Stabilization for partially dissipative laminated beams with non-constant coefficients

Abstract

We address the stability of solutions corresponding to weakly damped two-layered Timoshenko beams with space variable coefficients. The mathematical model is composed of a system of coupled wave equations and describes the slip effect produced by a thin adhesive layer joining the beams. For the considered PDE system, combined with different boundary conditions, the main result is that the decay rates for the energy depend on local equality of the wave propagation velocity. The latter is achieved by introducing new observability estimates for the present model. We also study the problem with constant coefficients. Here, two main results are obtained: (i) for mixed boundary condition type, we characterize the exponential decay rate by assuming the equality of the wave propagation velocity and (ii) without imposing this assumption, we establish optimal rational decay rates.