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  "body": "\n## Definition\n\nLet $a \u003c b$, $T \u003e 0$, and let $f_0, f_1 \\colon \\R^n \\to \\R^n$ and $L_0, L_1 \\colon \\R^n \\to \\R$ be of class $C^1$.\nConsider the [[control system|ZPDTVZQN5K]] $x' = f_0(x) + u f_1(x)$ with scalar control $u(t) \\in [a, b]$ and running cost $\\int_0^T (L_0(x) + u L_1(x)) \\, dt$.\nLet $(x, u, p, \\lambda_0)$ be an extremal, that is, a quadruple that satisfies the conclusions of [[Pontryagin's maximum principle|7RPV2T29CF]].\nThe [[Hamiltonian|RACVWG5QTK]] along it is $H = H_0 + u \\Phi$ with $H_0 = p \\cdot f_0(x) - \\lambda_0 L_0(x)$, where the *switching function* is\n$$\\Phi(t) = p(t) \\cdot f_1(x(t)) - \\lambda_0 L_1(x(t)).$$\nThe control $u$ is *singular* on an interval $J \\subseteq [0, T]$ of positive length if $\\Phi(t) = 0$ for all $t \\in J$.\nAn extremal arc on such an interval $J$ is a *singular arc*.\n\n## Remarks\n\nWhere $\\Phi(t) \u003e 0$ the maximum condition forces $u(t) = b$, and where $\\Phi(t) \u003c 0$ it forces $u(t) = a$.\nWhere $\\Phi$ vanishes only at isolated times, the control is [[bang–bang|GK1Z3Q1WRZ]].\nExample: minimize $\\int_0^2 x^2 \\, dt$ for $x' = u$, $x(0) = 1$, $u \\in [-1, 1]$, with free final state.\nThe optimal control is $u = -1$ on $[0, 1]$ and $u = 0$ on $[1, 2]$, with $x(t) = 1 - t$ on $[0, 1]$ and $x = 0$ on $[1, 2]$.\nHere $H = p u - x^2$, $\\Phi = p$, and the [[costate equation|WS9MRAZMFW]] is $p' = 2x$ with $p(2) = 0$.\nSo $p = 0$ on $[1, 2]$, and the control is singular there, with the interior value $u = 0$.\nNon-example: in the same problem, the control is not singular on $[0, 1)$, since $p(t) = -(1 - t)^2 \u003c 0$ there.\nTo find a singular control, one differentiates $\\Phi$ in time until $u$ appears and solves for $u$.\n",
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