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

¹ 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.
² Institute of Mathematics, PO Box 68530, Federal University of Rio de Janeiro, UFRJ, 21945-970 Rio de Janeiro, RJ, Brasil. ademir@acd.ufrj.br.
³ Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. enrique.zuazua@uam.es.

Received: 30 November 2001
Revised: 12 March 2002

Abstract

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ε > 0 and study its asymptotic behavior for t large, as ε → 0. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ε. In order for this to be true the damping mechanism has to have the appropriate scale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam models for which the energy tends to zero exponentially as well. This is done both in the case of internal and boundary damping. We address the same problem for plates with internal damping.