Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method
Lissy, Pierre; Roventa, Ionel (2018), Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method. https://basepub.dauphine.fr/handle/123456789/18469
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01865920
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Department of Mathematics [UCV]
Abstract (EN)We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method.
Subjects / Keywordsbiorthogonal families; moment problem; fractional Laplacian; hyperbolic equations; control approximation
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