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Spectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations

hal.structure.identifierInstitut de Recherche Mathématique de Rennes [IRMAR]
dc.contributor.authorBEAUCHARD, Karine
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorJAMING, Philippe
hal.structure.identifierInstitut de Recherche Mathématique de Rennes [IRMAR]
dc.contributor.authorPRAVDA-STAROV, Karel
dc.date.accessioned2024-04-04T03:06:17Z
dc.date.available2024-04-04T03:06:17Z
dc.date.issued2021
dc.identifier.issn0039-3223
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/193321
dc.description.abstractEnSome recent works have shown that the heat equation posed on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform. In the present work, we prove new uncertainty principles for finite combinations of Hermite functions and establish an analogue of the Logvinenko-Sereda theorem with an explicit control of the constant with respect to the energy level of the Hermite functions as eigenfunctions of the harmonic oscillator for thick control subsets. This spectral inequality allows to derive the null-controllability in any positive time from thick control regions for para-bolic equations associated with a general class of hypoelliptic non-selfadjoint quadratic differential operators. More generally, the spectral inequality for finite combinations of Hermite functions is actually shown to hold for any measurable control subset of positive Lebesgue measure, and some quantitative estimates of the constant with respect to the energy level are given for two other classes of control subsets including the case of non-empty open control subsets.
dc.description.sponsorshipCentre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
dc.language.isoen
dc.publisherInstytut Matematyczny - Polska Akademii Nauk
dc.title.enSpectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations
dc.typeArticle de revue
dc.identifier.doi10.4064/sm191205-12-10
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halMathématiques [math]/Analyse classique [math.CA]
bordeaux.journalStudia Mathematica
bordeaux.page1-43
bordeaux.volume260
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01766300
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01766300v1
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