Parameter estimation and stabilisation for a one-dimensional wave equation with boundary output constant disturbance and non-collocated control

We are concerned with the parameter estimation and stabilisation of a one-dimensional wave equation with an unknown constant disturbance suffered by boundary observation at one end and the non-collocated control at another end. An adaptive observer is designed in terms of measured velocity with unknown constant disturbance. 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.