Comptes Rendus
no. 2
Partial Differential Equations/Optimal Control
Controllability of some coupled parabolic systems by one control force
[Contrôlabilité de quelques systèmes paraboliques couplés avec un seul contrôle]

Présenté par : Gilles Lebeau

Manuel González-Burgos1 ; Rosario Pérez-García1
1 EDAN, Facultad de Matemáticas, Universidad de Sevilla, Apartado 1160, 41080 Sevilla, Spain
Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 125-130.

Dans cette Note, on présente une nouvelle approche qui permet de prouver de nouveaux résultats de contrôlabilité pour quelques systèmes paraboliques couplés considerés dans un domaine borné Ω de RN et contrôlés par un seul contrôle distribué. On analyse, comme exemple modèle, la contrôlabilité nulle d'un système linéaire de champ de phases. D'abord, on contrôle le système par deux contrôles. Ensuite, on élimine le contrôle artificiel introduit. Des estimations globales de Carleman et la régularité parabolique sont employées.

In this Note we present a new approach which allows one to prove new controllability results for some coupled parabolic systems considered in a bounded domain Ω of RN when one controls by a unique distributed control. We analyze, as a model example, the null controllability of a linear phase field system. First, one controls the system by two controls. Then, one eliminates the introduced fictitious control. Global Carleman estimates and the parabolic regularity are used.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.11.025
Manuel González-Burgos 1 ; Rosario Pérez-García 1

1 EDAN, Facultad de Matemáticas, Universidad de Sevilla, Apartado 1160, 41080 Sevilla, Spain 
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Manuel González-Burgos; Rosario Pérez-García. Controllability of some coupled parabolic systems by one control force. Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 125-130. doi : 10.1016/j.crma.2004.11.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.025/

[1] F. Ammar Khodja; A. Benabdallah; C. Dupaix; I. Kostin Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., Volume 42 (2003) no. 5, pp. 1661-1680

[2] O. Bodart; M. González-Burgos; R. Pérez-García Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, C. R. Acad. Sci. Paris, Volume 335 (2002) no. 8, pp. 677-682

[3] C. Caginalp An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal., Volume 92 (1986), pp. 205-245

[4] C. Fabre; J.-P. Puel; E. Zuazua Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, Volume 125 (1995) no. 1, pp. 31-61

[5] M. González-Burgos, R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, in preparation

[6] M. González-Burgos, R. Pérez-García, Controllability of some nonlinear coupled parabolic systems by one control force in unbounded domains, in preparation

[7] R. Pérez-García, Thesis, Universidad de Sevilla, Sevilla, Spain, 2004

[8] E. Zuazua Exact controllability for the semilinear wave equation, J. Math. Pures Appl., Volume 69 (1990) no. 1, pp. 1-31

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