On Hamilton-Jacobi equations for neutral-type differential games

Abstract

We consider a two-person zero-sum differential game in which a motion of the dynamical system is described by neutral-type functional-differential equations in Hale’s form and the quality index estimates a motion history realized up to the terminal instant of time and includes integral estimations of control realizations of the players. The formalization of the game in the class of pure positional strategies is given, the corresponding notions of the value functional and optimal control strategies of the players are defined. For the value functional, we derive a Hamilton-Jacobi type equation with coinvariant derivatives. It is proved that, if a solution of this equation satisfies certain smoothness conditions, then it coincides with the value functional. On the other hand, it is proved that, at the points of coinvariant differentiability, the value functional satisfies the derived Hamilton-Jacobi equation. Therefore, this equation can be called the Hamilton-Jacobi-Bellman-Isaacs equation for neutral-type systems.