Decay rates related to the stabilization of hybrid elastic systems

The nature of vibrations of a number of physically motivated mathematical models of hybrid elastic systems is considered. This is a system consisting of two types of materials: an elastic material occupying the interior of the system, and another material which may or may not be elastic, occupying all or part of the boundary of the system. Together, these two materials are subject to vibrations, each vibrating according to a different law, but interacting with each other. By applying feedback stabilization to the boundary material, a dissipative system is obtained, and with it an associated contraction semigroup. Pointwise convergence is achieved of the (energy) norm to zero as time approaches infinity, but at no specific rate. This seems to be a characteristic property of these systems. Another characteristic property is that by assuming some additional smoothness of the initial data, an inverse power decay is obtained as time approaches infinity