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On the analysis and control of hyperbolic systems associated with vibrating networks

Published online by Cambridge University Press:  14 November 2011

J. E. Lagnese ,
G. Leugering  and
E. J. P. G. Schmidt
J. E. Lagnese
Affiliation:
Department of Mathematics, Georgetown University, Washington DC 20057, U.S.A.
G. Leugering
Affiliation:
Department of Mathematics, Georgetown University, Washington DC 20057, U.S.A.
E. J. P. G. Schmidt
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A2K6
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Abstract

In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 77 - 104
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Ali Mehmeti, F. and Nicaise, S.. Nonlinear interaction problems. J. Nonlinear Anal, (to appear).Google Scholar
2Bresse, J. A. C.. Cours de Mechanique Applique (Paris: Mallet Bachelier, 1859).Google Scholar
3Garabedian, P. R.. Partial Differential Equations (New York: John Wiley and Sons, 1964).Google Scholar
4Lagnese, J. E., Leugering, G. and Schmidt, E. J. P. G.. Control of planar networks of Timoshenko beams. SIAM J. Control Optim. (to appear).Google Scholar
5Lagnese, J. E., Leugering, G. and Schmidt, E. J. P. G.. Modelling of dynamic networks of thin thermoelastic beams. Math. Meth. Applied Sci. (to appear).Google Scholar
6Lions, J. L.. Contrôlabilitée Exacte, Perturbations et Stabilisation de Systemes Distribués, Tome 1, Contrôlabilité Exacte, Collection RMA 8 (Paris: Masson, 1988).Google Scholar
7Lumer, G.. Espaces ramifiés et diffusions sur les réseaux topologiques. C. R. Acad. Sci. Paris Ser. I 303 (1986), 443–446.Google Scholar
8Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
9Rauch, J. and Taylor, M.. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974), 79–86.CrossRefGoogle Scholar
10Russell, D. L.. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978), 639–739.CrossRefGoogle Scholar
11Schmidt, 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
12Schmidt, E. J. P. G.. On an energy estimate and exact boundary controllability for hyperbolic systems in one space variable (Report #91-12, Department of Mathematics and Statistics, McGill University, 1991).Google Scholar
13Schmidt, E. J. P. G. and , Wei Ming. On the modelling and analysis of networks of vibrating strings and masses. (Report #91-13, Department of Mathematics and Statistics, McGill University, 1991).Google Scholar
14von Below, J.. Classical solvability of linear parabolic equations on networks. J. Differential Equations 72(1988), 316–337.CrossRefGoogle Scholar