Optimal mixed strategies in a dynamic game

In this paper we treat a specific two-person, zero-sum, dynamic game of the type
x_{k + 1} = f(x_{k}, u_{k}, w_{k})
. The optimal solutions for this game (i.e, a saddle point) have to be sought in the class of mixed (synonymously, randomized) strategies. For this particular game a theory of optimality of mixed strategies is developed and a hierarchy of problems of increasing generality, within this particular game, is solved. The specific game considered is one of the most classic of the problems in game theory. A gun is firing at a moving object. How best should the object move in order to reach a certain destination? Conversely, where should the gun fire in order to prevent the object from reaching its destination? This problem occurs in different guises in a variety of situations. The moving object could, for example, be a ship or a tank. The optimal strategies of the two palyers have perforce to be mixed.