Parameter estimation and stabilization for an unstable one-dimensional wave equation with boundary input harmonic disturbances

This paper is concerned with the parameter estimation and stabilization of a one-dimensional wave equation with instability suffered at one end and uncertainty of harmonic disturbances at the controlled end. An adaptive observer is designed in terms of measured position at one end and velocity at the other end. The backstepping method for infinite-dimensional system is adopted in the design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity.