A note on the calculation of optimal strategies for the minimization of fuel consumption in the control of trains

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimize fuel consumption. It is assumed that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. During each phase of the journey the power developed by the locomotive is determined by the rate of fuel supply. For each given sequence of throttle settings it has been shown that fuel consumption is minimized by finding an optimal duration for each phase. This suboptimal strategy has been called a strategy of optimal type. In this note we will show that a comparison of suboptimal strategies allows us to find an idealized strategy of optimal type that minimizes fuel consumption.>