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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^m$ be a nonempty set, and let $f \\colon \\R^n \\times U \\to \\R^n$ and $L \\colon \\R^n \\times U \\to \\R$ be continuous, with partial derivatives $\\partial f / \\partial x$ and $\\partial L / \\partial x$ that are continuous on $\\R^n \\times U$.\nLet $T \u003e 0$ and $x_0, x_1 \\in \\R^n$ be fixed.\nConsider the optimal control problem: minimize\n$$J(u) = \\int_0^T L(x(t), u(t)) \\, dt$$\nover the measurable bounded controls $u \\colon [0, T] \\to U$ whose trajectory of the [[control system|ZPDTVZQN5K]] $x' = f(x, u)$ with $x(0) = x_0$ satisfies $x(T) = x_1$.\nFor $\\lambda_0 \\in \\R$ and $p \\in \\R^n$, let $H(x, u, p, \\lambda_0) = p \\cdot f(x, u) - \\lambda_0 L(x, u)$.\nLet $u^*$ be an optimal control with trajectory $x^*$.\nThen there are a number $\\lambda_0 \\in \\set{0, 1}$ and an [[absolutely continuous|ZT6F4C2YHK]] function $p \\colon [0, T] \\to \\R^n$, with $(\\lambda_0, p(t)) \\ne (0, 0)$ for all $t$, such that:\n\n1. $p'(t) = -\\dfrac{\\partial H}{\\partial x}(x^*(t), u^*(t), p(t), \\lambda_0)$ for almost every $t \\in [0, T]$;\n2. $H(x^*(t), u^*(t), p(t), \\lambda_0) = \\max_{v \\in U} H(x^*(t), v, p(t), \\lambda_0)$ for almost every $t \\in [0, T]$;\n3. the function $t \\mapsto H(x^*(t), u^*(t), p(t), \\lambda_0)$ is equal almost everywhere to a constant.\n\n## Remarks\n\nThe proof compares the optimal control with needle variations, which replace $u^*$ by a constant value $v \\in U$ on a short interval.\nThe first-order effects of these variations on the endpoint and the cost form a convex cone.\nOptimality means that this cone does not meet a certain direction.\nA separating hyperplane gives $(\\lambda_0, p(T))$, and $p$ is then transported backward by the adjoint equation of condition 1.\nExample: minimize $\\int_0^1 \\frac{1}{2} u^2 \\, dt$ for $x' = u$, $x(0) = 0$, $x(1) = 1$, $U = \\R$.\nHere $H = p u - \\lambda_0 u^2 / 2$ and $p' = 0$, so $p$ is constant; if $\\lambda_0 = 0$, condition 2 forces $p = 0$, which is excluded, so $\\lambda_0 = 1$ and $u = p$ is constant, which gives $u \\equiv 1$ and $x(t) = t$.\nThe theorem gives necessary conditions only: an extremal that satisfies them need not be optimal.\nThe case $\\lambda_0 = 0$, called abnormal, can occur, and then the conditions do not involve the cost at all.\n\n## Explanations\n\n### L1 human\n\nTo steer a system in the cheapest way, one cannot test every possible control.\nThe maximum principle says that an optimal control must, at each moment, maximize a single function $H$ over the allowed control values.\nThe function $H$ combines the dynamics and the cost with a vector $p(t)$ of shadow prices that solves its own differential equation.\nThis turns an optimization over functions into a pointwise maximization plus a boundary value problem.\n\n### L2 human\n\nThe maximum principle extends the Euler–Lagrange equations of the calculus of variations to controls with values in a constraint set $U$, such as $[-1, 1]$.\nWhen $H$ is linear in $u$ and $U$ is a box, condition 2 forces $u$ to take extreme values, which gives [[bang–bang controls|GK1Z3Q1WRZ]] except on singular arcs.\nThe state equation $x' = \\partial H / \\partial p$ and the adjoint equation form a Hamiltonian system with boundary conditions at both ends, usually solved by shooting methods.\nVariants with free final time add the condition that the constant in condition 3 is zero.\nVariants with a free endpoint and no terminal cost add the transversality condition $p(T) = 0$.\n",
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