In this paper, we study the optimal time problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter. To begin with, we build an exact solution. The dependence of this solution as regards the scaling parameter naturally opens the way to study the existence and uniqueness of an optimal time control. If, moreover, one assumes the null controllability, it enables to establish a bang-bang type property.
Mots clés : Optimal time control problem, null controllability, bang-bang property, heat equation, Scaling parameter
@article{RO_2017__51_4_1289_0,
author = {Benalia, Karim and David, Claire and Oukacha, Brahim},
title = {An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {1289--1299},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/ro/2017006},
mrnumber = {3783945},
zbl = {1393.35070},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ro/2017006/}
}
TY - JOUR AU - Benalia, Karim AU - David, Claire AU - Oukacha, Brahim TI - An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2017 SP - 1289 EP - 1299 VL - 51 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2017006/ DO - 10.1051/ro/2017006 LA - en ID - RO_2017__51_4_1289_0 ER -
%0 Journal Article %A Benalia, Karim %A David, Claire %A Oukacha, Brahim %T An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter %J RAIRO - Operations Research - Recherche Opérationnelle %D 2017 %P 1289-1299 %V 51 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2017006/ %R 10.1051/ro/2017006 %G en %F RO_2017__51_4_1289_0
Benalia, Karim; David, Claire; Oukacha, Brahim. An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 4, pp. 1289-1299. doi : 10.1051/ro/2017006. http://archive.numdam.org/articles/10.1051/ro/2017006/
, and , On the “bang-bang” control problem. Quart. Appl. Math. 14 (1956) 11–18. | DOI | MR | Zbl
, Time-optimal control of solutions of operational differenital equations. J. Soc. Ind. Appl. Math. Ser. A Control 2 (1964) 54–59. | DOI | MR | Zbl
, A remark on the “bang-bang” principle for linear control systems in infinite dimensional spaces. SIAM J. Control 6 (1968) 109–113. | DOI | MR | Zbl
H.O. Fattorini and H.O. Fattorini. Infinite dimensional linear control systems. The time optimal and norm optimal problems. Vol. 201 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam (2005). | MR | Zbl
J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Avant propos de P. Lelong. Dunod, Paris (1968). | MR | Zbl
Tucsnak and Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2009). | MR | Zbl
and , Maximum principle and bang-bang property of time optimal controls for Schrödinger type systems. SIAM J. Control Optim. 51 (2013) 4016–4038. | DOI | MR | Zbl
, and , Time optimal boundary controls for the heat equation. J. Function. Anal. 263 (2012) 25–49. | DOI | MR | Zbl
, null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47 (2008) 1701–1720. | DOI | MR | Zbl
J.-Y. Chemin and Cl. David, Sur la construction de grandes solutions pour des équations de Schrödinger de type “masse critique”, Séminaire Laurent Schwartz – EDP et applications (2013).
and , From an initial data to a global solution of the nonlinear Schrödinger equation: a building process. Int. Math. Res. Not. 2016 (2016) 2376–2396. | DOI | MR | Zbl
and , High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 112 (1999) 131–175. | DOI | MR | Zbl
J.-P. Puel, Contrôle et équations aux dérivées partielles. Journées mathématiques X-UPS (1999) 169–188. | MR | Zbl
Unicité du prolongement des solutions pour quelques opérateurs paraboliques. Mem. Coll. Sci. Univ. Kyoto, Ser. A31 (1958) 219–239. | Zbl
and , Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl
J.-M. Coron, Quelques résultats sur la commandabilité et la stabilisation des systèmes non linéaires. Journées mathématiques X-UPS (1999) 123–168. | MR | Zbl
, Uber Systeme von Linearen Partiellen Differentialgleichungen Erster Ordnung. Math. Ann. 117 (1939) 98–105. | DOI | MR | Zbl
, Qishu Yan Bang-bang property of time optimal controls for some semilinear heat equation. J. Optim. Theory Appl. 165 (2015) 263-278.
E. Trélat. Contrôle optimal. Mathématiques Concrètes (Concrete Mathematics). Théorie & applications (Theory and applications). Vuibert, Paris (2005). | MR | Zbl
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