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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^m$ be nonempty, and let $f \\colon \\R^n \\times U \\times \\R \\to \\R^n$, $L \\colon \\R^n \\times U \\times \\R \\to \\R$, $\\phi \\colon \\R \\times \\R^n \\times \\R \\times \\R^n \\to \\R$ be continuous.\nLet $c \\colon \\R^n \\times U \\times \\R \\to \\R^q$ give path constraints and $b \\colon \\R \\times \\R^n \\times \\R \\times \\R^n \\to \\R^r$ give boundary conditions.\nA *trajectory optimization problem* asks for times $t_0 \u003c t_f$, a measurable control $u \\colon [t_0, t_f] \\to U$ and an absolutely continuous state $x \\colon [t_0, t_f] \\to \\R^n$ that minimise\n$$\\phi(t_0, x(t_0), t_f, x(t_f)) + \\int_{t_0}^{t_f} L(x(t), u(t), t) \\, dt$$\nsubject to:\n\n1. the [[control system|ZPDTVZQN5K]] $x'(t) = f(x(t), u(t), t)$ for almost every $t$;\n2. the path constraints $c(x(t), u(t), t) \\le 0$ componentwise for almost every $t$;\n3. the boundary conditions $b(t_0, x(t_0), t_f, x(t_f)) = 0$.\n\n*Trajectory optimization* is the numerical solution of such problems, which yields an open-loop control and its trajectory.\n\n## Remarks\n\nThe problem is an [[optimal control|T3V6ZNAMDN]] problem in Bolza form with constraints; the word stresses the computation of one optimal trajectory.\nIndirect methods solve the conditions of [[Pontryagin's maximum principle|7RPV2T29CF]]; direct methods discretise $x$ and $u$ and solve a finite-dimensional nonlinear program.\n\nExample: $n = 2$, $x = (x_1, x_2)$, $f = (x_2, u)$, $U = \\R$, $L = u^2$, $\\phi = 0$, no path constraints, $t_0 = 0$, $t_f = 1$, and boundary conditions $x(0) = (0, 0)$, $x(1) = (1, 0)$.\nThe control $u(t) = 6 - 12t$ gives $x_2(t) = 6t - 6t^2$ and $x_1(t) = 3t^2 - 2t^3$, which meet the boundary conditions, with cost $\\int_0^1 (6 - 12t)^2 \\, dt = 12$.\nThe maximum principle forces the optimal $u$ to be affine in $t$, and this is the only affine control that meets the boundary conditions, so it is the solution.\n\nNon-example: finding a feedback law $u = \\kappa(x)$ that is good for all initial states is a synthesis problem, not a trajectory optimization problem in this sense.\n\n## Explanations\n\n### L1 human\n\nTrajectory optimization finds one best motion of a system, together with the controls that produce it, from given start and end conditions.\n",
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