Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability

TL;DR

This paper analyzes the mathematical properties, stability, and controllability of a complex coupled Rao-Nakra sandwich beam system with dynamic boundary conditions, delays, and damping, using advanced functional analysis techniques.

Contribution

It establishes well-posedness, exponential energy decay, and null controllability for the full three-equation coupled system with dynamic boundary conditions and delays.

Findings

Solutions exist and are unique for the damped system.

System energy decays exponentially over time.

The system is null controllable via boundary controls.

Abstract

In this work, we investigate the well-posedness, stabilization, and boundary controllability of a linear Rao-Nakra type sandwich beam. The system consists 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 dynamical 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 significant difficulty due to the dynamic boundary conditions, resolved via appropriately constructed Lyapunov functionals and intermediate observability inequalities.