Optimal train control: Analysis of a new local optimization principle

It is known that the optimal driving strategy for a train takes the form of a power-speedhold-coast-brake strategy unless the track contains steep grades. In such cases the predominant speedhold mode must be interrupted by phases of power on steep uphill sections and phases of coast on steep downhill sections. The Freightmiser device is used by Pacific National to provide on-board advice to train drivers about energy efficient driving strategies. Freightmiser uses a fast and efficient numerical algorithm to solve a key local energy minimization problem and hence find the optimal switching points. Although the numerical algorithm converges to a feasible solution there is no direct proof that the solution is unique. We explain the basic ideas behind the local energy minimization principle and use an extended perturbation analysis to derive various equivalent forms of the necessary conditions.