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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^m$ be a nonempty set, let $a \u003c b$, and let $u \\colon [a, b] \\to U$ be a measurable control.\nA *switching time* of $u$ is a point $s \\in [a, b]$ with this property.\nOn no neighbourhood of $s$ in $[a, b]$ is $u$ almost everywhere equal to a constant.\nThe control $u$ is *chattering* if its set of switching times is infinite but has only finitely many accumulation points.\n\n## Remarks\n\nThe set of switching times is closed, and on each interval between consecutive switching times the control is almost everywhere constant.\nSince $[a, b]$ is compact, the switching times of a chattering control have an accumulation point.\nThe condition on accumulation points excludes, for example, continuous nonconstant controls such as $u(t) = t$, for which every point is a switching time.\nExample: on $[0, 1]$ let $u(t) = (-1)^k$ for $1 - 2^{-k} \\le t \u003c 1 - 2^{-k-1}$, $k = 0, 1, 2, \\ldots$, and $u(1) = 1$.\nEach point $1 - 2^{-k}$ with $k \\ge 1$ is a switching time, since $u$ takes the value $1$ on one side and $-1$ on the other.\nSo $u$ is a chattering [[bang–bang control|GK1Z3Q1WRZ]], and its switching times accumulate at $1$.\nNon-example: $u(t) = -1$ for $t \u003c 1/2$ and $u(t) = 1$ for $t \\ge 1/2$ has only one switching time, $1/2$.\nSome authors also use the word for a sequence of fast-switching ordinary controls that approximates a relaxed control.\n",
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