Uniform Exponential Stability and Control Convergence of Semi-discrete Scheme for a Timoshenko Beam

This paper considers numerical approximations of a Timoshenko beam under boundary control. The continuous system under boundary feedback is known to be exponentially stable. Firstly, the continuous system is transformed into an equivalent first-order port-Hamiltonian formulation. A basically order reduction finite difference scheme is applied to derive a family of semi-discretized systems. Secondly, a completely new method which is based on a mixed discrete observability inequality involving final state observability and exact observability is developed to prove the uniform exponential stability of the discrete systems. More interestingly, the proof for the stability of discrete systems is almost parallel to that of the continuous counterpart. Thirdly, the solutions of the semi-discretized systems are shown to be strongly convergent to the solution of the original system through Trotter-Kato theorem. Finally, both exact controllability of continuous system and the discrete systems are proved in light of Russell’s “controllability via stability” principle and the explicit controls are derived. Moreover, the discrete controls are shown in first time to be convergent to the continuous control by proposed approach.