Optimal control of gantry crane for minimum payload oscillations

We consider optimal control of gantry crane for transporting a payload with minimum oscillation of the payload. The system consists of a driving dc motor, and a payload which is suspended by a flexible cable. The dynamic model of the system is given by a coupled system of ordinary and partial differential equation along with another ordinary differential equation that serves as the boundary condition. Concepts of optimal control theory have been used to derive a control law that transports the payload with minimum oscillation in a finite time. Necessary conditions of optimality in the form of a coupled system of state equation, adjoint equation, and an inequality are presented.