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We are concerned with the stability of a 1-D coupled Rayleigh beam-string transmission system. We obtain the polynomial decay ratet-1{t^{-1}}or the exponential decay rate for the given transmission system whether the frictional damping is only effective in the beam part or the string part, respectively. This paper generalizes the recent result in [Y.-F. Li, Z.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett. 78 2018, 51–58]. The main ingredient of the proof is some careful analysis for the Rayleigh beam and string transmission system.

Stabilization of a 1-D transmission problem for the Rayleigh beam and string with localized frictional damping