Uniform convergence of semi-discrete scheme for output regulation of 1-D wave equation

In this paper, we investigate the uniform convergence of a semi-discrete scheme for output regulation of a system governed by a one-dimensional wave equation. The disturbances and reference signals stem from an exosystem, infiltrating the system through all channels. The exponential convergence of the continuous partial differential equation (PDE) system is firstly established using the Lyapunov functional approach. Utilizing the order reduction approach, we develop a semi-discrete finite difference scheme for the continuous PDE closed-loop system and demonstrate that this semi-discrete scheme exhibits uniform internal exponential stability, regardless of the step size, in complete alignment with its PDE counterpart. Consequently, the tracking errors for the discrete systems exhibit uniform exponential convergence.