Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability
George J. Bautista, Roberto de A. Capistrano-Filho, Boumediene Chentouf, Oscar Sierra Fonseca, Juan Límaco
TL;DR
The paper develops a rigorous framework for a linear Rao–Nakra sandwich beam with three coupled PDEs and dynamic boundary conditions, addressing well-posedness, stabilization, and boundary controllability. It proves existence and uniqueness for the delayed system via Kato’s abstract Cauchy framework, and demonstrates exponential energy decay using a Lyapunov functional that accounts for time-varying delays. For the boundary-control problem, it establishes well-posedness with semigroup methods and achieves null controllability through the Hilbert Uniqueness Method, supported by intermediate observability inequalities for the adjoint system. Collectively, the results provide a solid mathematical foundation for stabilization and boundary control of complex three-field Rao–Nakra beam models with dynamic boundary effects.
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-dependent weights and 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 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.