Optimal Control
An Ingham type proof for the boundary observability of a Nd wave equation
Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 63-68.

The boundary observability of the wave equation has been studied by many authors. A method of choice is to use the multiplier method (cf. Komornik (1994)). Recently, in Ramdani et al. (2005), a first Fourier based proof is given in the case where the domain is a square, thanks to a new Hautus type test. We give here a new self-contained proof with an Ingham type approach in the more general case where the domain is a product of intervals; this leads, in contrary to the proof in Ramdani et al., to explicit time and constants. However, we do not reach the optimal time which can be obtained for this problem by the multiplier method.

L'observabilité frontière de l'équation des ondes a été étudiée par de nombreux auteurs. Une méthode de choix est d'utiliser la méthode des multiplicateurs (cf. Komornik (1994)). Récemment, dans Ramdani et al. (2005), une première preuve basée sur les séries de Fourier a été donnée dans le cas où le domaine est un carré grâce à un test de type Hautus. On donne ici une nouvelle preuve auto-contenue par une approche de type Ingham, dans le cas plus général où le domaine est un produit d'intervalles ; on obtient alors un temps et des constantes explicites, contrairement à la preuve de Ramdani et al. (2005). Cependant, on n'atteint pas le temps optimal, qui peut être obtenu pour ce problème par la méthode des multiplicateurs.

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DOI: 10.1016/j.crma.2008.11.002
Mehrenberger, Michel 1

1 Institut de recherche mathématique avancée, Université Louis-Pasteur, UMR CNRS 7501, 7, rue René-Descartes, 67084 Strasbourg, France
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Mehrenberger, Michel. An Ingham type proof for the boundary observability of a $ N-d$ wave equation. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 63-68. doi : 10.1016/j.crma.2008.11.002. http://www.numdam.org/articles/10.1016/j.crma.2008.11.002/

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