no. 1-2
Partial Differential Equations/Optimal Control
A new Carleman inequality for parabolic systems with a single observation and applications
[Une nouvelle inégalité de Carleman pour des systèmes paraboliques avec une seule observation et applications]

Présenté par : Gilles Lebeau

Benabdallah, Assia1 ; Cristofol, Michel1 ; Gaitan, Patricia1 ; de Teresa, Luz2
1 Laboratoire d'analyse topologie probabilités, CNRS UMR 6632, universités d'Aix-Marseille, 39, rue F. Joliot Curie 1, 13453 Marseille cedex 13, France
2 Instituto de Matemáticas, Universidad Nacional Autónoma de México
Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 25-29.

On établit des inégalités d'observabilité pour des systèmes linéaires de réaction–diffusion–convection d'ordre 2 et des systèmes linéaires de réaction–diffusion d'ordre 3, basées sur des estimations de Carleman. Elles permettent de démontrer des résultats de contrôlabilité aux trajectoires par une seule force localisée en espace ainsi que des résultats d'identification de n coefficients par (n−2) observations.

In this Note, we present Carleman estimates for linear reaction–diffusion–convection systems of two equations and linear reaction–diffusion systems of three equations. These estimates are the key for proving controllability results for semilinear reaction–diffusion–convection systems of order 2 and reaction–diffusion systems of order 3. They allow us to derive results for identification of n coefficients by (n−2) observations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.11.001
Benabdallah, Assia 1 ; Cristofol, Michel 1 ; Gaitan, Patricia 1 ; de Teresa, Luz 2

1 Laboratoire d'analyse topologie probabilités, CNRS UMR 6632, universités d'Aix-Marseille, 39, rue F. Joliot Curie 1, 13453 Marseille cedex 13, France
2 Instituto de Matemáticas, Universidad Nacional Autónoma de México 
@article{CRMATH_2010__348_1-2_25_0,
     author = {Benabdallah, Assia and Cristofol, Michel and Gaitan, Patricia and de Teresa, Luz},
     title = {A new {Carleman} inequality for parabolic systems with a single observation and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {25--29},
     publisher = {Elsevier},
     volume = {348},
     number = {1-2},
     year = {2010},
     doi = {10.1016/j.crma.2009.11.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.11.001/}
}
TY  - JOUR
AU  - Benabdallah, Assia
AU  - Cristofol, Michel
AU  - Gaitan, Patricia
AU  - de Teresa, Luz
TI  - A new Carleman inequality for parabolic systems with a single observation and applications
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 25
EP  - 29
VL  - 348
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.11.001/
DO  - 10.1016/j.crma.2009.11.001
LA  - en
ID  - CRMATH_2010__348_1-2_25_0
ER  - 
%0 Journal Article
%A Benabdallah, Assia
%A Cristofol, Michel
%A Gaitan, Patricia
%A de Teresa, Luz
%T A new Carleman inequality for parabolic systems with a single observation and applications
%J Comptes Rendus. Mathématique
%D 2010
%P 25-29
%V 348
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.11.001/
%R 10.1016/j.crma.2009.11.001
%G en
%F CRMATH_2010__348_1-2_25_0
Benabdallah, Assia; Cristofol, Michel; Gaitan, Patricia; de Teresa, Luz. A new Carleman inequality for parabolic systems with a single observation and applications. Comptes Rendus. Mathématique, Tome 348 (2010) no. 1-2, pp. 25-29. doi : 10.1016/j.crma.2009.11.001. http://www.numdam.org/articles/10.1016/j.crma.2009.11.001/

[1] 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

[2] Ammar-Khodja, F.; Benabdallah, A.; Dupaix, C.; Kostine, I. Null controllability of some systems of parabolic type by one control force, E.S.A.I.M.: Control, Optimization and Calculus of Variations, Volume 11 (2005), pp. 426-448

[3] Ammar Khodja, F.; Benabdallah, A.; Gonzàlez-Burgos, M.; Dupaix, C. Controllability for a class of reaction–diffusion systems: Generalized Kalman's condition, J. Evolution Equations, Volume 9 (2009) no. 2, p. 267

[4] Benabdallah, A.; Cristofol, M.; Gaitan, P.; Yamamoto, M. Inverse problem for a parabolic system with two components by measurements of one component, Appl. Anal., Volume 88 (2009) no. 5, pp. 583-710

[5] A. Benabdallah, M. Cristofol, P. Gaitan, L. de Teresa, New Carleman inequalities and controllability results for parabolic systems, in preparation

[6] Cristofol, M.; Gaitan, P.; Ramoul, H. Inverse problems for a 2×2 reaction–diffusion system using a Carleman estimate with one observation, Inverse Problems, Volume 22 (2006), pp. 1561-1573

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

[8] González-Burgos, M.; Pérez-García, R. Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., Volume 46 (2006) no. 2, pp. 123-162

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

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

Cité par Sources :