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  "body": "\n## Definition\n\nLet $x' = f(x, u)$, $y = h(x)$ be a [[control system|ZPDTVZQN5K]] on $\\R^n$ with output map $h \\colon \\R^n \\to \\R^p$.\nLet $x_0, \\tilde{x}_0 \\in \\R^n$.\nFor an admissible control $u$, let $x$ and $\\tilde{x}$ be the trajectories with $x(0) = x_0$ and $\\tilde{x}(0) = \\tilde{x}_0$.\nThe states $x_0$ and $\\tilde{x}_0$ are *indistinguishable* if, for every $u$, $h(x(t)) = h(\\tilde{x}(t))$ for all $t \\ge 0$ at which both trajectories are defined.\nThe system is *observable* if any two indistinguishable initial states are equal.\n\n## Remarks\n\nConsider a linear system $x' = A x + B u$, $y = C x$.\nThe output difference of two initial states does not depend on $u$.\nThe system is observable if and only if, for any $T \u003e 0$, the output $y$ on $[0, T]$ determines $x(0)$.\nObservability of the linear system is equivalent to $\\rank \\begin{pmatrix} C \\\\ CA \\\\ \\vdots \\\\ CA^{n-1} \\end{pmatrix} = n$.\nExample: $x_1' = x_2$, $x_2' = 0$, $y = x_1$ is observable, since $y(t) = x_1(0) + t \\, x_2(0)$ gives $x_1(0) = y(0)$ and $x_2(0) = y'(0)$.\nNon-example: the same system with output $y = x_2$ is not observable, since $(0, 0)$ and $(1, 0)$ give the same output $y \\equiv 0$.\n",
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