Analysis of optimal train control problem on a non-steep track with speed limit

Modern control theory has been successfully applied to the single train optimal control problem, especially the Pontryagin's maximum principle. The uniqueness of optimal strategies on a non-steep track and steep track without speed limit has been established. And corresponding constructive algorithms have been proposed. However, the cases with speed limit and steep track have not been thoroughly solved because the switching points of different modes may not occur at the singular cruising mode. In this paper, we analyze the feasible transitions with the speed limit on a non-steep track using the Hamiltonian analysis, Picard's uniqueness theorem and a useful comparison lemma. We exclude the steep case to limit the range of feasible transition points. We analyze several typical cases with speed limit and give some preliminary conclusions to construct the algorithm. In some special cases, the optimal transition is unique which would simplify the solving process. Finally we make the simulation to describe the effects of the above analysis.