[Contrôlabilité frontière à zéro des équations linéaires de type diffusion–réaction]
Il s'agit de la contrôlabilité frontière à zéro des équations linéaires de type diffusion–réaction dans un domaine borné de . 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.
Accepté le :
Publié le :
@article{CRMATH_2010__348_19-20_1083_0,
author = {Hamdi, Adel and Mahfoudhi, Imed},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.09.019/}
}
TY - JOUR AU - Hamdi, Adel AU - Mahfoudhi, Imed 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2010.09.019/ DO - 10.1016/j.crma.2010.09.019 LA - en ID - CRMATH_2010__348_19-20_1083_0 ER -
%0 Journal Article %A Hamdi, Adel %A Mahfoudhi, Imed %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 %U http://www.numdam.org/articles/10.1016/j.crma.2010.09.019/ %R 10.1016/j.crma.2010.09.019 %G en %F CRMATH_2010__348_19-20_1083_0
Hamdi, Adel; Mahfoudhi, Imed. Boundary null-controllability of linear diffusion–reaction equations. Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1083-1086. doi : 10.1016/j.crma.2010.09.019. http://www.numdam.org/articles/10.1016/j.crma.2010.09.019/
[1] Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Research Institute of Math., Global Anal. Research Center, Seoul National University, 1996
[2] 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] 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] 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] Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal., Volume 141 (1998), pp. 297-329
[6] Contrôlabilité Exacte Pertubations et Stabilisation de Systèmes Distribués, Tome 1: Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, vol. 8, Masson, Paris, 1988
[7] Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., Volume 30 (1988) no. 1, pp. 1-68
[8] Contrôlabilité frontière des équations linéaires de type diffusion–réaction, Mémoire de recherche, LMI-INSA de Rouen, France, 2010
[9] 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] Partial Differential Equations, Springer, 1991
[12] Basic Linear Partial Differential Equations, Academic Press, 1975
Cité par Sources :
