[Contrôlabilité à zéro uniforme de l'équation de la chaleur avec densité périodique rapidement oscillante]
Nous considérons l'équation de la chaleur avec densité périodique rapidement oscillante, et un contrôle interne dans un domaine borné. Nous établissons d'abord des estimations fines de convergence dependant explicitement de la donnée initiale pour l'équation non contrôlée ; ces estimations sont nouvelles, et leur démonstration repose sur une régularisation judicieuse de la donnée initiale. Puis nous utilisons ces estimations pour démontrer que l'équation initiale est uniformément contrôlable à zéro, pourvu q'un contrôle interne, supplémentaire, evanescent, et convenablement choisi, soit ajouté à cette équation. Ce résultat de contrôlabilité uniforme est le premier dans le cadre multidimensionnel pour l'équation de la chaleur avec densité rapidement oscillante. Enfin nous montrons que la suite des contrôles converge vers le contrôle à zero optimal de l'équation limite lorsque la période tend vers zéro.
We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero.
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@article{CRMATH_2009__347_13-14_779_0,
author = {Louis Tebou},
title = {Uniform null controllability of the heat equation with rapidly oscillating periodic density},
journal = {Comptes Rendus. Math\'ematique},
pages = {779--784},
publisher = {Elsevier},
volume = {347},
number = {13-14},
year = {2009},
doi = {10.1016/j.crma.2009.04.030},
language = {en},
}
Louis Tebou. Uniform null controllability of the heat equation with rapidly oscillating periodic density. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 779-784. doi : 10.1016/j.crma.2009.04.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.030/
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