On the Stabilization of the Disk-Beam System via Torque and Direct Strain Feedback Controls

This note deals with the stabilization problem of a rotating disk-beam system. The proposed feedback law consists of a nonlinear torque control exerted on the disk, whereas a direct strain control is applied at the beam with a rigid offset. As a consequence, we end up with a nondissipative nonlinear hybrid PDE-ODE system and hence the existence and stability of solutions of the system cannot be addressed by using standard methods such as energy-multiplier or frequency-domain. To overcome this situation, we adopt the Riesz basis methodology and show that the closed-loop system is well-posed and stable as long as the angular velocity of the disk is less than a well-defined value related to the beam. The results are illustrated through a wide set of numerical examples.