Uniform boundary stabilization of the dynamical von Kármán and Timoshenko equations for plates

Abstract

The full nonlinear dynamic von Kárm´n system depending on a small parameter ε > 0 is considered. We study the asymptotic behaviour of the total energy associated with the model for large t and ε → 0. Introducing appropriate boundary feedback, we show that the total energy of a solution of the corresponding damped model decays exponentially as t → +∞, uniformly with respect to the parameter ε > 0. As ε → 0, we obtain a damped plate model for which the energy also tends to zero exponentially. The limit system can be viewed as new variant of the so-called Timoshenko model. It consists of a second-order hyperbolic equation for transversal vibrations of the plate coupled with a first-order ordinary differential equation whose solution appears as coefficient of the plate model and takes into account (when ε → 0) the contribution of the tangential components.