Comptes Rendus
no. 3-4
Partial Differential Equations/Optimal Control
Controllability of the Ginzburg–Landau equation
[Contrôlabilité de l'équation de Ginzburg–Landau]

Présenté par : Haïm Brezis

Lionel Rosier1, 2 ; Bing-Yu Zhang3
1 Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática, Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
2 Institut Élie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
3 Department of Mathematical Sciences, University of Cincinnati,Cincinnati, OH 45221, USA
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 167-172.

Cette Note est dévolue à l'étude de la contrôlabilité frontière, ou interne, de l'équation complexe de Ginzburg–Landau. Des résultats de contrôlabilité à zéro sont obtenus au moyen d'une inégalité de Carleman et d'une analyse basée sur la théorie des opérateurs sectoriels.

This Note investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Null-controllability results are derived from a Carleman estimate and an analysis based upon the theory of sectorial operators.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.11.031
Lionel Rosier 1, 2 ; Bing-Yu Zhang 3

1 Centro de Modelamiento Matemático (CMM) and Departamento de Ingeniería Matemática, Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
2 Institut Élie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France
3 Department of Mathematical Sciences, University of Cincinnati,Cincinnati, OH 45221, USA 
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Lionel Rosier; Bing-Yu Zhang. Controllability of the Ginzburg–Landau equation. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 167-172. doi : 10.1016/j.crma.2007.11.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.031/

[1] J.L. Boldrini, E. Fernandez-Cara, S. Guerrero, On the controllability of the Ginzburg–Landau equation, in preparation

[2] E. Fernandez-Cara Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., Volume 2 (1997), pp. 87-103

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

[4] A.V. Fursikov; O.Y. Imanuvilov Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1996

[5] D. Henry Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981

[6] C.D. Levermore; M. Oliver Distribution-valued initial data for the complex Ginzburg–Landau equation, Comm. Partial Differential Equations, Volume 22 (1997), pp. 39-48

[7] C.D. Levermore; M. Oliver Berkeley, CA, 1994, Lectures in Appl. Math., vol. 31, Amer. Math. Soc., Providence, RI (1996), pp. 141-190

[8] A. Mielke The complex Ginzburg–Landau equation on large and unbounded domains: Sharper bounds and attractors, Nonlinearity, Volume 10 (1997), pp. 199-222

[9] L. Rosier, B.-Y. Zhang, Null controllability of the complex Ginzburg–Landau equation, submitted for publication

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