Comptes Rendus
Partial Differential Equations/Optimal Control
Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition
Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 543-548.

In this Note, we study the controllability of a class of parabolic systems of the form Yt=(DΔ+A)Y+Bχωu with Dirichlet conditions on the boundary of a bounded domain Ω, where ωΩ is a subdomain. Here D,AL(Rn), BL(Rm;Rn) and we prove that the algebraic Kalman condition extends to such systems.

Dans cette Note, on étudie la contrôlabilité d'une classe de systèmes paraboliques de la forme Yt=(DΔ+A)Y+Bχωu avec des conditions de Dirichlet sur le bord d'un domaine borné Ω, où ωΩ est un sous-domaine. Ici D,AL(Rn), BL(Rm;Rn) et on montre que la condition algébrique de Kalman s'étend à de tels systèmes.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.10.023
Farid Ammar-Khodja 1; Assia Benabdallah 2; Cédric Dupaix 1; Manuel González-Burgos 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
@article{CRMATH_2007__345_10_543_0,
     author = {Farid Ammar-Khodja and Assia Benabdallah and C\'edric Dupaix and Manuel Gonz\'alez-Burgos},
     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},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.023},
     language = {en},
}
TY  - JOUR
AU  - Farid Ammar-Khodja
AU  - Assia Benabdallah
AU  - Cédric Dupaix
AU  - Manuel González-Burgos
TI  - Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 543
EP  - 548
VL  - 345
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2007.10.023
LA  - en
ID  - CRMATH_2007__345_10_543_0
ER  - 
%0 Journal Article
%A Farid Ammar-Khodja
%A Assia Benabdallah
%A Cédric Dupaix
%A Manuel González-Burgos
%T Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition
%J Comptes Rendus. Mathématique
%D 2007
%P 543-548
%V 345
%N 10
%I Elsevier
%R 10.1016/j.crma.2007.10.023
%G en
%F CRMATH_2007__345_10_543_0
Farid Ammar-Khodja; Assia Benabdallah; Cédric Dupaix; Manuel González-Burgos. Controllability for a class of reaction–diffusion systems: the generalized Kalman's condition. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 543-548. doi : 10.1016/j.crma.2007.10.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.023/

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

[2] F. Ammar-Khodja; A. Benabdallah; C. Dupaix; I. Kostine Null controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., Volume 11 (2005) no. 3, pp. 426-448

[3] E. Fernández-Cara; E. Zuazua The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, Volume 5 (2000) no. 4–6, pp. 465-514

[4] A. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University, Korea, 1996

[5] M. González-Burgos; R. Pérez-García Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptotic Anal., Volume 46 (2006), pp. 123-162

[6] M. González-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Systems Control Lett., submitted for publication

[7] E.L. Ince Ordinary Differential Equations, Dover Publications, New York, 1944

[8] R.E. Kalman; P.L. Falb; M.A. Arbib Topics in Mathematical Control Theory, McGraw-Hill Book Co., New York, 1969

[9] H. Leiva Controllability of a system of parabolic equations with non-diagonal diffusion matrix, IMA J. Math. Control Inform., Volume 22 (2005) no. 2, pp. 187-199

[10] D.L. Russell Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., Volume 20 (1978) no. 4, pp. 639-739

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

Cited by Sources:

Comments - Policy


Articles of potential interest

On the boundary controllability of non-scalar parabolic systems

Enrique Fernández-Cara; Manuel González-Burgos; Luz de Teresa

C. R. Math (2009)


A new Carleman inequality for parabolic systems with a single observation and applications

Assia Benabdallah; Michel Cristofol; Patricia Gaitan; ...

C. R. Math (2010)


Controllability of cascade coupled systems of multi-dimensional evolution PDEs by a reduced number of controls

Fatiha Alabau-Boussouira

C. R. Math (2012)