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From the reachable space of the heat equation to Hilbert spaces of holomorphic functions

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorHARTMANN, Andreas
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorKELLAY, Karim
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorTUCSNAK, Marius
dc.date.accessioned2024-04-04T03:09:27Z
dc.date.available2024-04-04T03:09:27Z
dc.date.created2017-07-27
dc.date.issued2020-06-15
dc.identifier.issn1435-9855
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/193605
dc.description.abstractEnThis work considers systems described by the heat equation on the interval [0, π] with L^2 boundary controls and it studies the reachable space at some instant τ > 0. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has [0, π] as one of the diagonals. More precisely, in the case Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Smirnov space and it is contained in the Bergman space associated to the above mentioned square. The methodology, quite different of the one employed in previous literature, is a direct one. We first represent the input-to-state map as an integral operator whose kernel is a sum of Gaussians and then we study the range of this operator by combining the theory of Riesz bases for Smirnov spaces in polygons and the theory developed by Aikawa, Hayashi and Saitoh on the range of integral transforms, in particular those associated with the heat kernel.
dc.description.sponsorshipImpacts marchands, non marchands et structurels des réformes des politiques agricoles et agri-environnementales - ANR-05-PADD-0015
dc.language.isoen
dc.publisherEuropean Mathematical Society
dc.subject.enreachable space
dc.subject.encontrollability
dc.subject.enheat equation
dc.subject.enBergman spaces
dc.subject.enSmirnov spaces
dc.subject.enRiesz basis
dc.title.enFrom the reachable space of the heat equation to Hilbert spaces of holomorphic functions
dc.typeArticle de revue
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halMathématiques [math]/Optimisation et contrôle [math.OC]
dc.subject.halMathématiques [math]/Variables complexes [math.CV]
dc.subject.halMathématiques [math]/Analyse fonctionnelle [math.FA]
bordeaux.journalJournal of the European Mathematical Society
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01569695
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01569695v1
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