A Dynamic Programming Approach to the Nonparametric Problem in the Calculus of Variations

摘要

This memorandum presents an approach to the nonparametric simple problem of the calculus of variations that is in the spirit of dynamic programming. Under the assumptions that the functions that define minimizing curves exhibit a certain regularity in their behavior, differentiability properties of the optimal value function W are determined. Then the fundamental partial differential equation of dynamic programming is obtained, and the necessary conditions of the calculus of variations, with the exception of Jacobi's condition, are deduced as immediate corollaries. (Author)