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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 


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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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.

© 2006  Académie des sciences@@#104156@@
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