Uniformly exponential stability of semi-discrete scheme for a vibration cable with a tip mass under observer-based feedback control

In this paper, we consider uniformly exponential approximation for  a vibrating cable with tip mass under a non-collocated  output stabilizing feedback control. By designing an  observer-based output feedback control, the closed-loop system is composed of the coupled same type of PDEs  and ODEs. By order reduction method, we find a global Lyapunov functional for the closed-loop system. The  closed-loop system  is then semi-discretized by the  finite difference method. For the discrete systems, we also construct the  Lyapunov functions. The uniform exponential stability of the semi-discretized systems is then established analogously as the proof for  the continuous counterpart via an indirect Lyapunov functional approach.