Parameter estimation and velocity signal extraction for one-dimensional wave equation with harmonic corrupted boundary observation.

This article addresses parameter estimation and velocity signal extraction for an unstable one-dimensional wave equation subject to harmonic external disturbances with known frequencies but unknown amplitudes. The authors design an adaptive observer based on boundary displacement and corrupted boundary velocity measurements, combined with an output feedback controller employing the backstepping method for infinite-dimensional systems. They prove that the estimated parameters converge asymptotically to the true unknown values, the initial disturbance can be recovered, and the velocity signal is asymptotically reconstructed, while ensuring the closed-loop system's asymptotic stability via semigroup theory and Lyapunov methods. Numerical simulations for a single-frequency disturbance illustrate the effectiveness of the proposed scheme in stabilizing the system and accurately estimating parameters.