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  "body": "\n## Definition\n\nLet $N \\ge 2$ and $T \u003e 0$.\nFor each player $i = 1, \\dots, N$, let $U_i \\subseteq \\R^{m_i}$ be a nonempty compact set.\nLet $f \\colon [0, T] \\times \\R^n \\times U_1 \\times \\cdots \\times U_N \\to \\R^n$ be continuous and [[Lipschitz|0QNVQ4VD9T]] in its second variable.\nFor each $i$, let $L_i \\colon [0, T] \\times \\R^n \\times U_1 \\times \\cdots \\times U_N \\to \\R$ and $g_i \\colon \\R^n \\to \\R$ be continuous.\nA *differential game* on $[0, T]$ with initial state $x_0 \\in \\R^n$ consists of these data, used as follows.\n\n1. Each player $i$ chooses a measurable control $u_i \\colon [0, T] \\to U_i$.\n2. The controls drive the state by the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u_1(t), \\dots, u_N(t))$ with $x(0) = x_0$.\n3. Player $i$ wants to minimise its cost $J_i(u_1, \\dots, u_N) = \\int_0^T L_i(t, x(t), u_1(t), \\dots, u_N(t)) \\, dt + g_i(x(T))$.\n\nThe game is *two-player zero-sum* if $N = 2$ and $J_2 = -J_1$; then player $1$ minimises $J = J_1$ and player $2$ maximises it.\n\n## Remarks\n\nA full definition also says what information each player uses.\nIn open-loop play each player picks a control as a function of time alone.\nIn feedback play each player picks $u_i(t)$ as a function of $(t, x(t))$.\nIn the zero-sum case one often uses nonanticipating strategies, which choose the present control from the opponent's past controls.\nA solution concept is a Nash equilibrium, a tuple of strategies from which no player gains by deviating alone.\nIn the zero-sum case this is a [[saddle point|0JW6MDHAZD]] of $J$, and its value is the value of the game.\n\nExample: $n = 2$, $U_1$ the closed disc of radius $2$, $U_2$ the closed unit disc, $f(t, x, u_1, u_2) = u_2 - u_1$, $L_1 = 0$ and $g_1(x) = \\abs{x}$, with $g_2 = -g_1$.\nHere $x$ is the position of player $2$ relative to player $1$, and player $1$ wants to end close to player $2$.\nAll clauses hold: $f$ is continuous and does not depend on $x$, and the sets $U_i$ are compact.\n\nNon-example: in a game where the state changes only at the times $0, 1, 2, \\dots$ by a recurrence $x_{k+1} = F(x_k, u_{1,k}, u_{2,k})$, clause 2 fails.\nIt is a dynamic game in discrete time, not a differential game.\n\n## Explanations\n\n### L1 human\n\nSeveral players steer the same system at the same time, each with its own goal.\nThe rules of motion are a differential equation, so the game is played in continuous time.\n",
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