Optimal strategies for the control of a train

This paper describes a method for the calculation of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with non-constant gradient. The journey must be completed within a given time, and it is desirable to minimise the fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomotive. For each fixed finite sequence of control settings, we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. Several realistic examples are given with the results of associated numerical calculations. The examples demonstrate the profound effect of even a small gradient on a strategy of optimal type. We show that a strategy of optimal type with alternate phases of coast and maximum power can be used to approximate the idealised minimum cost strategy.