Optimal driving strategies for two successive trains on level track subject to a safe separation condition

When two trains travel along the same track in the same direction it is a common safety requirement that they should always be separated by at least one signal. If the signals are located at fixed positions dividing the track into different sections then the two trains cannot occupy the same section of track at the same time. It is desirable to find driving strategies that minimize the total energy consumption and allow the trains to reach their final destinations at the designated times. In this paper we impose a prescribed sequence of intermediate times that ensure adequate separation for two successive trains on level track. By varying these times we find necessary conditions defining the optimal sequence of prescribed intermediate times-a sequence that minimizes total energy consumption.