Optimal Positional Strategies in Differential Games for Neutral-Type Systems

The paper deals with a zero-sum differential game for a dynamical system described by neutral-type functional-differential equations in Hale’s form with initial conditions determined by piecewise continuous functions. It is proved that the differential game has a value and optimal positional (feedback) players’ strategies. If the value functional satisfies certain smoothness conditions, the optimal strategies are constructed based on its gradient. In the general case, such strategies are described using quasi-gradient constructions. The fact that the quasi-gradients under consideration require looking for extremum points only on a finite-dimensional set is the crucial contribution of this paper.