Adaptive error feedback tracking for a wave equation with a nonlocal term and multi-channel unknown harmonic disturbances

This paper addressed the output regulation problem for a one-dimensional (1-D) wave equation subject to a nonlocal term and multi-channel unknown disturbances. Motivated by the combined challenge of structural instability from nonlocal coupling and the realistic presence of multi-channel harmonic disturbances with unknown frequencies, this work aimed to integrate and simultaneously address both issues to meet more complex application scenarios. The nonlocal term caused energy growth and open-loop instability, requiring sequential stabilization and output regulation. The disturbances consisted of sinusoidal signals with unknown amplitudes and frequencies, where only an upper bound on the frequencies was known. Our approach constructed an auxiliary system to eliminate the nonlocal effect and employed a coordinate transformation that concentrated disturbances into the tracking error channel. An adaptive observer was then developed for online frequency identification, enabling output-feedback control using the tracking error and its derivative. Theoretical analysis established the well-posedness and state boundedness of the closed-loop system, while numerical simulations confirmed the effectiveness of the proposed approach and demonstrated exponential convergence of the tracking error to zero.