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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2006-08-04
Accepted: 2006-08-14
Published online: 2006-09-25

DOI: 10.1016/j.crma.2006.08.002

Author's affiliations:

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

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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},
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     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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