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  "body": "\n## Definition\n\nLet $x' = f(x, u)$ be a [[control system|SBQVGQPXT1]] on $\\R^n$ with control set $U \\subseteq \\R^m$.\nFix a time horizon $T \u003e 0$, an initial state $x_0 \\in \\R^n$ and a closed target set $S \\subseteq \\R^n$.\nLet $L \\colon \\R^n \\times U \\to \\R$ and $\\varphi \\colon \\R^n \\to \\R$ be [[continuous functions|0X3WGX95FE]], the running cost and the terminal cost.\nFor an [[admissible control|8M31QKB0GW]] $u$ for $(x_0, T, S)$ with trajectory $x_u$, the *cost* is\n$$J(u) = \\varphi\\big(x_u(T)\\big) + \\int_0^T L\\big(x_u(t), u(t)\\big) \\, dt.$$\nAn admissible control $u^*$ is an *optimal control* if $J(u^*) \\le J(u)$ for every admissible control $u$ for $(x_0, T, S)$.\nThe pair $(u^*, x_{u^*})$ is then an optimal pair, and $x_{u^*}$ is an optimal trajectory.\n\n## Remarks\n\nThe integral exists, because $x_u$ is continuous and $u$ is essentially bounded, so the integrand is a bounded measurable function.\nA cost with both terms is in Bolza form; $L = 0$ gives the Mayer form and $\\varphi = 0$ the Lagrange form.\nAn optimal control need not exist, and when it exists it need not be unique.\n\nExample: $x' = u$ on $\\R$ with $U = [-1, 1]$, $x_0 = 0$, $T = 1$, $S = \\R$, $L = 0$ and $\\varphi(x) = -x$.\nEvery admissible $u$ gives $x_u(1) = \\int_0^1 u(t) \\, dt \\le 1$, so $J(u) \\ge -1$.\nThe control $u^* = 1$ gives $J(u^*) = -1$, so it is optimal.\n\nNon-example: the control $u = 0$ in the same problem is admissible but not optimal, since $J(0) = 0 \u003e -1 = J(u^*)$.\n",
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