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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^m$ be a nonempty [[compact set|0R6EWFKRCF]] and $T \u003e 0$.\nLet $f \\colon [0, T] \\times \\R^n \\times U \\to \\R^n$ be continuous and Lipschitz in $x$ uniformly in $(t, u)$.\nLet $L \\colon [0, T] \\times \\R^n \\times U \\to \\R$ and $g \\colon \\R^n \\to \\R$ be continuous.\nFor $(t, x) \\in [0, T] \\times \\R^n$ and a measurable control $u \\colon [t, T] \\to U$, let $y$ be the trajectory of the [[control system|ZPDTVZQN5K]] $y' = f(s, y, u)$ with $y(t) = x$, and set\n$$J(t, x; u) = \\int_t^T L(s, y(s), u(s)) \\, ds + g(y(T)).$$\nThe *value function* of this [[optimal control|T3V6ZNAMDN]] problem is the function $V \\colon [0, T] \\times \\R^n \\to \\R$ given by\n$$V(t, x) = \\inf \\set{ J(t, x; u) : u \\colon [t, T] \\to U \\text{ measurable} }.$$\n\n## Remarks\n\nUnder these hypotheses the trajectories from $(t, x)$ stay in a bounded set, so $V$ is finite.\nBy definition $V(T, x) = g(x)$, and a control $u$ is optimal from $(t, x)$ exactly when $J(t, x; u) = V(t, x)$.\nExample: $n = m = 1$, $U = [-1, 1]$, $f(t, x, u) = u$, $L = 0$ and $g(x) = \\abs{x}$.\nFrom $(t, x)$ one can move toward $0$ at speed at most $1$, so $V(t, x) = \\max(\\abs{x} - (T - t), 0)$.\nThis $V$ is Lipschitz but not differentiable where $\\abs{x} = T - t$ and $t \u003c T$.\nNon-example: the cost $J(t, x; u_0)$ of one fixed control $u_0$ is in general not the value function.\nFor $u_0 = 0$ in the example it is $\\abs{x}$, which is larger than $V(t, x)$ when $x \\ne 0$ and $t \u003c T$.\nFor maximization problems the infimum is replaced by a supremum.\nFor infinite-horizon problems with discounting, $V$ depends only on the initial state.\n",
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