Weak stability of a laminated beam

In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.