Saddle Points for Linear Differential Games

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Abstract
It is proved that there is a saddle point over the relaxed controls, and so over the strategies defined on the relaxed controls, for differential games in which the trajectory variable appears linearly in the dynamical equation and payoff. This is a strong saddle point property, but the example of Berkovitz [1], of a game that does not have a saddle point in pure strategies, does have a saddle point in this sense. Saddle points over the chattering controls are obtained for linear games in which the opposing control variables appear separated. The introduction of relaxed controls into differential games is analogous to the introduction by von Neumann of mixed strategies into two person, zero sum games.

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