[Contrôle optimal en temps échantillonné pour l'équation de la chaleur]
Nous décrivons ici, dans un premier temps, un problème de contrôle optimal en temps pour l'équation de la chaleur, avec contrôles d'échantillons. Nous l'utilisons ensuite pour approcher un problème de contrôle optimal en temps, avec contrôles distribués. L'étude d'un tel problème de contrôle optimal en temps échantillonné n'est pas facile, car il peut exister une infinité de contrôles optimaux. Nous montrons qu'il existe des liens entre ce problème, un problème de contrôle échantillonné de norme minimale et un problème de minimisation. Nous obtenons divers résultats sur ces problèmes, qui nous permettent, non seulement d'établir des estimations d'erreur pour le temps et les contrôles optimaux dans le passage entre le problème de contrôle optimal en temps échantillonné et le problème de contrôle optimal en temps distribué, en fonction de la période d'échantillonnage, mais aussi de montrer que ces estimations sont, dans un certain sens, optimales.
In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls.
The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.
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@article{CRMATH_2017__355_12_1252_0,
author = {Wang, Gengsheng and Yang, Donghui and Zhang, Yubiao},
title = {Time optimal sampled-data controls for the heat equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {1252--1290},
publisher = {Elsevier},
volume = {355},
number = {12},
year = {2017},
doi = {10.1016/j.crma.2017.11.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.11.006/}
}
TY - JOUR AU - Wang, Gengsheng AU - Yang, Donghui AU - Zhang, Yubiao TI - Time optimal sampled-data controls for the heat equation JO - Comptes Rendus. Mathématique PY - 2017 SP - 1252 EP - 1290 VL - 355 IS - 12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.11.006/ DO - 10.1016/j.crma.2017.11.006 LA - en ID - CRMATH_2017__355_12_1252_0 ER -
%0 Journal Article %A Wang, Gengsheng %A Yang, Donghui %A Zhang, Yubiao %T Time optimal sampled-data controls for the heat equation %J Comptes Rendus. Mathématique %D 2017 %P 1252-1290 %V 355 %N 12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.11.006/ %R 10.1016/j.crma.2017.11.006 %G en %F CRMATH_2017__355_12_1252_0
Wang, Gengsheng; Yang, Donghui; Zhang, Yubiao. Time optimal sampled-data controls for the heat equation. Comptes Rendus. Mathématique, Tome 355 (2017) no. 12, pp. 1252-1290. doi : 10.1016/j.crma.2017.11.006. http://www.numdam.org/articles/10.1016/j.crma.2017.11.006/
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