no. 23-24
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

Liu, Xu1, 2 ; Zhang, Xu2, 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, Tome 347 (2009) no. 23-24, pp. 1379-1384.

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 C1 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 C1-norm of the principle part coefficients. The later is based on a new global Carleman estimate for the linear parabolic equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.017
Liu, Xu 1, 2 ; Zhang, Xu 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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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://archive.numdam.org/articles/10.1016/j.crma.2009.09.017/

[1] Barbu, V. Controllability of parabolic and Navier–Stokes equations, Sci. Math. Jpn., Volume 56 (2002), pp. 143-211

[2] Beceanu, M. Local exact controllability of the diffusion equation in one dimension, Abstr. Appl. Anal., Volume 14 (2003), pp. 793-811

[3] Doubova, A.; Fernández-Cara, E.; González-Burgos, M.; Zuazua, E. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., Volume 41 (2002), pp. 798-819

[4] Fabre, C.; Puel, J.P.; Zuazua, E. Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, Volume 125 (1995), pp. 31-61

[5] Fernández-Cara, E.; Zuazua, E. The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, Volume 5 (2000), pp. 465-514

[6] Fu, X. A weighted identity for partial differential operator of second order and its applications, C. R. Math. Acad. Sci. Paris, Volume 342 (2006), pp. 579-584

[7] Fursikov, A.V.; Imanvilov, O.Yu. Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Seoul, Korea, 1996

[8] Ladyzhenskaya, O.A.; Solonnikov, V.A.; Ural'ceva, N.N. Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, RI, 1968

[9] X. Liu, X. Zhang, Local controllability of multidimensional quasilinear parabolic equations, preprint

[10] X. Zhang, E. Zuazua, in preparation

[11] Zuazua, E. Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier Science, 2006, pp. 527-621

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