The minimization of the fuel costs in the train transportation

This paper presenting case studies in modern large scale constrained optimization, the purpose of which is to illustrate how recent advances in algorithms and modelling languages have made it easy to solve difficult optimization problems using of software. In this paper we use measure theory technique and iterative dynamic programming algorithm for solving of a trajectory optimization problem: how to drive a train so as to minimize fuel costs. In the first method, we optimal control problem to using of atomic measures change this one to an infinite dimensional linear programming problem and then we approximate the latter one to a finite dimensional linear programming problem, then by the optimal solution of the final problem we obtain sub-optimal controls and then by these controls we obtain the approximate solution of the original problem. In the second method, we will using dynamic programming as iteration for solving optimal control problem.