Exponential and polynomial stabilization of laminated beams with two history memories

In this paper, a laminated beam system with two history-type controls is studied. One of the controls acts on the effective rotation angle and the other on the slip equation. The latter control replaces the structural damping usually considered in this model in the literature. Using the semigroup of linear operators approach, we prove the system is globally well-posed. We establish the exponential stability of the system provided the equal-speed waves propagation holds. Otherwise, the system lacks exponential stability. The polynomial decay with rate \begin{document}$ t^{-\frac{1}{4}} $\end{document} is also proved using the theorem due to Borichev and Tomilov.