An applicable method for solving the shortest path problems

A theorem of Hardy, Littlewood, and Polya, first time is used to find the variational form of the well known shortest path problem, and as a consequence of that theorem, one can find the shortest path problem via quadratic programming. In this paper, we use measure theory to solve this problem. The shortest path problem can be written as an optimal control problem. Then the resulting distributed control problem is expressed in measure theoretical form, in fact an infinite dimensional linear programming problem. The optimal measure representing the shortest path problem is approximated by the solution of a finite dimensional linear programming problem.