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  "body": "\n## Definition\n\nLet $A \\subseteq \\R^{k}$ and $B \\subseteq \\R^{l}$ be nonempty compact sets and let $T \u003e 0$.\nLet $f \\colon \\R^n \\times A \\times B \\to \\R^n$, $h \\colon \\R^n \\times A \\times B \\to \\R$ and $g \\colon \\R^n \\to \\R$ be continuous.\nConsider the two-player zero-sum [[differential game|THSHGNJNPS]] with dynamics $x' = f(x, a, b)$ and payoff $\\int_t^T h(x(s), a(s), b(s)) \\, ds + g(x(T))$.\nThe player with control $a(s) \\in A$ minimises the payoff, and the player with control $b(s) \\in B$ maximises it.\nThe *upper* and *lower Hamiltonians* are, for $x, p \\in \\R^n$,\n$$H^+(x, p) = \\min_{a \\in A} \\max_{b \\in B} \\left\\{ f(x, a, b) \\cdot p + h(x, a, b) \\right\\}, \\qquad H^-(x, p) = \\max_{b \\in B} \\min_{a \\in A} \\left\\{ f(x, a, b) \\cdot p + h(x, a, b) \\right\\}.$$\nThe *Hamilton–Jacobi–Isaacs equations* are the terminal value problems\n$$V_t + H^\\pm(x, \\nabla_x V) = 0 \\quad \\text{in } (0, T) \\times \\R^n, \\qquad V(T, x) = g(x),$$\nfor an unknown function $V \\colon [0, T] \\times \\R^n \\to \\R$, one with $H^+$ and one with $H^-$.\nThe game satisfies *Isaacs' condition* if $H^+ = H^-$, and then the two equations coincide.\n\n## Remarks\n\nAlways $H^- \\le H^+$.\nIf $B$ has one point, both equal $\\min_{a} \\{ f \\cdot p + h \\}$ and the equation is the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]], a special [[Hamilton–Jacobi equation|5TRXQKP7DY]].\nAssume also that $f$, $h$ and $g$ are bounded and Lipschitz in $x$.\nThen the upper and lower values of the game, defined with nonanticipating strategies, are the [[viscosity solutions|H1ERJ3T75T]] of these equations.\nUnder Isaacs' condition the two values are equal, so the game has a value.\n\nExample: if $f(x, a, b) = f_1(x, a) + f_2(x, b)$ and $h(x, a, b) = h_1(x, a) + h_2(x, b)$, then the expression splits into a part that depends only on $a$ and a part that depends only on $b$.\nSo $\\min_a \\max_b$ and $\\max_b \\min_a$ are both the sum of $\\min_a$ of the first part and $\\max_b$ of the second, and Isaacs' condition holds.\n\nNon-example: $n = 1$, $A = B = [-1, 1]$, $f(x, a, b) = (a - b)^2$ and $h = 0$.\nFor $p = 1$, $\\max_b (a - b)^2 = (1 + \\abs{a})^2$, so $H^+(x, 1) = 1$, while $\\min_a (a - b)^2 = 0$, so $H^-(x, 1) = 0$.\nIsaacs' condition fails, and the two equations are different.\n\n## Explanations\n\n### L1 human\n\nThe equation says how the value of the game changes in time when both players play their best against each other at each instant.\n",
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