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Comptes Rendus Mathématique
Volume 343, n° 7
pages 493-498 (octobre 2006)
Doi : 10.1016/j.crma.2006.08.002
Received : 4 August 2006 ;  accepted : 14 August 2006
A mixed formulation and exact controllability approach for the computation of the periodic solutions of the scalar wave equation. (I): Controllability problem formulation and related iterative solution
Sur le calcul des solutions périodiques en temps de lʼéquation des ondes scalaire via formulation mixte et exacte contrôlabilité. (I) : Formulation et résolution itérative du problème de contrôle
 

Roland Glowinski a , Tuomo Rossi b
a Department of Mathematics, University of Houston, 4800, Calhoun, Houston, TX 77004, USA 
b Department of Mathematical Information Technology, University of Jyväskylä, PO Box 35, 40014 Jyväskylä, Finland 

Abstract

In this Note we discuss an exact controllability based method for the computation of the time-periodic solutions of a scalar wave equation with constant coefficients. We take advantage of an equivalent mixed formulation of the wave problem to derive a related controllability problem taking place in   (assuming that  ). Compared to previous work, where the controllability problem takes place in a subspace of  , we can compute the periodic solutions by solving the novel controllability problem by a conjugate gradient algorithm operating in  . The finite dimensional realization of the above algorithm does not require special preconditioning (as it is the case when the control space is contained in  , requiring then the solution of discrete elliptic problems to achieve preconditioning). The results of numerical experiments validating this novel approach will be presented in a further Note. To cite this article: R. Glowinski, T. Rossi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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Résumé

Dans cette Note, on étudie une méthode, basée sur la contrôlabilité exacte, pour le calcul des solutions périodiques en temps dʼune équation des ondes scalaire à coefficients constants. On y prend avantage dʼune formulation mixte équivalente du problème dʼondes pour se rammener à un problème de contrôlabilité posé dans   (on suppose que  ). Comparé à des travaux précédents, où le problème de contrôlabilité est posé dans un sous-espace de  , on peut calculer les solutions périodiques en résolvant le nouveau problème de contrôlabilité par un algorithme de gradient conjugué opérant dans  . Lʼ analogue discret de lʼalgorithme ci-dessus ne demande pas de préconditionnement sophistiqué (comme cʼest le cas quand lʼespace de contrôle est contenu dans  , exigeant alors la résolution de problèmes elliptiques discrets pour préconditionner). Les résultats dʼessais numériques validant la nouvelle approche feront lʼobjet dʼune note ultérieure. Pour citer cet article : R. Glowinski, T. Rossi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.


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