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

¹ School of Mathematics and Physics North China Electrical Power University, Beijing 102206, PR China
² Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190
³ School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, PR China

^* Corresponding author: bzguo@iss.ac.cn

Received: 14 August 2022
Accepted: 26 July 2023

Abstract

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.