Chattering control

Remarks

The set of switching times is closed, and on each interval between consecutive switching times the control is almost everywhere constant. Since $[a, b]$ is compact, the switching times of a chattering control have an accumulation point. The condition on accumulation points excludes, for example, continuous nonconstant controls such as $u(t) = t$, for which every point is a switching time. Example: on $[0, 1]$ let $u(t) = (-1)^k$ for $1 - 2^{-k} \le t < 1 - 2^{-k-1}$, $k = 0, 1, 2, \ldots$, and $u(1) = 1$. Each 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. So $u$ is a chattering bang–bang control, and its switching times accumulate at $1$. Non-example: $u(t) = -1$ for $t < 1/2$ and $u(t) = 1$ for $t \ge 1/2$ has only one switching time, $1/2$. Some authors also use the word for a sequence of fast-switching ordinary controls that approximates a relaxed control.