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

Roland Glowinski; Tuomo Rossi

doi:10.1016/j.crma.2006.08.002

Numerical Analysis/Partial Differential Equations

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 l'équation des ondes scalaire via formulation mixte et exacte contrôlabilité. (I) : Formulation et résolution itérative du problème de contrôle]

Présenté par : Philippe G. Ciarlet

Roland Glowinski ¹ ; Tuomo Rossi ²

¹ Department of Mathematics, University of Houston, 4800, Calhoun, Houston, TX 77004, USA
² Department of Mathematical Information Technology, University of Jyväskylä, PO Box 35, 40014 Jyväskylä, Finland

Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 493-498.

Résumé
Abstract

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 $(L^{2} {(Ω))}^{d + 1}$ (on suppose que $Ω \subset R^{d}$ ). Comparé à des travaux précédents, où le problème de contrôlabilité est posé dans un sous-espace de $H^{1} (Ω) \times L^{2} (Ω)$ , on peut calculer les solutions périodiques en résolvant le nouveau problème de contrôlabilité par un algorithme de gradient conjugué opérant dans $(L^{2} {(Ω))}^{d + 1}$ . 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 $H^{1} (Ω) \times L^{2} (Ω)$ , 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.

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 $(L^{2} {(Ω))}^{d + 1}$ (assuming that $Ω \subset R^{d}$ ). Compared to previous work, where the controllability problem takes place in a subspace of $H^{1} (Ω) \times L^{2} (Ω)$ , we can compute the periodic solutions by solving the novel controllability problem by a conjugate gradient algorithm operating in $(L^{2} {(Ω))}^{d + 1}$ . 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 $H^{1} (Ω) \times L^{2} (Ω)$ , 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.

Reçu le : 2006-08-04
Accepté le : 2006-08-14
Publié le : 2006-09-25

DOI : 10.1016/j.crma.2006.08.002

Affiliations des auteurs :

Roland Glowinski ¹ ; Tuomo Rossi ²

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     author = {Roland Glowinski and Tuomo Rossi},
     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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {493--498},
     publisher = {Elsevier},
     volume = {343},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.08.002},
     language = {en},
}

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Roland Glowinski; Tuomo Rossi. 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. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 493-498. doi : 10.1016/j.crma.2006.08.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.08.002/

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