Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition

Ammar-Khodja, Farid; Benabdallah, Assia; Dupaix, Cédric; González-Burgos, Manuel

doi:10.1016/j.crma.2007.10.023

Partial Differential Equations/Optimal Control

Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition
[Contrôlabilité d'une classe de systèmes réaction–diffusion : la condition de Kalman généralisée]

Présenté par : Gilles Lebeau

Ammar-Khodja, Farid ¹ ; Benabdallah, Assia ² ; Dupaix, Cédric ¹ ; González-Burgos, Manuel ³

¹ Laboratoire de mathématiques UMR 6623, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
² CMI-LATP, UMR 6632, Université de Provence, technopôle Château-Gombert, 39, rue F. Joliot-Curie, 13453 Marseille cedex 13, France
³ Dpto. de Ecuaciones Diferentiales y Análisis Numérico, Universidad Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Comptes Rendus. Mathématique, Tome 345 (2007) no. 10, pp. 543-548.

Résumé
Abstract

Dans cette Note, on étudie la contrôlabilité d'une classe de systèmes paraboliques de la forme $Y_{t} = (D Δ + A) Y + B χ_{ω} u$ avec des conditions de Dirichlet sur le bord d'un domaine borné Ω, où $ω \subset Ω$ est un sous-domaine. Ici $D, A \in L (R^{n})$ , $B \in L (R^{m}; R^{n})$ et on montre que la condition algébrique de Kalman s'étend à de tels systèmes.

In this Note, we study the controllability of a class of parabolic systems of the form $Y_{t} = (D Δ + A) Y + B χ_{ω} u$ with Dirichlet conditions on the boundary of a bounded domain Ω, where $ω \subset Ω$ is a subdomain. Here $D, A \in L (R^{n})$ , $B \in L (R^{m}; R^{n})$ and we prove that the algebraic Kalman condition extends to such systems.

Reçu le : 2007-01-15
Accepté le : 2007-10-02
Publié le : 2007-11-09

DOI : 10.1016/j.crma.2007.10.023

Affiliations des auteurs :

Ammar-Khodja, Farid ¹ ; Benabdallah, Assia ² ; Dupaix, Cédric ¹ ; González-Burgos, Manuel ³

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     author = {Ammar-Khodja, Farid and Benabdallah, Assia and Dupaix, C\'edric and Gonz\'alez-Burgos, Manuel},
     title = {Controllability for a class of reaction{\textendash}diffusion systems: the generalized {Kalman's} condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {543--548},
     publisher = {Elsevier},
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Ammar-Khodja, Farid; Benabdallah, Assia; Dupaix, Cédric; González-Burgos, Manuel. Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition. Comptes Rendus. Mathématique, Tome 345 (2007) no. 10, pp. 543-548. doi : 10.1016/j.crma.2007.10.023. http://www.numdam.org/articles/10.1016/j.crma.2007.10.023/

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