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  "body": "\n## Summary\n\nThe value function of an optimal control problem gives the optimal cost as a function of the initial time and the initial state.\nIt is the central object of dynamic programming: it satisfies the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]], in the [[viscosity|H1ERJ3T75T]] sense when it is not smooth.\nIts gradient, where it exists, is the costate of [[Pontryagin's maximum principle|7RPV2T29CF]] up to sign.\nIn economics the same notion appears as the indirect utility or the optimal value of a maximization problem.\n\n## Explanations\n\n### L1 human\n\nImagine a map that shows, at each place and each time, the least cost to finish a journey from there.\nThat map is the value function.\nIf you know it, you can always choose the move that leads to the cheapest remaining cost.\nFor example, if you can walk at speed at most $1$ and pay the distance to the origin that is left at time $T$, the value at $(t, x)$ is $\\max(\\abs{x} - (T - t), 0)$.\n",
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