Optimal decay rate for a Rayleigh beam–string coupled system with frictional damping

This paper addresses the stability of a coupled system that consists of a conservative Rayleigh beam and a damped elastic string, the latter being equipped with frictional damping. By estimating the growth of the resolvent along the imaginary axis, we employ the frequency domain method to demonstrate that the coupled Rayleigh beam–string system exhibits polynomial stability, characterized by a decay rate $t^{-1/4}$. Furthermore, we establish the optimality of the decay rate, $t^{-1/4}$, through a rigorous spectral analysis. Finally, a numerical example is present to validate the theoretical findings.