Exact controllability of semilinear heat equations through a constructive approach
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| hal.structure.identifier | Centre National de la Recherche Scientifique [CNRS] | |
| dc.contributor.author | ERVEDOZA, Sylvain | |
| hal.structure.identifier | Laboratoire de Mathématiques Blaise Pascal [LMBP] | |
| hal.structure.identifier | Université Clermont Auvergne [UCA] | |
| dc.contributor.author | LEMOINE, Jérôme | |
| hal.structure.identifier | Laboratoire de Mathématiques Blaise Pascal [LMBP] | |
| hal.structure.identifier | Université Clermont Auvergne [UCA] | |
| dc.contributor.author | MUNCH, Arnaud | |
| dc.date.accessioned | 2024-04-04T02:45:32Z | |
| dc.date.available | 2024-04-04T02:45:32Z | |
| dc.date.issued | 2023-01-02 | |
| dc.identifier.issn | 2163-2480 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191477 | |
| dc.description.abstractEn | The 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.sponsorship | Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux - ANR-20-CE40-0009 | |
| dc.language.iso | en | |
| dc.publisher | American Institute of Mathematical Sciences (AIMS) | |
| dc.subject.en | AMS Classifications: 35K58 | |
| dc.subject.en | 93B05 Semilinear heat equation | |
| dc.subject.en | Null controllability | |
| dc.subject.en | Carleman estimates | |
| dc.subject.en | Fixed point | |
| dc.title.en | Exact controllability of semilinear heat equations through a constructive approach | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.3934/eect.2022042 | |
| dc.subject.hal | Mathématiques [math]/Optimisation et contrôle [math.OC] | |
| bordeaux.journal | Evolution Equations and Control Theory | |
| bordeaux.page | 567-599 | |
| bordeaux.volume | 12 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 2 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-03350534 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03350534v1 | |
| bordeaux.COinS | ctx_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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