Toward a theory of many player differential games.

We consider a differential game $\mathcal{G}$ between the players $1,2, \cdots ,N$ whose state is governed by the equation $\dot x = f(t,x,u_1 , \cdots ,u_N )$, where $u_i$ is a control vector belonging to player i, and suppose that each player i wishes to manipulate his $u_i$ in such a way as to maximize the functional \[J_i = K_i (t_f ,x(t_f )) + f_{t_0 }^{t_f } {L_i (t,x(t),u_i (t), \cdots ,u_N (t))dt.} \] Here $t_0$ and $t_f$. are, respectively, the times at which the game begins and ends. A strategy N-tuple $u_1 = \sigma _1^ * (t,x), \cdots ,u_N = \sigma _N^ * (t,x)$ is called an equilibrium point for $\mathcal{G}$ if the inequalities \[J_i (\sigma _1^ * , \cdots ,\sigma _N^ * ) \geqq J_i ( \cdots ,\sigma _{i - 1}^ * ,\sigma _i ,\sigma _{i + 1}^ * , \cdots )\] hold for each $i = 1, \cdots ,N$ and for each admissible strategy $u_i = \sigma _i (t,x)$ for player i. We seek methods of finding equilibrium points for the game $\mathcal{G}$.The principal results of the paper are three: a theorem to the effe...