On the Elimination of Infinite Memory Effects on the Stability of a Nonlinear Non-homogeneous Rotating Body-Beam System

The main concern of this article is to study the impact of the presence of an infinite memory term (boundary or distributed) on the stability of the rotating disk-beam system. On one hand, unlike the previous works, the compensating term is not of the same type as the infinite memory. On the other hand, the mass per unit length and the flexural rigidity of the beam are assumed to vary. Despite this situation, it is shown that if the standard torque control operates, then the system remains exponentially stable and the effect of the infinite memory term can be neutralized so that the beam's vibrations are suppressed and the disk rotates with a desired angular velocity. Of course, such an outcome is obtained under standard requirements on the memory kernel function and the desired constant angular velocity of the disk. The proof of the stability outcome relies on the resolvent method. Finally, numerical examples are provided to illustrate our outcomes.