The value and optimal strategies in a positional differential game for a neutral-type system

On a finite time interval, a differential game for the minimax–maximin of a given cost functional is considered. In this game, the motion of a conflict-controlled dynamical system is described by functional differential equations of neutral type in Hale's form. Under assumptions more general than those considered previously, a theorem on the existence of the value and saddle point of the game in classes of players' closed-loop control strategies with memory of the motion history is proved. The proof involves the technique of the corresponding path-dependent Hamilton–Jacobi equations with coinvariant derivatives and the theory of minimax (generalized) solutions of such equations. In order to construct optimal strategies, which constitute a saddle point of the game, a recent result on the existence and uniqueness of a suitable minimax solution and a special Lyapunov–Krasovskii functional are used.