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  "body": "\n## Definition\n\nLet $x' = f(x, u)$ be a [[control system|ZPDTVZQN5K]] on $\\R^n$ with control set $U$.\nLet $x_0, x_1 \\in \\R^n$ and $T \u003e 0$.\nThe state $x_1$ is *reachable* from $x_0$ in time $T$ if some admissible control $u$ on $[0, T]$ has a trajectory with $x(0) = x_0$ and $x(T) = x_1$.\nThe system is *controllable* if for all $x_0, x_1 \\in \\R^n$ there is $T \u003e 0$ such that $x_1$ is reachable from $x_0$ in time $T$.\n\n## Remarks\n\nFor a linear system $x' = A x + B u$ with $U = \\R^m$, controllability in some time implies controllability in every time $T \u003e 0$.\nExample: the double integrator $x_1' = x_2$, $x_2' = u$ with $U = \\R$ is controllable.\nGiven $(a_0, b_0)$, $(a_1, b_1)$ and $T \u003e 0$, take the cubic polynomial $p$ with $p(0) = a_0$, $p'(0) = b_0$, $p(T) = a_1$, $p'(T) = b_1$; then $u = p''$ gives the trajectory $(x_1, x_2) = (p, p')$, which joins the two states.\nNon-example: the system $x_1' = u$, $x_2' = 0$ is not controllable, because $x_2$ stays constant along every trajectory, so $(0, 1)$ is not reachable from $(0, 0)$.\nWeaker notions are local controllability near an equilibrium and small-time local controllability.\n",
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