Feedback stabilization of rotating Timoshenko beam with adaptive gain

The problem of boundary feedback stabilization of rotating Timoshenko beam, arising from control of flexible robot arms, is studied in this paper. Fir st, under gain adaptive direct strain feedback controls, a counterexample i s given to show that the corresponding closed loop system is not asymptotic ally stable, which is contrary to traditional conjecture. The counterexampl e given in this paper also exemplifies an interesting result: certain two t wo-order linear partial differential equations with five homogeneous bounda ry conditions have non-trivial solutions. Then, with an additional boundary feedback control, the related energy of the closed loop system is proved t o be strongly stable, or more precisely, the configuration of the beam can be exponentially stabilized with some suitable non-linear boundary feedback controls with adaptive gain.