Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: A Riesz basis approach

In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, stiffness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered.