A Variational Approach to Differential Games

by Leonard David Berkovitz


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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.

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