Semigroup well-posedness of a voltage controlled active constrained layered (ACL) beam with magnetic effects

The layered smart composites involving a piezoelectric layer are traditionally activated by a voltage source, and the magnetic effects are totally ignored since these effects are relatively smaller in comparison to electrical and mechanical effects. However, recent results for even a single piezoelectric beam show that ignoring these effects may cause uncontrollable and unstabilizable systems. In this paper, a variational approach is used to derive a voltage-controlled Rao-Nakra type active constrained layer model. All magnetic effects due to the full set of Maxwell's equations are included. It is shown that the proposed model can be put into the semigroup formulation, i.e. ẋ = Ax + Bu; and is well-posed in the corresponding energy space. Moreover, the observed quantity due to the observation operator B* (corresponding to the control operator B with the voltage control) is totally electrical.