A Minimal State Approach to Dynamic Stabilization of the Rotating Disk-Beam System with Infinite Memory

This note is dedicated to the investigation of the stabilization problem of the well-know rotating disk-beam system. We suggest a linear feedback law which consists of a torque control applied on the disk and a dynamic force control with infinite memory term exerted on the beam. Thereafter, sufficient conditions on the angular velocity of the disk and the memory kernel are derived to assure the existence and uniqueness of solutions of the closed-loop system. The main ingredient of the proof of such a well-posedness result is to adopt the method used for viscoelastic systems, which mainly consists in utilizing the concept of minimal state variables. Moreover, it is shown that the beam vibrations are forced to exponentially decay to zero while the disk keeps rotating with a fixed angular velocity. This desirable property, which can be interpreted as the exponential stability of the system, is achieved by means of Lyapunov-energy method.