“…Remark 4: By comparing (31) and (32), it is evident that the time rate of change of the rope energy evaluated using the definition of differentiation, (31), and the material derivative of the mechanical energy, (32), yield the same result and, henceforth, either approach can be used when evaluating the time-rate of the energy of an axially moving material. Now, based upon (25) and using the fact that, for suppressing the transverse vibration, the rigid-body-like longitudinal motion of the rope can be omitted from the stability analysis point of view of the closed-loop system, a Lyapunov function candidate is considered as follows: positive (controller) gains and d w is the target position of the gantry, which is normally given as a fixed value (i.e., 0).…”