  A three-layer active constrained layer (ACL) beam model, consisting of a
piezoelectric elastic layer, a stiff layer, and a constrained viscoelastic
layer, is obtained for cantilevered boundary conditions by using the reduced
Rao-Nakra sandwich beam assumptions through a consistent variational approach.
The Rao-Nakra sandwich beam assumptions keeps the longitudinal and rotational
inertia terms. We consider electrostatic, quasi-static and fully dynamic
assumptions due to Maxwell's equations. For that reason, we first include all
magnetic effects for the piezoelectric layer. Two PDE models are obtained; one
for the charge-controlled case and one for the current-controlled case. These
two cases are considered separately since the underlying control operators are
very different in nature. For both cases, the semigroup formulations are
presented, and the corresponding Cauchy problems are shown to be well- posed in
the natural energy space.
