Existence of an optimal strategy for stochastic train energy-efficient operation problem

When we formulate the optimal train control model for optimizing the operation strategy, it is necessary to consider the uncertain disturbances arising from the whether, route and locomotive rolling conditions. Stochastic energy-efficient operation has proved to be an efficient approach to reduce the effect of uncertainty by estimating the resistance coefficients as random variables. The purpose of this paper is to prove the existence of an optimal operation strategy. First, we prove the existence of a feasible operation strategy satisfying the nonnegativity, boundary conditions, trip distance and the motion equation. Furthermore, we prove the Lipschitz continuity for the stochastic objective function. Finally, we prove the existence theorem for an optimal operation strategy.