Dynamic programming and the calculus of variations.

Stuart E. Dreyfus

Expert InsightsPublished 1960

A study showing that the functional equation technique of dynamic programming yields formal derivations of such classical necessary conditions of the calculus of variations as the Euler-Lagrange equations, the Weierstrass and Legendre conditions, natural boundary conditions, a transversality condition, and the Erdmann corner conditions. The more general "problem of Bolza"-in which the final time is defined implicitly and in which the expression to be extremized is the sum of an integral, and a function evaluated at the end point-is also considered. The principal necessary condition, usually called the "multiplier rule," is deduced. Necessary conditions are derived for the case in which the decision variables are restricted by inequality constraints. Finally, it is shown that the functional equation characterization readily yields the Hamilton-Jacobi partial differential equation of classical mechanics.

Document Details

Citation

Chicago Manual of Style

Dreyfus, Stuart E., Dynamic programming and the calculus of variations. Santa Monica, CA: RAND Corporation, 1960. https://www.rand.org/pubs/papers/P1464.html.
BibTeX RIS

This publication is part of the RAND paper series. The paper series was a product of RAND from 1948 to 2003 that captured speeches, memorials, and derivative research, usually prepared on authors' own time and meant to be the scholarly or scientific contribution of individual authors to their professional fields. Papers were less formal than reports and did not require rigorous peer review.

This document and trademark(s) contained herein are protected by law. This representation of RAND intellectual property is provided for noncommercial use only. Unauthorized posting of this publication online is prohibited; linking directly to this product page is encouraged. Permission is required from RAND to reproduce, or reuse in another form, any of its research documents for commercial purposes. For information on reprint and reuse permissions, please visit www.rand.org/pubs/permissions.

RAND is a nonprofit institution that helps improve policy and decisionmaking through research and analysis. RAND's publications do not necessarily reflect the opinions of its research clients and sponsors.