A symmetric alternating direction method of multipliers for controlling the movement of multiple trains on a railway line

We consider the design of control inputs for multiple trains on a single railway line with several stations. A discrete time system model is firstly established to approximate the dynamics of a single moving train. Combining this with the headway constraints between adjacent trains and the station punctuality constraints, a distributed optimal control problem is formulated for the design of control inputs for multiple trains on a single railway line. We then propose an algorithm based on the framework of symmetric alternating direction multiplier to solve the distributed optimal control problem. The pseudocode of the algorithm is finally provided for the interested reader.