Comptes Rendus
no. 7
Spectral boundary controllability of networks of strings
[Contrôlabilité spectrale de réseaux de cordes vibrantes]
René Dáger1 ; Enrique Zuazua1
1 Departamento de Matemática, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 545-550.

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 T0=2∑i=1Mℓi, 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.

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.

Reçu le :
Publié le :
DOI : 10.1016/S1631-073X(02)02314-2
René Dáger 1 ; Enrique Zuazua 1

1 Departamento de Matemática, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain 
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René Dáger; Enrique Zuazua. Spectral boundary controllability of networks of strings. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 545-550. doi : 10.1016/S1631-073X(02)02314-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02314-2/

[1] S.A. Avdonin; W. Moran Simultaneous control problems for systems of elastic strings and beams, Systems Control Lett., Volume 44 (2001) no. 2, pp. 147-155

[2] S.A. Avdonin; M. Tucsnak Simultaneous controllability in sharp time for two elastic strings, ESAIM:COCV, Volume 6 (2001), pp. 259-273

[3] C. Baiocchi, V. Komornik, P. Loreti, Ingham–Beurling type theorems with weakened gap conditions, Acta Math. Hungar, to appear

[4] J. von Below, Parabolic network equations, Habilitation thesis, Eberhard-Karls-Universität, Tübingen, 1993

[5] R. Dáger; E. Zuazua Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001) no. 7, pp. 621-626

[6] R. Dáger; E. Zuazua Controllability of tree-shaped networks of strings, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001) no. 12, pp. 1087-1092

[7] R. Dáger; E. Zuazua Controllability of star-shaped networks of strings (A. Bermúdez et al., eds.), Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation, SIAM Proceedings, 2000, pp. 1006-1010

[8] A. Haraux; S. Jaffard Pointwise and spectral control of plate vibrations, Rev. Mat. Iberoamericana, Volume 7 (1991) no. 1, pp. 1-24

[9] J. Lagnese; G. Leugering; E.J.P.G. Schmit Modelling, Analysis and Control of Multi-Link Flexible Structures, Systems Control Found. Appl., Birhäuser, Basel, 1994

[10] J.-L. Lions Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Vol. 1, Masson, Paris, 1988

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

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