[Contrôlabilité exacte à zéro dʼune equation de la chaleur semi-linéaire par une méthode des moindres carrés]
Cette Note concerne la détermination effective de contrôles à zéro pour une équation de la chaleur semi-linéaire, dans le cas légèrement surlinéaire. Sous des conditions de croissances optimales, lʼexistence de contrôles a été obtenue dans [E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, Ann. Inst. Henri Poincaré Analyse non linéaire 17 (5) (2000) 583] par un argument de point fixe ; voir aussi [V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications 42 (1) (2000) 73]. Précisément, des inégalités de Carleman et le théorème de Kakutani impliquent lʼexistence de points fixes pour un opérateur de contrôle linéarisé associé. En pratique, la difficulté est dʼextraire des itérés de Picard une sous-suite convergente. Cette note propose et analyse une reformulation du problème par une approche de type moindres carrés : on montre que celle-ci garantit une construction explicite de points fixes.
This Note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, Ann. Inst. Henri Poincaré Analyse non linéaire 17 (5) (2000) 583] via a fixed point reformulation; see also [V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications 42 (1) (2000) 73]. More precisely, Carleman estimates and Kakutaniʼs theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points.
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@article{CRMATH_2011__349_15-16_867_0,
author = {Fern\'andez-Cara, Enrique and M\"unch, Arnaud},
title = {Numerical null controllability of a semi-linear heat equation via a least squares method},
journal = {Comptes Rendus. Math\'ematique},
pages = {867--871},
publisher = {Elsevier},
volume = {349},
number = {15-16},
year = {2011},
doi = {10.1016/j.crma.2011.07.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.07.014/}
}
TY - JOUR AU - Fernández-Cara, Enrique AU - Münch, Arnaud TI - Numerical null controllability of a semi-linear heat equation via a least squares method JO - Comptes Rendus. Mathématique PY - 2011 SP - 867 EP - 871 VL - 349 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.07.014/ DO - 10.1016/j.crma.2011.07.014 LA - en ID - CRMATH_2011__349_15-16_867_0 ER -
%0 Journal Article %A Fernández-Cara, Enrique %A Münch, Arnaud %T Numerical null controllability of a semi-linear heat equation via a least squares method %J Comptes Rendus. Mathématique %D 2011 %P 867-871 %V 349 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.07.014/ %R 10.1016/j.crma.2011.07.014 %G en %F CRMATH_2011__349_15-16_867_0
Fernández-Cara, Enrique; Münch, Arnaud. Numerical null controllability of a semi-linear heat equation via a least squares method. Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 867-871. doi : 10.1016/j.crma.2011.07.014. http://www.numdam.org/articles/10.1016/j.crma.2011.07.014/
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