Stabilization of a transmission problem for two Rayleigh beams and a string with damping

We are concerned with the stability of a coupled Rayleigh beam–string–Rayleigh beam transmission system with damping. We first show the exponential stability in the undamped–damped–undamped situation, which means that the frictional damping is only effective in the string part. This result generalizes the recent result in Martinez et al. (J Diff Equ 2022:1-19, 2022). Next, we prove the polynomial stability in the damped–undamped–damped situation, which means that the structural dampings are effective in the beam parts. The main ingredient of the proof is some careful analysis for the Rayleigh beams and string transmission system.