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  "body": "\n## Summary\n\nThe Hamilton–Jacobi–Isaacs equation is the partial differential equation for the value of a two-player zero-sum [[differential game|THSHGNJNPS]].\nIt is the analogue for games of the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] of optimal control, with a min-max in place of a minimum.\nUnder standard hypotheses the upper and lower values of the game are its [[viscosity solutions|H1ERJ3T75T]].\nWhen the min and the max commute, the game has a value and the equation characterises it.\n\n## Explanations\n\n### L1 human\n\nIn a game between two players with opposite goals, each one reacts to the other at every instant.\nThe value of the game, as a function of time and position, satisfies one equation.\nIn it, the minimiser and the maximiser each make their best choice against the other at the present instant.\nFor example, if each player controls its own part of the motion and pays its own part of the cost, the order of the two choices does not matter.\nThen the equation has a single form, and solving it gives both the value and the best feedback of each player.\n",
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