Comptes Rendus
Partial Differential Equations/Calculus of Variations
On the exact controllability of a system of mixed order with essential spectrum
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 629-634.

We consider the exact boundary controllability problem for the system y+Acy=0 where Ac is a self-adjoint operator in H whose (non empty) essential spectrum is {0}, which implies the non-exact controllability. We construct the HUM control for all initial condition in the subspace Hc×V˜0 of H×V. Each subspace Hc and V˜0 is spanned by the eigenvectors associated to the eigenvalues of Ac satisfying the separation condition (Σ), and this property seems general. We also study the behavior of the control when c goes to zero.

On s'intéresse à la contrôlabilité exacte frontière d'un système du type y+Acy=0Ac est un opérateur dans l'espace H dont le spectre essentiel (non vide) est {0}, ce qui implique la non contrôlabilité exacte. On construit le contrôle HUM pour toute donnée initiale dans le sous-espace Hc×V˜0 de H×V. Les sous-espaces Hc et V˜0 sont engendrés par les vecteurs propres associés aux valeurs propres de Ac satisfaisant la condition de séparation (Σ), propriété qui semble générale. Nous étudions également le comportement du contrôle lorsque c tend vers zéro.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.009
Farid Ammar-Khodja 1; Giuseppe Geymonat 2; Arnaud Münch 1

1 Laboratoire de mathématiques, Université de Franche-Comté, UMR CNRS 6623, 25030 Besançon, France
2 Laboratoire de mécanique et de génie civil, Université de Montpellier II, UMR CNRS 5508, 34095 Montpellier, France
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     title = {On the exact controllability of a system of mixed order with essential spectrum},
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Farid Ammar-Khodja; Giuseppe Geymonat; Arnaud Münch. On the exact controllability of a system of mixed order with essential spectrum. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 629-634. doi : 10.1016/j.crma.2008.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.009/

[1] F. Ammar-Khodja, G. Geymonat, A. Münch, in preparation

[2] G. Geymonat; P. Loreti; V. Valente Exact controllability of thin elastic hemispherical shell via harmonic analysis, Boundary Value Problems for Partial Differential Equations and Applications, Masson, 1993

[3] G. Geymonat; P. Loreti; V. Valente Spectral problems for thin shells and exact controllability, Spectral Analysis of Complex Structures, Travaux en cours, vol. 49, Hermann, Paris, 1995, pp. 35-57

[4] G. Geymonat; V. Valente A noncontrollability result for systems of mixed order, SIAM J. Control Optim, Volume 39 (2000) no. 3, pp. 661-672

[5] G. Grubb; G. Geymonat The essential spectrum of elliptic boundary value problem, Math. Ann., Volume 227 (1977), pp. 247-276

[6] V. Komornik; P. Loreti Fourier Series in Control Theory, Springer Monographs in Mathematics, 2004

[7] J.-L. Lions Contrôlabilité exacte, perturbations et stabilisations de systèmes distribués, 2 tomes, Masson, Paris, 1988

[8] J. Sanchez Hubert; E. Sanchez-Palencia Coques élastiques minces : propriétés asymptotiques, Masson, Paris, 1997

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