Comptes Rendus
Partial Differential Equations/Optimal Control
Null controllability of a degenerated reaction–diffusion system in cardiac electro-physiology
[Nulle contrôlabilité dʼun système dégénéré de réaction–diffusion dans lʼélectique-physiologie du cœur]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 587-590.

Ce travail est consacré à lʼanalyse de la nulle contrôlabilité dʼun système dégénéré de réaction–diffusion non-linéaire modélisant lʼactivité électrique du cœur. Notre contrôle agit dans un sous-domaine fixe du domaine du coeur. Nous prouvons la nulle contrôlabilité de notre modèle en établissant en particulier une estimation de Carleman pour lʼéquation dégénéré. Des estimations globales de type Carleman et la régularité parabolique sont employées.

This Note is devoted to the analysis of the null controllability of a nonlinear reaction–diffusion system, approximating a parabolic–elliptic system, modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precise estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.06.006

Mostafa Bendahmane 1 ; Felipe Wallison Chaves-Silva 2

1 Laboratoire de institut mathematiques de Bordeaux, Université Victor-Segalen Bordeaux 2, place de la Victoire, 33076 Bordeaux, France
2 BCAM – Basque Center for Applied Mathematics, Alameda Mazzaredo 14, 48009 Bilbao, Basque Country, Spain
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Mostafa Bendahmane; Felipe Wallison Chaves-Silva. Null controllability of a degenerated reaction–diffusion system in cardiac electro-physiology. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 587-590. doi : 10.1016/j.crma.2012.06.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.006/

[1] M. Bendahmane, F.W. Chaves-Silva, Controllability of a degenerate reaction–diffusion system in electrocardiology, Preprint.

[2] E. Fernandez-Cara; E. Zuazua Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. Henri Poincare, Analise non lineaire, Volume 17 (2000) no. 5, pp. 583-616

[3] A.V. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, Seoul National University, RIM, Global Analysis Research Center, Seoul, 1996

[4] I.L. Glicksberg A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points, Proc. AMS, Volume 3 (1952) no. 9, pp. 170-174

[5] S. Guerrero Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control and Optimization, Volume 46 (2007) no. 2, pp. 379-394

[6] S. Guerrero; G. Lebeau Singular Optimal Control for a Transport–Diffusion equation, Comm. Partial Diff. Equations, Volume 32 (2007), pp. 1813-1836

[7] F.A. Khodja; A. Benabdallah; C. Dupaix Null-controllability of some reaction–diffusion systems with one control force, J. Math. Anal. Appl., Volume 320 (2006), pp. 928-943

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