Design of controls to move a flexible body such as a robot arm while minimizing unwanted vibrations has been described in many papers and presented in many forms. For the vibration reduction issue alone, it is shown that almost all the proposed designs can be formulated as optimal controls of either the fixed final time or the minimum time type. Furthermore, it is shown that under reasonable assumptions the two types have the same solution and are thus equivalent. Continuous time, tapped-delay-line input shaping filters, and discrete controls are considered. It is shown that the discrete equivalent of the general vibration reduction problem is a convex problem for the fixed final time case and quasi-convex for the free final time problem. The two formulations are compared in terms of computation complexity as well as practical implementation issues.