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Exact controllability of semilinear heat equations through a constructive approach

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
hal.structure.identifierCentre National de la Recherche Scientifique [CNRS]
dc.contributor.authorERVEDOZA, Sylvain
hal.structure.identifierLaboratoire de Mathématiques Blaise Pascal [LMBP]
hal.structure.identifierUniversité Clermont Auvergne [UCA]
dc.contributor.authorLEMOINE, Jérôme
hal.structure.identifierLaboratoire de Mathématiques Blaise Pascal [LMBP]
hal.structure.identifierUniversité Clermont Auvergne [UCA]
dc.contributor.authorMUNCH, Arnaud
dc.date.accessioned2024-04-04T02:45:32Z
dc.date.available2024-04-04T02:45:32Z
dc.date.issued2023-01-02
dc.identifier.issn2163-2480
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/191477
dc.description.abstractEnThe exact distributed controllability of the semilinear heat equation ∂ty − ∆y + f (y) = v 1ω posed over multi-dimensional and bounded domains, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/(|r| ln 3/2 |r|) β for some β small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/ ln 3/2 |r| β for some β small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.
dc.description.sponsorshipNouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux - ANR-20-CE40-0009
dc.language.isoen
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)
dc.subject.enAMS Classifications: 35K58
dc.subject.en93B05 Semilinear heat equation
dc.subject.enNull controllability
dc.subject.enCarleman estimates
dc.subject.enFixed point
dc.title.enExact controllability of semilinear heat equations through a constructive approach
dc.typeArticle de revue
dc.identifier.doi10.3934/eect.2022042
dc.subject.halMathématiques [math]/Optimisation et contrôle [math.OC]
bordeaux.journalEvolution Equations and Control Theory
bordeaux.page567-599
bordeaux.volume12
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue2
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-03350534
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03350534v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Evolution%20Equations%20and%20Control%20Theory&rft.date=2023-01-02&rft.volume=12&rft.issue=2&rft.spage=567-599&rft.epage=567-599&rft.eissn=2163-2480&rft.issn=2163-2480&rft.au=ERVEDOZA,%20Sylvain&LEMOINE,%20J%C3%A9r%C3%B4me&MUNCH,%20Arnaud&rft.genre=article


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