Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-03T20:22:35.038Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact controllability for a model of a multidimensional flexible structure

Published online by Cambridge University Press:  14 November 2011

Jean Pierre Puel  and
Enrique Zuazua
Jean Pierre Puel
Affiliation:
Departement de Mathématiques, Université d'Orléans; and Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Enrique Zuazua
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Synopsis

A simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 2 , 1993 , pp. 323 - 344
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bardos, C., Lebeau, G. and Rauch, J.. Contrôle et stabilisation dans les systèmes hyperboliques, Appendix II in J. L. Lions [11], pp. 492–538.Google Scholar
2Cazenave, T. and Haraux, A.. Introduction aux problémes d'éevolution semi-linéaires, Mathématiques & Applications 1 (Paris: Ellipses, 1990).Google Scholar
3Chen, G., Delfour, M. C., Krall, A. M. and Payre, G.. Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim. 25 (1987), 526–546.Google Scholar
4Ciarlet, Ph.. Méthodes asymptotiques pour les plaques et pour les jonctions dans les multi structures élastiques (Paris: Masson, RMA, 1991).Google Scholar
5Ciarlet, Ph., Le Dret, H. and Nzengwa, R.. Junctions between 3d and 2d linearly elastic structures. J. Math. Pures Appl. 68 (1989), 261–295; C. R. Acad. Sci. Paris Sér 1. Math. 305 (1987), 55–58.Google Scholar
6Geymonat, G.. Personal communication.Google Scholar
7Grisvard, P.. Contrô1abilité exacte des solutions de l'équation des ondes en présence de singularités. J. Math. Pures Appl. 68 (1989), 215–259.Google Scholar
8Haraux, A.. Une remarque sur la stabilisation de certains systèmes du deuxieme ordre en temps. Portugal Math 46 (1989), 245–258.Google Scholar
9Komornik, V. and Zuazua, E.. A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990), 33–54.Google Scholar
10Le Dret, H.. Jonctions dans les multi structures (Paris: Masson, RMA, 1991).Google Scholar
11Lions, J. L.. Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Controlabilité exacte (Paris: Masson, RMA8, 1988).Google Scholar
12Lions, J. L.. Contrôle de structures souples multidimensionnelles: Remarques introductives. Matapli 24 (1990), 27–42.Google Scholar
13Lions, J. L.. Contrôle des systèmes distribués singuliers (Paris: Gauthier-Villars, 1983).Google Scholar
14Nicaise, S.. Exact controllabilitée of a pluridimensional coupled problem. Rev. Mat. Univ. Complutense Madrid (to appear).Google Scholar
15Puel, J. P. and Zuazua, E.. Contrôlabilité exacte et stabilisation d'un modèle de structure vibrante multidimensionnelle. C.R. Acad. Sci. Paris Sér. 1 Math. 314 (1992), 121–125.Google Scholar
16Puel, J. P. and Zuazua, E.. Exact Controllability for Some Models of Multidimensional Vibrating Structures. In Mathematics, Climate and Environment, eds Diaz, J. I. and Lions, J. L. (Paris: Masson, to appear).Google Scholar
17Schmidt, E. J. P. G.. On the modelling and exact controllability of networks of vibrating strings. SIAM J. Control Optim. 30 (1992), 229–245.CrossRefGoogle Scholar
18Ralston, J.. Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823.Google Scholar