Measurable strategies in differential games

Nonlinear approach-evasion differential games are considered in which the initial data depend on the time. These games are investigated in the class of strategies that are functions of three variables, namely, the time, the phase variable, and the current value of the other player's control, and are measurable jointly with respect to the time and the phase variable. The ideas of the Pontryagin methods in differential games and Krasovskii's ideas on extremal aiming are developed, and it is shown that measurable strategies have broad applicability.
It is proved that measurable strategies are compatible with differential equations with discontinuous right-hand side, and general theorems on the existence of solving measurable strategies in approach-evasion problems are proved, along with some auxiliary assertions. It is shown that the saddle point condition in the small game ensures the existence of solving measurable strategies. An example is given.
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