In this paper we study numerically the cost of the null controllability of a linear control parabolic 1-D equation as the diffusion coefficient tends to 0. For this linear control parabolic 1-D equation, we know from a prior work by J.-M. Coron and S. Guerrero (2005), that, when the diffusion coefficient tends to 0, for a small controllability time, the norm of the optimal control tends to infinity and that, if the controllability time is large enough, this norm tends to 0. For controllability times which are not covered by this work, we estimate numerically the norm of the optimal control as the diffusion coefficient tends to 0.
Dans cette Note nous étudions de manière numérique le coût de la contrôlabilité à zéro d'une équation de convection diffusion unidimensionnelle quand le coefficient de diffusion tend vers 0. Nous savons, d'après un travail antérieur de J.-M. Coron et S. Guerrero (2005), que pour un temps de contrôlabilité trop petit la norme du contrôle optimal tend vers l'infini quand le terme de diffusion tend vers zéro et que, par contre, cette norme tend vers 0 si le temps de contrôlabilité est assez grand. Pour des valeurs du temps de contrôle qui ne sont pas couvertes par ce travail, nous étudions numériquement la norme du contrôle optimal quand le coefficient de diffusion tend vers 0.
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@article{CRMATH_2009__347_15-16_927_0, author = {Salem, Ali}, title = {A numerical study of the null boundary controllability of a convection diffusion equation}, journal = {Comptes Rendus. Math\'ematique}, pages = {927--932}, publisher = {Elsevier}, volume = {347}, number = {15-16}, year = {2009}, doi = {10.1016/j.crma.2009.05.002}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.05.002/} }
TY - JOUR AU - Salem, Ali TI - A numerical study of the null boundary controllability of a convection diffusion equation JO - Comptes Rendus. Mathématique PY - 2009 SP - 927 EP - 932 VL - 347 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.05.002/ DO - 10.1016/j.crma.2009.05.002 LA - en ID - CRMATH_2009__347_15-16_927_0 ER -
%0 Journal Article %A Salem, Ali %T A numerical study of the null boundary controllability of a convection diffusion equation %J Comptes Rendus. Mathématique %D 2009 %P 927-932 %V 347 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.05.002/ %R 10.1016/j.crma.2009.05.002 %G en %F CRMATH_2009__347_15-16_927_0
Salem, Ali. A numerical study of the null boundary controllability of a convection diffusion equation. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 927-932. doi : 10.1016/j.crma.2009.05.002. http://www.numdam.org/articles/10.1016/j.crma.2009.05.002/
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