On the local controllability of a class of multidimensional quasilinear parabolic equations

Xu Liu; Xu Zhang

doi:10.1016/j.crma.2009.09.017

Partial Differential Equations/Optimal Control

On the local controllability of a class of multidimensional quasilinear parabolic equations
[Contrôlabilité locale d'une classe d'équations quasi-linéaires paraboliques multidimensionnelles]

Présenté par : Pierre-Louis Lions

Xu Liu ^1,² ; Xu Zhang ^2,³

¹ School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
² Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
³ Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1379-1384.

Résumé
Abstract

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 $C^{1}$ 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 $C^{1}$ -norm of the principle part coefficients. The later is based on a new global Carleman estimate for the linear parabolic equation.

Reçu le : 2009-04-15
Accepté le : 2009-09-21
Publié le : 2009-10-29

DOI : 10.1016/j.crma.2009.09.017

Affiliations des auteurs :

Xu Liu ^{1, 2} ; Xu Zhang ^{2, 3}

¹ School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
² Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
³ Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

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     author = {Xu Liu and Xu Zhang},
     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},
}

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Xu Liu; Xu Zhang. On the local controllability of a class of multidimensional quasilinear parabolic equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1379-1384. doi : 10.1016/j.crma.2009.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.017/

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