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Comptes Rendus Mathématique
Volume 334, n° 7
pages 545-550 (2002)
Doi : S1631-073X(02)02314-2
Received : 4 February 2002 ; 
Spectral boundary controllability of networks of strings
Contrôlabilité spectrale de réseaux de cordes vibrantes

René Dáger , Enrique Zuazua
Departamento de Matemática, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain 

Note presented by Philippe G. Ciarlet


In this Note we give a necessary and sufficient condition for the spectral controllability from one simple node of a general network of strings that undergoes transversal vibrations in a sufficiently large time. This condition asserts that no eigenfunction vanishes identically on the string that contains the controlled node. The proof combines the Beurling-Malliavin's theorem and an asymptotic formula for the eigenvalues of the network. The optimal control time may be characterized as twice the sum of the lengths of all the strings of the network. To cite this article: R. Dáger, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 545-550.

The full text of this article is available in PDF format.

On considère un réseau général de cordes vibrantes et on étudie le problème du contrôle spectral moyennant des contrôles agissant sur une extrémité libre du réseau. Moyennant une généralisation des théorèmes de Beurling-Malliavin et à l'aide d'une formule asymptotique des valeurs propres du réseau, on donne une condition nécessaire et suffisante pour la contrôlabilité approchée et spectrale au temps T 0 =2∑i =1 M i , où les ℓi sont les longueurs des cordes du réseau. Cette condition exige qu'aucune fonction propre ne s'annule identiquement le long de la corde où le contrôle agit. Pour citer cet article : R. Dáger, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 545-550.

The full text of this article is available in PDF format.

 This work has been partially supported by grants PB96-0663 of the DGES (Spain) and the EU TMR project “Homogenization and Multiple Scales”.

© 2002  Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All Rights Reserved.
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