Comptes Rendus
Partial Differential Equations/Optimal Control
Boundary null-controllability of linear diffusion–reaction equations
[Contrôlabilité frontière à zéro des équations linéaires de type diffusion–réaction]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1083-1086.

Il s'agit de la contrôlabilité frontière à zéro des équations linéaires de type diffusion–réaction dans un domaine borné de R2. Nous transformons la détermination du contrôle de type HUM en la minimisation d'une fonctionnelle continue et strictement convexe. Dans le cas d'un domaine rectangulaire où le tenseur de diffusion est représenté par une matrice diagonale, nous exprimons explicitement le contrôle recherché dans une base orthonormée construite par les fonctions propres d'un problème de Sturm–Liouville.

This Note deals with the boundary null-controllability of linear diffusion–reaction equations in a 2D bounded domain. We transform the determination of the sought HUM boundary control into the minimization of a continuous and strictly convex functional. In the case of a rectangular domain where the diffusion tensor is represented by a diagonal matrix, we establish a procedure based on the inner product method that uses a complete orthonormal family of Sturm–Liouville's eigenfunctions to express explicitly the sought control.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.09.019

Adel Hamdi 1 ; Imed Mahfoudhi 1

1 Laboratoire de mathématiques LMI, INSA de Rouen, avenue de l'université, BP 8, 76801 St-Etienne-du-Rouvray cedex, France
@article{CRMATH_2010__348_19-20_1083_0,
     author = {Adel Hamdi and Imed Mahfoudhi},
     title = {Boundary null-controllability of linear diffusion{\textendash}reaction equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1083--1086},
     publisher = {Elsevier},
     volume = {348},
     number = {19-20},
     year = {2010},
     doi = {10.1016/j.crma.2010.09.019},
     language = {en},
}
TY  - JOUR
AU  - Adel Hamdi
AU  - Imed Mahfoudhi
TI  - Boundary null-controllability of linear diffusion–reaction equations
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1083
EP  - 1086
VL  - 348
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2010.09.019
LA  - en
ID  - CRMATH_2010__348_19-20_1083_0
ER  - 
%0 Journal Article
%A Adel Hamdi
%A Imed Mahfoudhi
%T Boundary null-controllability of linear diffusion–reaction equations
%J Comptes Rendus. Mathématique
%D 2010
%P 1083-1086
%V 348
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2010.09.019
%G en
%F CRMATH_2010__348_19-20_1083_0
Adel Hamdi; Imed Mahfoudhi. Boundary null-controllability of linear diffusion–reaction equations. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1083-1086. doi : 10.1016/j.crma.2010.09.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.019/

[1] A. Fursikov; O. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Research Institute of Math., Global Anal. Research Center, Seoul National University, 1996

[2] R. Glowinski; C.H. Li; J.L. Lions A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods, Japan J. Appl. Math., Volume 7 (1990), pp. 1-76

[3] A. Hamdi The recovery of a time-dependent point source in a linear transport equation: application to surface water pollution, Inverse Problems, Volume 25 (2009) no. 7, pp. 75006-75023

[4] A. Hamdi Identification of point sources in two-dimensional advection–diffusion–reaction equation: application to pollution sources in a river: Stationary case, Inverse Probl. Sci. Eng., Volume 15 (2007) no. 8, pp. 855-870

[5] G. Lebeau; E. Zuazua Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal., Volume 141 (1998), pp. 297-329

[6] J.L. Lions Contrôlabilité Exacte Pertubations et Stabilisation de Systèmes Distribués, Tome 1: Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, vol. 8, Masson, Paris, 1988

[7] J.L. Lions Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., Volume 30 (1988) no. 1, pp. 1-68

[8] I. Mahfoudhi Contrôlabilité frontière des équations linéaires de type diffusion–réaction, Mémoire de recherche, LMI-INSA de Rouen, France, 2010

[9] M. Negreanu; E. Zuazua Uniform boundary controllability of a discrete 1-D wave equation, Systems Control Lett., Volume 48 (2003) no. 3–4, pp. 261-280

[10] J.M. Rasmussen, Boundary control of linear evolution PDEs—continuous and discrete, PhD Thesis, Technical University of Denmark, 2004.

[11] J. Rauch Partial Differential Equations, Springer, 1991

[12] F. Trèves Basic Linear Partial Differential Equations, Academic Press, 1975

Cité par Sources :

Commentaires - Politique