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hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorMazari, Idriss
HAL ID: 751069
dc.date.accessioned2022-01-07T14:37:26Z
dc.date.available2022-01-07T14:37:26Z
dc.date.issued2022
dc.identifier.issn0362-546X
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22410
dc.language.isoenen
dc.subjectShape optimisationen
dc.subjectOptimal controlen
dc.subjectParabolic PDEsen
dc.subjectQuantitative inequalitiesen
dc.subject.ddc515en
dc.titleQuantitative estimates for parabolic optimal control problems under L∞ and L1 constraintsen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenIn this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition (uf)t−Δuf=f, f being the control. We seek to maximise the functional JT(f):=12∫(0;T)×Ωu2f or, for some ϵ>0, JϵT(f):=12∫(0;T)×Ωu2f+ϵ∫Ωu2f(T,⋅) and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain Ω is a ball. In that case, if f satisfies L1 and L∞ constraints and does not depend on time, we propose a shape derivative approach that shows that, for any competitor f=f(x) satisfying the same constraints, we have JT(f∗)−JT(f)≳∥f−f∗∥2L1(Ω), f∗ being the maximiser. Through our proof of this time-independent case, we also show how to obtain coercivity norms for shape hessians in such parabolic optimisation problems. We also consider the case where f=f(t,x) satisfies a global L∞ constraint and, for every t∈(0;T), an L1 constraint. In this case, assuming ϵ>0, we prove an estimate of the form JϵT(f∗)−JϵT(f)≳∫T0aϵ(t)∥f(t,⋅)−f∗(t,⋅)∥2L1(Ω) where aϵ(t)>0 for any t∈(0;T). The proof of this result relies on a uniform bathtub principle.en
dc.relation.isversionofjnlnameNonlinear Analysis
dc.relation.isversionofjnlvol215en
dc.relation.isversionofjnldate2022-02
dc.relation.isversionofdoi10.1016/j.na.2021.112649.en
dc.relation.isversionofjnlpublisherElsevieren
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2022-01-07T14:35:49Z
hal.author.functionaut


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