Time optimal sampled-data controls for the heat equation

Gengsheng Wang; Donghui Yang; Yubiao Zhang

doi:10.1016/j.crma.2017.11.006

Optimal control

Time optimal sampled-data controls for the heat equation
[Contrôle optimal en temps échantillonné pour l'équation de la chaleur]

Présenté par : Jean-Michel Coron

Gengsheng Wang ^1,² ; Donghui Yang ³ ; Yubiao Zhang ¹

¹ Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China
² School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China
³ School of Mathematical Sciences, Central South University, Changsha, 410075, China

Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1252-1290.

Résumé
Abstract

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.

Reçu le : 2017-02-01
Accepté le : 2017-11-09
Publié le : 2017-11-21

DOI : 10.1016/j.crma.2017.11.006

Affiliations des auteurs :

Gengsheng Wang ^{1, 2} ; Donghui Yang ³ ; Yubiao Zhang ¹

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     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},
}

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Gengsheng Wang; Donghui Yang; Yubiao Zhang. Time optimal sampled-data controls for the heat equation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 12, pp. 1252-1290. doi : 10.1016/j.crma.2017.11.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.006/

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