Exact boundary controllability and stabilizability of a degenerated Timoshenko beam

This paper investigates the boundary controllability and stabilizability of a Timoshenko beam subject to degeneracy at one end, while control is applied at the opposite boundary. Degeneracy in this context is measured by the real parameters for $\mu_a\in [0,2)$ for $a\in\{K,EI\}$, where $K(x)$ denotes shear stiffness and $EI(x)$ bending stiffness. We differentiate between weak degeneracy $\mu_a\in [0,1)$ and strong degeneracy $\mu_a\in [1,2)$, which may occur independently in shear and bending. Our study establishes observability inequalities for both weakly and strongly degenerate equations under Dirichlet, Robin, and Neumann boundary conditions. Using energy multiplier techniques and the Hilbert Uniqueness Method (HUM), we derive conditions for exact boundary controllability and show that appropriate boundary state and velocity feedback controls at the non-degenerate end can stabilize the system exponentially. Extending results previously obtained for the 1-dimensional wave equation in \cite{AlabauCannarsaLeugering2017}, this study highlights new control strategies and stabilization effects specific to the degenerate Timoshenko beam system, addressing challenges pertinent to real-world structural damping and control applications.