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Stabilization of Berger–Timoshenko's equation as limitof the uniform stabilization of the von Kármán systemof beams and plates

Published online by Cambridge University Press:  15 September 2002

G. Perla Menzala ,
Ademir F. Pazoto  and
Enrique Zuazua
G. Perla Menzala
Affiliation:
National Laboratory of Scientific Computation, LNCC/MCT, Rua Getúlio Vargas 333, Quitandinha, Petrópolis, RJ, CEP 25651-070, RJ, Brasil and Institute of Mathematics, UFRJ, P.O. Box 68530, Rio de Janeiro, RJ, Brasil. perla@lncc.br.
Ademir F. Pazoto
Affiliation:
Institute of Mathematics, PO Box 68530, Federal University of Rio de Janeiro, UFRJ, 21945-970 Rio de Janeiro, RJ, Brasil. ademir@acd.ufrj.br.
Enrique Zuazua
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. enrique.zuazua@uam.es.
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Abstract

We consider a dynamical one-dimensional nonlinear von Kármán model for beamsdepending on a parameter ε > 0 and studyits asymptotic behavior for t large, as ε → 0. Introducing appropriate dampingmechanisms we show that the energy of solutionsof the corresponding damped models decayexponentially uniformly with respect to theparameter ε. In order for this to be true thedamping mechanism has to have the appropriatescale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam modelsfor which the energy tends to zero exponentiallyas well. This is done both in the case ofinternal and boundary damping. We address the sameproblem for plates with internal damping.

Keywords

Uniform stabilizationsingular limitvon Kármán systembeamsplates.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 657 - 691
Copyright
© EDP Sciences, SMAI, 2002

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