A study of a class of differential games having pure strategy solutions, using results and techniques from the calculus of variations. These games are related to two Bolza problems with differential inequalities as added side conditions. Necessary conditions that must hold along an optimal path are derived from the theory of the related Bolza problems. These conditions are (1) a multiplier rule, together with transversality conditions and jump conditions, (2) a local min-max condition that is related to the Weierstrass condition, and (3) an analogue of the Clebsch condition. The continuity and differentiability properties of the value of the game are derived, and it is shown that wherever the value is differentiable, it satisfies an analogue of the Hamilton-Jacobi equation. Sufficient conditions are given in terms of the notion of a field and a local min-max condition.
This report is part of the RAND Corporation Paper series. The paper was a product of the RAND Corporation 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.
The RAND Corporation 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.