ERVEDOZA, Sylvain; LEMOINE, Jérôme; MUNCH, Arnaud

doi:10.3934/eect.2022042

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ERVEDOZA, Sylvain
Institut de Mathématiques de Bordeaux [IMB]
Centre National de la Recherche Scientifique [CNRS]

LEMOINE, Jérôme
Laboratoire de Mathématiques Blaise Pascal [LMBP]
Université Clermont Auvergne [UCA]

MUNCH, Arnaud
Laboratoire de Mathématiques Blaise Pascal [LMBP]
Université Clermont Auvergne [UCA]

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Article de revue

Este ítem está publicado en

Evolution Equations and Control Theory. 2023-01-02, vol. 12, n° 2, p. 567-599

American Institute of Mathematical Sciences (AIMS)

Resumen en inglés

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.< Leer menos

Palabras clave en inglés

AMS Classifications: 35K58

93B05 Semilinear heat equation

Null controllability

Carleman estimates

Fixed point

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191477

DOI

http://dx.doi.org/10.3934/eect.2022042

Proyecto ANR

Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux - ANR-20-CE40-0009

Orígen

Importado de Hal

Centros de investigación

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251