Optimal velocity in the race over variable slope trace.

The minimum-time running problem is reconsidered. The time of covering a given distance is minimized. The function that should be found is the runner's velocity that varies with the distance. The Hill-Keller model of motion is employed. It is based on the Newton second law and an equation of power balance. The new element of the current approach is that the trace slope angle varies with the distance. The problem is formulated and solved in optimal control applying the Chebyshev direct pseudospectral method. The essential finding is that the optimal velocity during the cruise is constant regardless of the local slope of the terrain. Such result is valid if the inequality constraints imposed on the propulsive force or the energy are not active.