On a differential interception game

We consider an interception game and discuss, using this particular example, aspects of the formalization of differential games that is being developed in Yekaterinburg. We establish both the functional character of the value of a game as a function of the current history of a process and the positional character of optimal strategies. We develop a unification of the original game, thus relating the descriptive theorem on the value and the saddle point of the game to the formalism of Hamilton–Jacobi equations and further to generalized minimax solutions to these equations. We emphasize the relationship between the unified form of a differential game and L. S. Pontryagin's concept of differential game. On the basis of an auxiliary boundary value problem for a degenerate parabolic equation, we develop and substantiate minimax and maximin controls in a feedback scheme with a stochastic guide. Finally, we discuss an algorithm for calculating the value and the saddle point of a game.