Controllability of the Kirchhoff System for Beams as a Limit of the Mindlin-Timoshenko System

We consider the dynamical one-dimensional Mindlin-Timoshenko system for beams. We analyze how its controllability properties depend on the modulus of elasticity in shear $k$. In particular we prove that the exact boundary controllability property of the Kirchhoff system may be obtained as a singular limit, as $k\rightarrow\infty,$ of the partial controllability of projections over a sharp subspace of solutions generated by the eigenfunctions that converge, as $k\rightarrow\infty$, towards the spectrum of the Kirchhoff system.