Boundary stabilization of an overhead crane with beam model

In this paper, we study the boundary feedback stabilization problem of a hybrid system consisting of a flexible beam attached to the platform moving a long a straight rail and carrying at the free end a load which is deplaced in a horizontal plan. The model proposed in this paper fits a large real applications such as an overhead crane with beam. Using the Riesz basis approach of general second-differential equation systems with non separated boundary conditions, it is shown that the Riesz basis property holds for the system and as consequence, the exponential stability is concluded. To verify the theoritical developments, numerical study of the spectrum is performed by Legendre approximation, also thenumerical simulations are presented to show the effectivnesse of the proposed control.