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- W2084063650 abstract "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 effect that the “value functions” are in a certain sense everywhere differentiable, a system of partial differential equations which they must satisfy and a sufficiency result which guarantees that, under certain circumstances, the “Dynamic Programming” method suggested by the preceding differential equations does indeed yield the equilibrium points of the game. In the final two sections, a significant class of games is solved explicitly, and a method of characteristics is given for the solution of the “Hamilton–Jacobi” partial differential equations." @default.
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- W2084063650 date "1969-05-01" @default.
- W2084063650 modified "2023-10-11" @default.
- W2084063650 title "Toward a Theory of Many Player Differential Games" @default.
- W2084063650 cites W2038021222 @default.
- W2084063650 doi "https://doi.org/10.1137/0307013" @default.
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