[Contrôlabilité spectrale de réseaux de cordes vibrantes]
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.
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@article{CRMATH_2002__334_7_545_0,
author = {D\'ager, Ren\'e and Zuazua, Enrique},
title = {Spectral boundary controllability of networks of strings},
journal = {Comptes Rendus. Math\'ematique},
pages = {545--550},
publisher = {Elsevier},
volume = {334},
number = {7},
year = {2002},
doi = {10.1016/S1631-073X(02)02314-2},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02314-2/}
}
TY - JOUR AU - Dáger, René AU - Zuazua, Enrique TI - Spectral boundary controllability of networks of strings JO - Comptes Rendus. Mathématique PY - 2002 SP - 545 EP - 550 VL - 334 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02314-2/ DO - 10.1016/S1631-073X(02)02314-2 LA - en ID - CRMATH_2002__334_7_545_0 ER -
%0 Journal Article %A Dáger, René %A Zuazua, Enrique %T Spectral boundary controllability of networks of strings %J Comptes Rendus. Mathématique %D 2002 %P 545-550 %V 334 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02314-2/ %R 10.1016/S1631-073X(02)02314-2 %G en %F CRMATH_2002__334_7_545_0
Dáger, René; Zuazua, Enrique. Spectral boundary controllability of networks of strings. Comptes Rendus. Mathématique, Tome 334 (2002) no. 7, pp. 545-550. doi : 10.1016/S1631-073X(02)02314-2. http://www.numdam.org/articles/10.1016/S1631-073X(02)02314-2/
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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”.
