Adaptive output tracking for a hybrid wave-ODE system subject to unknown sinusoidal disturbance

In this paper, we are concerned with the output tracking problem for a hybrid wave-ODE system coupling an elastic string with a rigid body at one end. The control force acts on the rigid body and the output is at the other end where is subject to an unknown sinusoidal disturbance. By constituting auxiliary system and internal model dynamic, an adaptive method is developed to identify the unknown frequency of disturbance. An error-based adaptive dynamic compensator is proposed to achieve exponential output tracking. The well-posedness and stability of the closed-loop system are proved by employing the semigroup theory. The numerical simulations are presented to confirm the effectiveness of the proposed control strategy.