Equivalence between observability at the boundary and exponential stabilization for an ACL beam actuated by a voltage source without magnetic effects

In this paper, we consider a one-dimensional dissipative system of a Rayleigh beam coupled with two wave equations, modeling an active constrained layer beam which consists of a stiff layer, a viscoelastic layer and a piezoelectric layer actuated by a voltage source without magnetic effects. We prove the equivalence between observability at the boundary and exponential stability previously proved by Yang and Wang (J Math Anal Appl 448:1204-1227, 2017). This is achieved by using some ingenious calculus techniques and the multiplier method to establish two observable lemmas for the conservative system and the auxiliary system, respectively, so as to overcome the complexity of equations and the difficulties caused by the appearance of the coupling term.