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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^m$ be a nonempty set and $I \\subseteq \\R$ an interval.\nConsider an [[optimal control|T3V6ZNAMDN]] problem with [[control system|ZPDTVZQN5K]] $x' = f(t, x, u)$ and running cost $\\int L(t, x, u) \\, dt$, where $f \\colon I \\times \\R^n \\times U \\to \\R^n$ and $L \\colon I \\times \\R^n \\times U \\to \\R$.\nThe *Hamiltonian* of the problem is the function $H \\colon I \\times \\R^n \\times U \\times \\R^n \\times \\R \\to \\R$ given by\n$$H(t, x, u, p, \\lambda_0) = p \\cdot f(t, x, u) - \\lambda_0 L(t, x, u).$$\nThe vector $p \\in \\R^n$ is the costate, and the number $\\lambda_0 \\ge 0$ is the cost multiplier.\nIn the normal case $\\lambda_0 = 1$ one writes $H(t, x, u, p) = p \\cdot f(t, x, u) - L(t, x, u)$.\n\n## Remarks\n\nExample: for $x' = u$ in $\\R$ with running cost $\\int \\frac12 (x^2 + u^2) \\, dt$, the normal Hamiltonian is $H = p u - \\frac12 (x^2 + u^2)$.\nFor fixed $(x, p)$ it is a concave function of $u$ with maximum at $u = p$.\nThe state equation is recovered as $x' = \\partial H / \\partial p$.\nMany texts use the opposite sign convention $H = L + p \\cdot f$, under which optimal controls minimize $H$; this gives the minimum principle.\nThe two conventions differ by the change $p \\mapsto -p$ and an overall sign.\nA non-example is the Lagrangian $L(t, x, u)$ alone: it does not involve the costate, so it does not carry the dynamics.\n",
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