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  "body": "\n## Problem\n\nLet $U \\subseteq \\R^m$ be a nonempty set, and let $t_0 \u003c t_1$ be fixed times.\nLet $f \\colon [t_0, t_1] \\times \\R^n \\times U \\to \\R^n$ and $L \\colon [t_0, t_1] \\times \\R^n \\times U \\to \\R$ be continuous.\nLet $\\varphi \\colon \\R^n \\times \\R^n \\to \\R$ be continuous, and let $S \\subseteq \\R^n \\times \\R^n$ be a closed set of allowed endpoint pairs.\nAn [[admissible control|8M31QKB0GW]] is a measurable map $u \\colon [t_0, t_1] \\to U$ whose trajectory $x$ of the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u(t))$ is absolutely continuous on $[t_0, t_1]$ and satisfies $(x(t_0), x(t_1)) \\in S$.\nThe *Bolza problem* asks to minimize the [[functional|1EQM0DJ4NC]]\n$$J(u) = \\varphi(x(t_0), x(t_1)) + \\int_{t_0}^{t_1} L(t, x(t), u(t)) \\, dt$$\nover all admissible controls $u$.\n\n## Remarks\n\nThe cost has two parts: an endpoint term $\\varphi$ and a running term given by the integral of $L$.\nThe case $L = 0$ is the Mayer problem, and the case $\\varphi = 0$ is the Lagrange problem; see [[Mayer and Lagrange problems|1J8TYT7C49]].\nThe three forms are equivalent: adding the state $y' = L(t, x, u)$ with $y(t_0) = 0$ turns the Bolza cost into the endpoint cost $\\varphi(x(t_0), x(t_1)) + y(t_1)$.\nIn the classical calculus of variations the control is the velocity, $f(t, x, u) = u$, and the problem is to minimize $\\varphi(x(t_0), x(t_1)) + \\int_{t_0}^{t_1} L(t, x, x') \\, dt$ with possible side conditions.\nNecessary conditions for a minimizer are given by [[Pontryagin's maximum principle|7RPV2T29CF]], with transversality conditions coming from $\\varphi$ and $S$.\nExistence of a minimizer needs extra hypotheses, such as compactness of $U$ and convexity of the set of velocities and costs; see [[Filippov's existence theorem|XEX7QZV1BS]].\n\n## Explanations\n\n### L1 human\n\nA Bolza problem asks for the best way to steer a system when one pays both along the way and at the end.\nThe running cost might be fuel spent, and the end cost might be a penalty for missing a target.\nMany optimal control problems in engineering and economics take this form.\n",
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