Minimal time of controllability of two parabolic equations with disjoint control and coupling domains

Ammar Khodja, Farid; Benabdallah, Assia; González-Burgos, Manuel; de Teresa, Luz

doi:10.1016/j.crma.2014.03.004

Partial differential equations/Optimal control

Minimal time of controllability of two parabolic equations with disjoint control and coupling domains
[Temps minimal de contrôlabilité de deux équations paraboliques avec des domaines de contrôle et de couplage disjoints]

Présenté par : Gilles Lebeau

Ammar Khodja, Farid ¹ ; Benabdallah, Assia ² ; González-Burgos, Manuel ³ ; de Teresa, Luz ⁴

¹ Laboratoire de mathématiques de Besançon, université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
² Aix–Marseille Université, CNRS, Centrale Marseille, l2M, UMR 7373, 13453 Marseille, France
³ Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
⁴ Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U. 04510 D.F., Mexico

Comptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 391-396.

Résumé
Abstract

On considère deux équations paraboliques couplées par une matrice $A (x) = q (x) A_{0}$ , où $A_{0}$ est un bloc de Jordan d'ordre 1, et contrôlées par un seul contrôle localisé en espace ou frontière. Le support du coefficient de couplage, q, et celui du contrôle peuvent être disjoints. Nous mettons en évidence un temps minimal de contrôlabilité à 0, $T_{0} (q) \in [0, + \infty]$ .

We consider two parabolic equations coupled by a matrix $A (x) = q (x) A_{0}$ , where $A_{0}$ is a Jordan block of order 1, and controlled by a single localized function, or by a single boundary control. The support of the coupling coefficient, q, and the control domain may be disjoint. We exhibit an explicit minimal time of null-controllability, $T_{0} (q) \in [0, + \infty]$ .

Reçu le : 2014-01-10
Accepté le : 2014-03-05
Publié le : 2014-04-03

DOI : 10.1016/j.crma.2014.03.004

Affiliations des auteurs :

Ammar Khodja, Farid ¹ ; Benabdallah, Assia ² ; González-Burgos, Manuel ³ ; de Teresa, Luz ⁴

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     author = {Ammar Khodja, Farid and Benabdallah, Assia and Gonz\'alez-Burgos, Manuel and de Teresa, Luz},
     title = {Minimal time of controllability of two parabolic equations with disjoint control and coupling domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--396},
     publisher = {Elsevier},
     volume = {352},
     number = {5},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.03.004/}
}

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Ammar Khodja, Farid; Benabdallah, Assia; González-Burgos, Manuel; de Teresa, Luz. Minimal time of controllability of two parabolic equations with disjoint control and coupling domains. Comptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 391-396. doi : 10.1016/j.crma.2014.03.004. http://www.numdam.org/articles/10.1016/j.crma.2014.03.004/

Bibliographie
Cité par

[1] Alabau-Boussouira, F. Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE's by a single control, Math. Control Signals Systems, Volume 26 (2014) no. 1, pp. 1-46

[2] Alabau-Boussouira, F.; Léautaud, M. Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 7–8, pp. 395-400

[3] Alabau-Boussouira, F.; Léautaud, M. Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9), Volume 99 (2013) no. 5, pp. 544-576

[4] Ammar Khodja, F.; Benabdallah, A.; Dupaix, C. Null-controllability of some reaction–diffusion systems with one control force, J. Math. Anal. Appl., Volume 320 (2006) no. 2, pp. 928-943

[5] Ammar Khodja, F.; Benabdallah, A.; González-Burgos, M.; de Teresa, L. Controllability of some systems of parabolic equations, Málaga (2012) http://personal.us.es/manoloburgos/es/conferencias/

[6] Ammar Khodja, F.; Benabdallah, A.; González-Burgos, M.; de Teresa, L. A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013) no. 19–20, pp. 743-746

[7] F. Ammar Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, Controllability of parabolic systems with disjoint control and coupling domains, in preparation.

[8] Boyer, F.; Olive, G. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields (2014) (in press) | HAL

[9] Dehman, B.; Le Rousseau, J.; Léautaud, M. Controllability of two coupled wave equations on a compact manifold, Arch. Ration. Mech. Anal. (2014) (in press) | HAL

[10] Fattorini, H.O.; Russell, D.L. Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292

[11] Fernández-Cara, E.; González-Burgos, M.; de Teresa, L. Boundary controllability of parabolic coupled equations, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1720-1758

[12] González-Burgos, M.; de Teresa, L. Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math., Volume 67 (2010) no. 1, pp. 91-113

[13] Kavian, O.; de Teresa, L. Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var., Volume 16 (2010) no. 2, pp. 247-274

[14] Rosier, L.; de Teresa, L. Exact controllability of a cascade system of conservative equations, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 5–6, pp. 291-296

[15] de Teresa, L. Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, Volume 25 (2000) no. 1–2, pp. 39-72

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