The cost differential for an optimal train journey on level track

The classic optimal train control problem is to drive a train on a track with known gradient over a fixed distance and within a specified time in such a way as to minimize tractive energy consumption. On level track the optimal strategies take two basic forms—a truncated strategy of optimal type with phases of maximum acceleration, coast and maximum brake which is typical of shorter metropolitan journeys, and an extended strategy of optimal type with phases of maximum acceleration, speedhold at the optimal driving speed, coast to the optimal braking speed, and maximum brake which is typical of longer journeys by freight trains and intercity passenger trains. The cost of these optimal strategies is uniquely determined by the journey distance and journey time. In this paper we extend a previously known formula for the partial rate of change of cost with respect to journey time to a formula for the full rate of change of cost that also incorporates the partial rate of change of cost with respect to journey distance.