Stability of a nonuniform Rayleigh beam with indefinite damping

This is a continuation of our earlier work [J.M. Wang, G.Q. Xu, S.P. Yung, Exponential stability for variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach, Systems Control Lett. 51 (1) (2004) 33-50] on the study of a nonhomogeneous Rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks. We continue to address a conjecture of Guo [Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl. 112(3) (2002) 529-547] in this paper and demonstrate how the damping term can affect the decay rate asymptotically. By a detailed spectral analysis, we obtain a necessary condition for the stability and establish the Riesz basis property as well as the spectrum determined growth condition for the system. Furthermore, when the damping is indefinite, we provide a condition on how "negative" the damping can be without destroying the exponential stability.