Uniformly exponentially stable approximations for Timoshenko beams

In this note, Timoshenko beams with interior damping and boundary damping are studied from  the viewpoints of  control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform exponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is  that the semi-discretization systems, which  are derived from an exponentially stable continuous beam by some semi-discretization schemes,  are uniformly exponentially stable with respect to the discretized parameter.  To investigate uniform exponential stability of continuous  and discrete systems, five completely different methods,  which are  stability theory of port-Hamiltonian system,  direct method  of Lyapunov functional, perturbation theory of $C_{0}$-semigroup, spectral analysis of unbounded operator and frequency  standard of exponential stability for contractive semigroup, are involved.  Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of  $C_{0}$-semigroup of contractions, is established  to verify the uniform exponential stability of semi-discretization systems derived from coupled system.  The effectiveness of the numerical approximating  algorithms is verified by numerical simulations.