Stabilization of memory type for a rotating disk-beam system

In this article, a rotating disk-beam system is considered. Specifically, the system consists of a flexible beam and a rigid disk which rotates with a time-varying angular velocity. The beam, free at one end and clamped at the other one to the center of the disk, is supposed to rotate with the disk in another plane perpendicular to that of the disk. To stabilize the system, we propose a feedback law which consists of a control torque applied on the disk, while either a boundary or distributed internal control with memory is exerted on the beam. Then, it is shown, in both cases, that the closed-loop system is stabilized under suitable conditions on the angular velocity and the memory terms.