[Contrôlabilité locale d'une classe d'équations quasi-linéaires paraboliques multidimensionnelles]
Dans cette Note, nous étudions la contrôlabilité locale vers zéro et son coût pour une classe d'équations quasi-linéaires paraboliques multidimensionnelles avec une condition homogène de Dirichlet et un contrôle interne. À la différence des résultats connus dans le cas monodimensionel, nous avons besoin de considérer le problème dans le cadre des solutions classiques. Le point clé consiste à améliorer la régularité de la fonction contrôle pour des données régulières. Ceci découle d'une nouvelle inégalité d'observabilité pour les équations linéaires paraboliques dans laquelle la constante d'observabilité est explicite vis-à-vis de la norme des coefficients de la partie principale. À cette fin, on établit une nouvelle inégalité de Carleman globale pour les équations linéaires paraboliques.
In this Note, we study the local null controllability and the cost estimate for a class of multidimensional quasilinear parabolic equations with homogeneous Dirichlet boundary conditions and an arbitrary located internal controller. Unlike the known result for one space dimension, we need to consider the problem in the frame of classical solutions. The key point is to improve the regularity of control function for smooth data, which is a consequence of a new observability inequality for linear parabolic equations with an explicit estimate on the observability constant in terms of the -norm of the principle part coefficients. The later is based on a new global Carleman estimate for the linear parabolic equation.
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@article{CRMATH_2009__347_23-24_1379_0,
author = {Liu, Xu and Zhang, Xu},
title = {On the local controllability of a class of multidimensional quasilinear parabolic equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {1379--1384},
publisher = {Elsevier},
volume = {347},
number = {23-24},
year = {2009},
doi = {10.1016/j.crma.2009.09.017},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.09.017/}
}
TY - JOUR AU - Liu, Xu AU - Zhang, Xu TI - On the local controllability of a class of multidimensional quasilinear parabolic equations JO - Comptes Rendus. Mathématique PY - 2009 SP - 1379 EP - 1384 VL - 347 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.09.017/ DO - 10.1016/j.crma.2009.09.017 LA - en ID - CRMATH_2009__347_23-24_1379_0 ER -
%0 Journal Article %A Liu, Xu %A Zhang, Xu %T On the local controllability of a class of multidimensional quasilinear parabolic equations %J Comptes Rendus. Mathématique %D 2009 %P 1379-1384 %V 347 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.09.017/ %R 10.1016/j.crma.2009.09.017 %G en %F CRMATH_2009__347_23-24_1379_0
Liu, Xu; Zhang, Xu. On the local controllability of a class of multidimensional quasilinear parabolic equations. Comptes Rendus. Mathématique, Tome 347 (2009) no. 23-24, pp. 1379-1384. doi : 10.1016/j.crma.2009.09.017. http://www.numdam.org/articles/10.1016/j.crma.2009.09.017/
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