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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^m$ be a nonempty [[compact set|0R6EWFKRCF]], let $T \u003e 0$, and let $f \\colon [0, T] \\times \\R^n \\times U \\to \\R^n$ be continuous.\nLet $x_0 \\in \\R^n$, let $S \\subseteq \\R^n$ be a closed set, and let $\\varphi \\colon \\R^n \\to \\R$ be continuous.\nCall a measurable map $u \\colon [0, T] \\to U$ an [[admissible control|8M31QKB0GW]] if the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u(t))$ with $x(0) = x_0$ has an absolutely continuous solution $x$ on $[0, T]$ with $x(T) \\in S$.\nAssume that:\n\n1. at least one admissible control exists;\n2. there is a constant $R$ such that $\\abs{x(t)} \\le R$ for every $t \\in [0, T]$ and every trajectory $x$ of an admissible control;\n3. for every $(t, x) \\in [0, T] \\times \\R^n$ the set of velocities $f(t, x, U) = \\set{f(t, x, v) : v \\in U}$ is [[convex|02AS45T3H9]].\n\nThen the [[Mayer problem|1J8TYT7C49]] of minimizing $\\varphi(x(T))$ over all admissible controls has a solution.\nThat is, there is an admissible control $u^*$ with trajectory $x^*$ such that $\\varphi(x^*(T)) \\le \\varphi(x(T))$ for every admissible trajectory $x$.\n\n## Remarks\n\nThe proof is a compactness argument.\nTake a minimizing sequence of trajectories; they are uniformly bounded with uniformly bounded derivatives, so the Arzelà–Ascoli theorem gives a subsequence that converges uniformly to an absolutely continuous limit $x^*$.\nThe convexity of the velocity sets shows that $x^*$ solves the [[differential inclusion|9ZSMF6NYKM]] $x'(t) \\in f(t, x(t), U)$ for almost every $t$.\nA measurable selection result, known as Filippov's lemma, then gives a measurable control $u^*$ with $x^{*\\prime}(t) = f(t, x^*(t), u^*(t))$, and continuity of $\\varphi$ shows that $u^*$ is optimal.\n\nConvexity cannot be dropped.\nTake $n = 2$, $U = \\set{-1, 1}$, $f(t, x, u) = (u, x_1^2)$, $x_0 = (0, 0)$, $T = 1$, $S = \\set{0} \\times \\R$ and $\\varphi(x) = x_2$, so that the cost is $\\int_0^1 x_1(t)^2 \\, dt$ with $x_1' = \\pm 1$ and $x_1(1) = 0$.\nControls that switch faster and faster keep $x_1$ close to $0$, so the infimum of the cost is $0$.\nA cost $0$ would need $x_1 \\equiv 0$, hence $x_1' = 0 \\notin \\set{-1, 1}$, so no optimal control exists.\nHere the velocity sets $\\set{(-1, x_1^2), (1, x_1^2)}$ are not convex.\n\nA Lagrange cost $\\int_0^T L(t, x, u) \\, dt$ is handled by adding the state $y' = L(t, x, u)$; the convexity hypothesis then concerns the sets $\\set{(f(t, x, v), L(t, x, v) + \\gamma) : v \\in U, \\gamma \\ge 0}$.\nHypothesis 2 holds, for example, when $\\abs{f(t, x, v)} \\le C(1 + \\abs{x})$ for a constant $C$, by Grönwall's inequality.\n\n## Explanations\n\n### L1 human\n\nAn optimal control problem can fail to have a best solution, because cheaper and cheaper controls may wiggle faster and faster without reaching a limit control.\nFilippov's theorem says that this cannot happen when the control values form a compact set and, at each state, the possible velocities form a convex set.\nThen a minimizing sequence settles down to a genuine optimal control.\n\n### L2 human\n\nThe theorem is the optimal control version of the direct method of the calculus of variations.\nCompactness comes from bounded velocities, and convexity of the velocity sets plays the role of lower semicontinuity.\nWhen convexity fails, one passes to relaxed controls, which convexify the velocity sets.\nThe relaxed problem then has a minimizer, and under mild extra hypotheses its value is the infimum of the original problem.\nExistence is the first step before using necessary conditions such as [[Pontryagin's maximum principle|7RPV2T29CF]], which only describe minimizers that exist.\nFor linear systems $x' = A x + B u$ with $U$ compact and convex, the velocity sets are automatically convex.\n",
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