Filter-based adaptive control of a wave equation through lower-order persistent excitation conditions

For the non-collocated output regulation for wave equations, this paper postulates that the measurable outputs encompass the tracking error and even its perturbed time derivative influenced by an unknown output disturbance. An adaptive control is designed directly using a set of filter systems, with persistent excitation conditions guaranteed in a lower-order manner. For the resulting closed-loop system, we prove that the tracking error converges to zero exponentially. Additionally, a simulation example is provided to demonstrate the effectiveness of the proposed control.