by the parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.2010 Mathematics Subject Classification. 49J20, 49K20, 35S15, 65R20, 65N30.

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“…Step It has been shown in [5] (see also [37]) that u solves (3.26) if and only if (3.25) holds and the claim is proved.…”

Section: 2mentioning

confidence: 83%

“…It has been shown in [37] that the operator −(−∆) s R generates a strongly continuous semigroup (e −t(−∆) s R ) t≥0 in L 2 (Ω). Hence, using semigroup theory, we can deduce that (3.31) has a unique weak solution w that belongs to L 2 ((0, T ); W s,2 Ω,κ ) ∩ H 1 ((0, T ); (W s,2 Ω,κ ) ) and is given by…”

Section: 2mentioning

confidence: 99%

External optimal control of fractional parabolic PDEs

2020

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“…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]. The case of the (possible) strong damping fractional wave equation has been investigated in [40].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Warma

Zamorano

2020

ESAIM: COCV

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We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval (−1, 1) associated with the fractional Laplace operator (−∂ 2x ) s , where 0 < s < 1. We show that there is a control function which is localized in a nonempty open set O ⊂ (R \ (−1, 1)), that is, at the exterior of the interval (−1, 1), such that the system is null controllable at any time T > 0 if and only if 1 2 < s < 1.2010 Mathematics Subject Classification. 35R11, 35S11, 35K20, 93B05, 93B07.

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“…1 Exterior controllability for the fractional Schrödinger equation. The concept of exterior controllability for evolution equations involving the fractional Laplacian has been recently introduced in several contributions (see [1,35,55,56,57]). This is the equivalent of the boundary controllability property for local partial differential equations.…”

Section: Conclusion and Open Problems And Perspectivesmentioning

confidence: 99%

“…Concerning wave-like models, the approximate controllability from the exterior of fractional wave equations has been proved in [35,57], while [6] treats the null-controllability of a one-dimensional fractional wave equation with memory. Nonetheless, as far as the author knows, there are currently no controllability results for the fractional Schrödinger equation.…”

mentioning

confidence: 99%

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

Biccari¹

2022

EECT

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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.

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Approximate Controllability from the Exterior of Space-Time Fractional Wave Equations

Cited by 17 publications

References 34 publications

External optimal control of fractional parabolic PDEs

External optimal control of fractional parabolic PDEs

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

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

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