On the boundary controllability of laminated Timoshenko beams

We study boundary controllability for a laminated Timoshenko beam with Neumann and Dirichlet boundary conditions, using a single control acting at one end. Controllability is obtained by proving an observability inequality for the adjoint system and then applying the Hilbert Uniqueness Method (HUM). The novelty of this work lies in a transparent spectral analysis tailored to the equal wave speed regime. Instead of asymptotic expansions, we use elementary function inequalities and Mean Value–type estimates. We provide a complete gap description of the spectrum: in the non-resonant case there is a uniform gap and Ingham's inequality yields full observability; in the resonant case a simple reordering shows that, although consecutive frequencies may cluster, every three-mode block is separated by a uniform gap, leading to observability on natural subspaces. These results, combined with HUM, give boundary controllability on the whole energy space in the non-resonant regime and on appropriate subspaces in the resonant regime.