# Stability and Stabilization Results for the Rao-Nakra Layered Beam System

Scott Hansen^{*}

The multilayer Rao-Nakra beam system is a multilayer generalization of the classical Rao-Nakra three-layer sandwich beam model, which describes longitudinal and transverse vibrations in a composite beam consisting of two relatively stiff outer ``face plates" and much more compliant inner ``core layer". The multilayer model consists of alternating stiff and compliant layers which are coupled through the shears of the compliant layers. \newline First we consider passive damping proportional to rate-of-shear in the compliant layers. In the case that all the wave speeds of the stiff layers are the same there is an infinite dimensional subspace of purely longitudinal motions that are undamped. If three of these wave speeds are distinct, then uniform exponential stability holds. If there are only two distinct wave speeds and an additional technical assumption holds, again uniform exponential stability holds. \newline Secondly we consider boundary feedback, without internal damping. Here we are able to show that uniform exponential stability holds for a variety of boundary conditions. The proof relies upon a decomposition of the semigroup into the sum of a uniformly stable semigroup and a compact perturbation together with a unique continuation result for an overdetermined eigensystem.

Mathematics Subject Classification: 93B52 74K10

Keywords: sandwich beam, boundary stabilization, exponential stability, exact controllability

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