Brachistochrone on a 1D Curved Surface Using Optimal Control

The brachistochrone for a steerable particle moving on a 1D curved surface in a gravity field is solved using an optimal control formulation with state feedback. The process begins with a derivation of a fourth-order open-loop plant model with the system input being the body yaw rate. Solving for the minimum-time control law entails introducing four costates and solving the Euler–Lagrange equations, with the Hamiltonian being stationary with respect to the control. Also, since the system is autonomous, the Hamiltonian must be zero. A two-point boundary value problem results with a transversality condition, and its solution requires iteration of the initial bearing angle so the integrated trajectory runs through the final point. For this choice of control, the Legendre–Clebsch necessary condition is not satisfied. However, the $k=1$ generalized Legendre–Clebsch necessary condition from singular control theory is satisfied for all numerical simulations performed, and optimality is assured. Simulations in MATLAB^® exercise the theory developed and illustrate application such as to ski racing and minimizing travel time over either a concave or undulating surface when starting from rest. Lastly, a control law singularity in particle speed is overcome numerically.