Spectral boundary controllability of networks of strings
[Contrôlabilité spectrale de réseaux de cordes vibrantes]
Comptes Rendus. Mathématique, Tome 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=1Mi, 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
Dáger, René 1 ; Zuazua, Enrique 1

1 Departamento de Matemática, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
@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/

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

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

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

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

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

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

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

Cité par Sources :

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