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  "body": "\n## Definition\n\nLet $T \u003e 0$, $x_0, x_1 \\in \\R^n$, let $U \\subseteq \\R^m$ be nonempty, and let $f \\colon \\R^n \\times U \\to \\R^n$ and $L \\colon \\R^n \\times U \\to \\R$ be continuously differentiable in $x$.\nConsider the problem of minimising $\\int_0^T L(x, u) \\, dt$ subject to $x' = f(x, u)$, $x(0) = x_0$, $x(T) = x_1$.\nLet $H(x, u, p) = p \\cdot f(x, u) - L(x, u)$ be the [[Hamiltonian|RACVWG5QTK]] in the normal case of [[Pontryagin's maximum principle|7RPV2T29CF]].\nAssume that for all $(x, p)$ the function $u \\mapsto H(x, u, p)$ has a unique maximiser $u^*(x, p)$ in $U$, and that $u^*$ is continuously differentiable.\nFor $p_0 \\in \\R^n$, let $(x(\\cdot\\,; p_0), p(\\cdot\\,; p_0))$ be the solution on $[0, T]$, if it exists, of\n$$x' = f(x, u^*(x, p)), \\qquad p' = -\\frac{\\partial H}{\\partial x}(x, u^*(x, p), p), \\qquad x(0) = x_0, \\qquad p(0) = p_0.$$\nThe *shooting function* is $S(p_0) = x(T; p_0) - x_1$.\nThe *indirect method by shooting* computes a zero $p_0^*$ of $S$, for example by [[Newton's method|07EXVNF9VP]], and returns the control $u(t) = u^*(x(t; p_0^*), p(t; p_0^*))$.\n\n## Remarks\n\nThe second equation is the [[costate equation|WS9MRAZMFW]].\nEvery normal extremal of the maximum principle with these endpoints comes from a zero of $S$, so the method computes candidates for optimality, not proven minimisers.\nThe method is a special case of the [[shooting method|ECT6RNF865]], with unknowns $p_0$ instead of an initial slope.\nFree final time or other boundary conditions add unknowns and equations to $S$, from the transversality conditions.\nMultiple shooting splits $[0, T]$ into pieces to reduce the sensitivity of $S$.\n\nExample: $n = m = 1$, $T = 1$, $f = u$, $L = u^2 / 2$, $U = \\R$, $x_0 = 0$ and $x_1 = 1$.\nHere $H = p u - u^2 / 2$, so $u^* = p$, and $p' = 0$.\nHence $p(t) = p_0$, $x(t) = p_0 t$ and $S(p_0) = p_0 - 1$.\nThe zero is $p_0^* = 1$, which gives the optimal control $u = 1$.\n\nNon-example: if $U = [-1, 1]$ and $L = 0$ with $f = u$, the maximiser of $H = p u$ is not unique when $p = 0$ and is not differentiable in $p$.\nThen the map $u^*$ of the definition does not exist, and the method must be adapted to the bang-bang structure.\n\n## Explanations\n\n### L1 human\n\nThe method guesses the initial costate, integrates the optimality conditions forward, and corrects the guess until the trajectory ends where it must.\n",
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