On a class of variational problems

The problem of determining the minimum of a functional is treated using the functional equation technique of the theory of dynamic programming. The problem is reduced to the solution of a system of ordinary differential equations satisfying one-point boundary conditions. The discrete case, corresponding to the minimization of a class of quadratic forms, is also treated by the same general method. A particular problem of this type arises in the treatment of the optimal inventory problem by Holt, Simon, and Modigliani.