Studies on the Rao–Nakra sandwich beam: well-posedness, dynamics, and controllability

In this work, we investigate two distinct control-theoretic problems associated with a linear Rao–Nakra-type sandwich beam. The first problem concerns stabilization under time-varying distributed damping and delayed boundary feedback, while the second addresses boundary controllability under dynamical boundary interactions. Both systems consist of three coupled equations that represent the longitudinal displacements of the outer layers and the transverse displacement of the composite beam, all of which are coupled with boundary conditions. In the first problem, time-varying interior damping and static boundary conditions with time-dependent delays are considered. Then, we establish the existence and uniqueness of solutions for the Cauchy problem associated with the damped system using semigroup theory and a classical result by Kato. Furthermore, employing a Lyapunov-based approach, we prove that the system's energy decays exponentially, despite the presence of time-varying weights and delays. In the second problem, we consider a boundary linear control system with dynamical boundary conditions, and prove its well-posedness. By deriving an observability inequality for the adjoint system and applying the Hilbert Uniqueness Method (HUM), we show that the system is null controllable. A key contribution of this work lies in handling the full three-equation coupled system, which involves a challenging difficulty due to the delayed or dynamical boundary conditions, resolved via appropriately constructed Lyapunov functionals and intermediate observability inequalities.