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  "body": "\n## Definition\n\nConsider a [[trajectory optimization|58895A7T4H]] problem on $[t_0, t_f]$ with dynamics $x' = f(x, u, t)$, cost $\\phi(x(t_f)) + \\int_{t_0}^{t_f} L(x, u, t) \\, dt$, and boundary conditions.\nMap the interval to $[-1, 1]$ by $t = \\frac{t_f - t_0}{2} \\tau + \\frac{t_f + t_0}{2}$.\nLet $N \\ge 2$ and let $-1 = \\tau_0 \u003c \\tau_1 \u003c \\cdots \u003c \\tau_N = 1$ be the Legendre–Gauss–Lobatto nodes, where $\\tau_1, \\dots, \\tau_{N-1}$ are the zeros of $P_N'$ and $P_N$ is the [[Legendre polynomial|QFYGJVA996]] of degree $N$.\nLet $w_0, \\dots, w_N$ be the corresponding quadrature weights, and let $\\ell_0, \\dots, \\ell_N$ be the [[Lagrange basis polynomials|237KR6FRCK]] of the nodes.\nLet $D$ be the $(N + 1) \\times (N + 1)$ matrix with $D_{kj} = \\ell_j'(\\tau_k)$.\nThe *Legendre pseudospectral method* replaces the problem by the nonlinear program with unknowns $x_0, \\dots, x_N \\in \\R^n$ and $u_0, \\dots, u_N \\in \\R^m$:\n\n1. minimise $\\phi(x_N) + \\frac{t_f - t_0}{2} \\sum_{k=0}^N w_k L(x_k, u_k, t(\\tau_k))$;\n2. subject to $\\sum_{j=0}^N D_{kj} x_j = \\frac{t_f - t_0}{2} f(x_k, u_k, t(\\tau_k))$ for $k = 0, \\dots, N$;\n3. and to the boundary conditions on $x_0$ and $x_N$ and the path constraints at the nodes.\n\nThe state is then approximated by $x(\\tau) \\approx \\sum_j x_j \\ell_j(\\tau)$.\n*Pseudospectral optimal control* is the family of such methods, with Legendre, Chebyshev or other Gaussian nodes.\n\n## Remarks\n\nIt is [[direct collocation|T58W34D60G]] with one interval and the nodes of a Gaussian rule as collocation points.\nThe Lobatto rule with $N + 1$ nodes integrates polynomials of degree up to $2N - 1$ exactly.\nVariants use Legendre–Gauss nodes, which omit both endpoints, or Legendre–Gauss–Radau nodes, which omit one endpoint.\n\nExample: $N = 2$ gives $P_2'(\\tau) = 3\\tau$, so the nodes are $-1, 0, 1$, with weights $\\frac{1}{3}, \\frac{4}{3}, \\frac{1}{3}$ (Simpson's rule).\nThe differentiation matrix is\n$$D = \\begin{pmatrix} -3/2 \u0026 2 \u0026 -1/2 \\\\ -1/2 \u0026 0 \u0026 1/2 \\\\ 1/2 \u0026 -2 \u0026 3/2 \\end{pmatrix}.$$\nFor instance, the first row gives the derivative at $-1$ of the parabola through $(-1, a)$, $(0, b)$, $(1, c)$, which is $(-3a + 4b - c) / 2$.\n\nNon-example: collocation at equally spaced nodes with one global polynomial of high degree is not a pseudospectral method, and it suffers from the Runge phenomenon.\n\n## Explanations\n\n### L1 human\n\nThe method approximates the whole trajectory by one polynomial through special nodes and enforces the dynamics at those nodes.\n",
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