Comptes Rendus
no. 19-20
Partial differential equations
A new relation between the condensation index of complex sequences and the null controllability of parabolic systems
[Une nouvelle relation entre lʼindice de condensation des suites complexes et la contrôlabilité à zéro des systèmes paraboliques]

Présenté par : the Editorial Board

Farid Ammar Khodja1 ; Assia Benabdallah2 ; Manuel González-Burgos3 ; Luz de Teresa4
1 Laboratoire de mathématiques, université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
2 Université de Provence, CMI, 39, rue Frédéric-Joliot-Curie, 13453 Marseille cedex 13, 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. México, Mexico
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 743-746.

On annonce un résultat qui relie lʼindice de condensation des suites complexes et la contrôlabilité à zéro des systèmes paraboliques. On montre quʼun temps minimal de contrôle est nécessaire. Ces résultats sont ensuite utilisés pour étudier la contrôlabilité à zéro par le bord de quelques systèmes paraboliques.

In this note, we present a new result that relates the condensation index of a sequence of complex numbers with the null controllability of parabolic systems. We show that a minimal time is required for controllability. The results are used to prove the boundary controllability of some coupled parabolic equations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.09.014
Farid Ammar Khodja 1 ; Assia Benabdallah 2 ; Manuel González-Burgos 3 ; Luz de Teresa 4

1 Laboratoire de mathématiques, université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
2 Université de Provence, CMI, 39, rue Frédéric-Joliot-Curie, 13453 Marseille cedex 13, 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. México, Mexico 
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     title = {A new relation between the condensation index of complex sequences and the null controllability of parabolic systems},
     journal = {Comptes Rendus. Math\'ematique},
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Farid Ammar Khodja; Assia Benabdallah; Manuel González-Burgos; Luz de Teresa. A new relation between the condensation index of complex sequences and the null controllability of parabolic systems. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 743-746. doi : 10.1016/j.crma.2013.09.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.014/

[1] F. Ammar-Khodja; A. Benabdallah; M. González-Burgos; L. de Teresa The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl. (9), Volume 96 (2011) no. 6, pp. 555-590

[2] F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, Optimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, submitted for publication.

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

[4] F. Luca; L. de Teresa Control of coupled parabolic systems and Diophantine approximations, SeMA J., Volume 61 (2013) no. 1, pp. 1-17

[5] J.R. Shackell Overconvergence of Dirichlet series with complex exponents, J. Anal. Math., Volume 22 (1969), pp. 135-170

[6] M. Tucsnak; G. Weiss Observation and Control for Operator Semigroups, Birkhäuser Adv. Texts. Basl. Lehrb., Birkhäuser Verlag, Basel, 2009

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