Asymptotic stabilization of vibrating composite plates

Abstract

This paper presents a solution to the boundary stabilization problem of free vibration of composite plates. It is assumed that an object such as a flange which has mass and mass moment of inertia can be attached to the free portion of the boundary of the plate. Both cases of with and without boundary object have been analyzed for other types of plates by many researches. In the presence of boundary mass and mass moment of inertia, two ordinary differential equations govern their motion. For both cases, the plate dynamics is presented by a linear fourth order partial differential equation (PDE). Linear boundary control laws are constructed to stabilize the composite plates asymptotically. The control forces and moments consist of feedbacks of the velocity and normal derivative of the velocity at the boundaries of the plate. Asymptotic stabilization of free vibration of composite plates in both cases (i.e. with and without boundary object) is achieved via boundary actions. Finally, it is shown that exponentially stability of the hybrid system cannot be fulfilled (in the other words the asymptotical stability is the best). Our main tools in this article are semigroup techniques, spectral theory of the linear operators and Lyapunov stability method.