Stabilization of the rotating disk-beam system with a delay term in boundary feedback

In this article, the stabilization problem of a rotating disk-beam system is addressed. It is assumed that the flexible beam is free at one end, whereas the other end is attached to the center of the rotating disk whose angular velocity is time-varying. The proposed feedback law consists of a torque control which acts on the disk, whereas a delayed boundary force control is exerted at the free end of the beam. Thereafter, it is proved that the presence of such controls in the system guarantees the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk as well as the feedback gain in the delay term. This result is illustrated by numerical examples.