[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]
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.
Accepté le :
Publié le :
@article{CRMATH_2012__350_11-12_591_0,
author = {Pierre Lissy},
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},
journal = {Comptes Rendus. Math\'ematique},
pages = {591--595},
publisher = {Elsevier},
volume = {350},
number = {11-12},
year = {2012},
doi = {10.1016/j.crma.2012.06.004},
language = {en},
}
TY - JOUR AU - Pierre Lissy TI - 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 JO - Comptes Rendus. Mathématique PY - 2012 SP - 591 EP - 595 VL - 350 IS - 11-12 PB - Elsevier DO - 10.1016/j.crma.2012.06.004 LA - en ID - CRMATH_2012__350_11-12_591_0 ER -
%0 Journal Article %A Pierre Lissy %T 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 %J Comptes Rendus. Mathématique %D 2012 %P 591-595 %V 350 %N 11-12 %I Elsevier %R 10.1016/j.crma.2012.06.004 %G en %F CRMATH_2012__350_11-12_591_0
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/
[1] Singular optimal control: a linear 1-D parabolic–hyperbolic example, Asymptot. Anal., Volume 44 (2005) no. 3–4, pp. 237-257
[2] Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 975-1017
[3] Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292
[4] Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996
[5] 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] Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations, Volume 32 (2007) no. 10–12, pp. 1813-1836
[7] Contrôle exact de lʼéquation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1–2, pp. 335-356
[8] 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] 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] New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, Volume 243 (2007) no. 1, pp. 70-100
Cité par Sources :
Commentaires - Politique
Remarks on local controllability for the Boussinesq system with Navier boundary condition
Cristhian Montoya
C. R. Math (2020)
Remarks on exact controllability for Stokes and Navier–Stokes systems
Enrique Fernández-Cara; Sergio Guerrero; Oleg Yurievich Imanuvilov; ...
C. R. Math (2004)
A numerical study of the null boundary controllability of a convection diffusion equation
Ali Salem
C. R. Math (2009)