no. 11-12
Partial Differential Equations/Optimal Control
A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation
[Un lien entre le coût des contrôles rapides pour lʼéquation de la chaleur 1-D et lʼuniforme contrôlabilité dʼune équation de transport-diffusion 1-D]

Présenté par : Gilles Lebeau

Lissy, Pierre1
1 UPMC Univ Paris 06, UMR 7598, laboratoire Jacques-Louis Lions, 75005, Paris, France
Comptes Rendus. Mathématique, Tome 350 (2012) no. 11-12, pp. 591-595.

Dans cette Note, on explique comment des résultats sur le coût de la contrôlabilité à 0 de lʼéquation de la chaleur en temps petit peuvent être utilisés pour majorer le coût de la contrôlabilité à 0 dʼune équation unidimensionelle de transport-diffusion dans la limite de viscosité évanescente. On améliore des résultats précédemment connus concernant le temps minimal nécessaire pour obtenir la décroissance exponentielle du coût du contrôle et on explique ce que donnerait en plus la conjecture habituelle concernant le coût du contrôle en temps petit de lʼéquation de la chaleur.

In this Note, we explain how results on the cost of the null-controllability of the one-dimensional heat equation in small time can be used to bound from above the cost of the null-controllability of a one-dimensional transport-diffusion equation in the vanishing viscosity limit. We improve previous results about the minimal time needed to obtain the exponential decrease of the cost of the control and explain what would provide the usual conjecture concerning the cost of fast controls for the heat equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.06.004
Lissy, Pierre 1

1 UPMC Univ Paris 06, UMR 7598, laboratoire Jacques-Louis Lions, 75005, Paris, France 
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Lissy, Pierre. A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation. Comptes Rendus. Mathématique, Tome 350 (2012) no. 11-12, pp. 591-595. doi : 10.1016/j.crma.2012.06.004. http://www.numdam.org/articles/10.1016/j.crma.2012.06.004/

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