Nonlinear feedback solution to a bounded Brachistochrone problem in a reduced state space

The optimal nonlinear feedback law for both an unbounded and bounded Brachistochrone problem is given. These control laws describe the Brachistochrone problem in a very simple manner. These laws are derived by using a dimensionless set of state variables which spans a lower dimensional space than the original state space. Also using this reduced state space a two-dimensional instead of three-dimensional field of extremals is constructed for a space bounded Brachistochrone problem. This illustrates that the state space may be reduced so that the storage of a field of extremal trajectories will not exceed the storage capacity of a practical on-board computer. In this way, optimal feedback control may become a practical technique for a larger class of nonlinear systems.