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 a platform moving along a straight rail and carrying at the free end a load which is free to move in a horizontal plane. The model proposed in this paper fits a large class of real-life applications such as an overhead crane with a 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 a consequence, the exponential stability is concluded. To verify the theoretical developments, numerical study of the spectrum is performed by Legendre approximation, also the numerical simulations are presented to show the effectiveness of the proposed control.