On the spectrum-determined growth condition of a vibration cable with a tip mass

We show that the spectrum-determined growth condition holds for the closed-loop system of a vibration cable with a tip mass and linear boundary feedback control. The optimal decay rate of the energy for a case left unsolved previously is determined, and the asymptotic expansion of the associated semigroup is obtained. As a consequence of the approach, we show, in a different point of view, the lack of uniform exponential stability of the system with only boundary velocity feedback control.