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

Pierre Lissy

doi:10.1016/j.crma.2012.06.004

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

Pierre Lissy ¹

¹ UPMC Univ Paris 06, UMR 7598, laboratoire Jacques-Louis Lions, 75005, Paris, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 591-595.

Résumé
Abstract

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 : 2012-05-25
Accepté le : 2012-06-14
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.06.004

Affiliations des auteurs :

Pierre Lissy ¹

¹ UPMC Univ Paris 06, UMR 7598, laboratoire Jacques-Louis Lions, 75005, Paris, France

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     title = {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},
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Pierre Lissy. 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, Volume 350 (2012) no. 11-12, pp. 591-595. doi : 10.1016/j.crma.2012.06.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.004/

Bibliographie
Cité par

[1] Jean-Michel Coron; Sergio Guerrero Singular optimal control: a linear 1-D parabolic–hyperbolic example, Asymptot. Anal., Volume 44 (2005) no. 3–4, pp. 237-257

[2] Sylvain Ervedoza; Enrique Zuazua Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 975-1017

[3] Hector O. Fattorini; David L. Russell Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292

[4] Andrei V. Fursikov; Oleg Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996

[5] Olivier Glass A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J. Funct. Anal., Volume 258 (2010) no. 3, pp. 852-868

[6] Sergio Guerrero; Gilles Lebeau Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, Volume 32 (2007) no. 10–12, pp. 1813-1836

[7] Gilles Lebeau; Luc Robbiano Contrôle exact de lʼéquation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1–2, pp. 335-356

[8] Luc Miller Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, Volume 204 (2004) no. 1, pp. 202-226

[9] Ali Salem A numerical study of the null boundary controllability of a convection diffusion equation, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009) no. 15–16, pp. 927-932

[10] Gérald Tenenbaum; Marius Tucsnak New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, Volume 243 (2007) no. 1, pp. 70-100

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