Comptes Rendus
no. 15-16
Partial Differential Equations/Optimal Control
Numerical null controllability of a semi-linear heat equation via a least squares method
[Contrôlabilité exacte à zéro dʼune equation de la chaleur semi-linéaire par une méthode des moindres carrés]

Présenté par : Roland Glowinski

Enrique Fernández-Cara1 ; Arnaud Münch2
1 Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Laboratoire de mathématiques, université Blaise-Pascal (Clermont-Ferrand 2), UMR CNRS 6620, campus des Cézeaux, 63177 Aubière, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 867-871.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.014
Enrique Fernández-Cara 1 ; Arnaud Münch 2

1 Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Laboratoire de mathématiques, université Blaise-Pascal (Clermont-Ferrand 2), UMR CNRS 6620, campus des Cézeaux, 63177 Aubière, France 
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Enrique Fernández-Cara; Arnaud Münch. Numerical null controllability of a semi-linear heat equation via a least squares method. Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 867-871. doi : 10.1016/j.crma.2011.07.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.07.014/

[1] V. Barbu Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications, Volume 42 (2000) no. 1, pp. 73-89

[2] T. Cazenave; A. Haraux Introduction aux problèmes dʼévolutions semi-linéaires, Mathématiques et Applications, Ellipses, Paris, 1989

[3] J.-M. Coron; E. Trélat Global steady-state controllability of one dimensional semilinear heat equations, SIAM J. Control Optim., Volume 43 (2004) no. 2, pp. 549-569

[4] E. Fernandez-Cara Null controllability of the semi-linear heat equation, Esaim: COCV (1997) no. 2, pp. 87-103

[5] E. Fernandez-Cara, A. Münch, Numerical null controllability of a semi-linear 1-d heat equation, Preprint.

[6] E. Fernandez-Cara; E. Zuazua Null and approximate controllability for weakly blowing up semi-linear heat equation, Ann. Inst. Henri Poincaré Analyse non linéaire, Volume 17 (2000) no. 5, pp. 583-616

[7] A.V. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University, Korea, 1996 (pp. 1–163)

[8] I. Lasiecka; R. Triggiani Exact controllability of semilinear abstract systems with applications to waves and plates boundary control, Appl. Math. Optim., Volume 23 (1991), pp. 109-154

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