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- W2056908288 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 (L2(Ω))d+1 (assuming that Ω⊂Rd). Compared to previous work, where the controllability problem takes place in a subspace of H1(Ω)×L2(Ω), we can compute the periodic solutions by solving the novel controllability problem by a conjugate gradient algorithm operating in (L2(Ω))d+1. 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 H1(Ω)×L2(Ω), 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). Dans cette Note, on étudie une méthode, basée sur la contrôlabilité exacte, pour le calcul des solutions périodiques en temps d'une équation des ondes scalaire à coefficients constants. On y prend avantage d'une formulation mixte équivalente du problème d'ondes pour se rammener à un problème de contrôlabilité posé dans (L2(Ω))d+1 (on suppose que Ω⊂Rd). Comparé à des travaux précédents, où le problème de contrôlabilité est posé dans un sous-espace de H1(Ω)×L2(Ω), on peut calculer les solutions périodiques en résolvant le nouveau problème de contrôlabilité par un algorithme de gradient conjugué opérant dans (L2(Ω))d+1. L' analogue discret de l'algorithme ci-dessus ne demande pas de préconditionnement sophistiqué (comme c'est le cas quand l'espace de contrôle est contenu dans H1(Ω)×L2(Ω), exigeant alors la résolution de problèmes elliptiques discrets pour préconditionner). Les résultats d'essais numériques validant la nouvelle approche feront l'objet d'une note ultérieure. Pour citer cet article : R. Glowinski, T. Rossi, C. R. Acad. Sci. Paris, Ser. I 343 (2006)." @default.
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- W2056908288 date "2006-10-01" @default.
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- W2056908288 title "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" @default.
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- W2056908288 doi "https://doi.org/10.1016/j.crma.2006.08.002" @default.
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