Uniform exponential stability of semi-discrete scheme for observer-based control of 1-D wave equation

Abstract

In this paper, we consider the uniformly exponential stability of a semi-discretized coupled system generated by a 1-D wave equation under a stabilizing output feedback control. Since the control and observation are non-collocated, we first design an observer to recover the state. The closed-loop system under the observer-based feedback is composed of a two coupled wave equations. First, we transform the closed-loop system into an equivalent system by order reduction method, and the transformed system is shown to be exponentially stable by constructing a suitable Lyapunov function. As a consequence, two equivalent semi-discrete finite difference schemes are developed corresponding to the transformed system and original coupled system. Parallel to the proof of the continuous counterpart, the Lyapunov function is constructed to prove the uniformly exponential stability of the semi-discrete scheme for the transformed system, which leads to the uniformly exponential stability of the semi-discrete scheme for the original coupled system. The weak convergence of the discrete solution to the continuous counterpart is also briefly presented.