Stability of Variable Coefficients Rayleigh Beams With Indefinite Damping Under a Force Control

In this paper, we continue our study of the damped Rayleigh beam system. The purpose of this work is to add two additional force controls in moment and rotation and see if this remains exponentially stable. Thus we obtain a four-control system. Using the Riesz basis approach, we prove the stability of nonhomogeneous Rayleigh beam via an indefinite damping and the boundary control feedbacks in position, velocity, moment and rotation. We transform the system to an evolution problem and prove that the linear operator of the Cauchy problem generates a semigroup of contractions. Subsequently, we show how the damping term can affect the decay rate asymptotically.