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  "body": "\n## Definition\n\nLet $x' = f(x, u)$ be a [[control system|ZPDTVZQN5K]] whose control set $U \\subseteq \\R^m$ is a compact [[convex set|02AS45T3H9]], for example a box $\\prod_{i=1}^m [a_i, b_i]$ with $a_i \u003c b_i$.\nAn admissible control $u \\colon [0, T] \\to U$ is *bang–bang* if $u(t)$ is an extreme point of $U$ for almost every $t \\in [0, T]$.\nFor a box, this means $u_i(t) \\in \\set{a_i, b_i}$ for each $i$ and almost every $t$.\nThe times at which $u$ jumps between extreme points are its switching times.\n\n## Remarks\n\nExample: for $U = [-1, 1]$, the control $u(t) = 1$ on $[0, 1)$ and $u(t) = -1$ on $[1, 2]$ is bang–bang, with one switching time $t = 1$.\nFor the double integrator $x_1' = x_2$, $x_2' = u$ starting at rest at $x_1 = -1$, it brings the state to rest at $x_1 = 0$ at time $2$, and this is the minimum time with $\\abs{u} \\le 1$.\nNon-example: $u(t) = \\sin t$ on $[0, \\pi]$ with $U = [-1, 1]$ is not bang–bang, since $\\abs{u(t)} \u003c 1$ except at $t = \\pi / 2$.\nConsider a linear system $x' = A x + B u$ whose control set $U$ is a compact convex polytope.\nThe bang–bang principle says that every state reachable at time $T$ is also reachable at time $T$ by a bang–bang control.\n",
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