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  "body": "\n## Summary\n\nA chattering control is a control that switches between values infinitely many times in a bounded time interval.\nIts switching times accumulate at some point.\nChattering controls can be optimal, as in [[Fuller's phenomenon|RSZS9SFFZ7]], and they also appear as approximations of [[relaxed controls|SFMD8RKRSE]].\nThey show that the [[bang–bang|GK1Z3Q1WRZ]] structure of an optimal control may involve infinitely many switches.\n\n## Explanations\n\n### L1 human\n\nImagine a heater that must be fully on or fully off, and that should bring a room to an exact temperature as well as possible.\nThe best strategy may switch on and off more and more often as the target gets close, with shorter and shorter periods.\nThe switching times pile up at one moment, like the points $1/2, 3/4, 7/8, \\ldots$ piling up at $1$.\nA control with such infinitely many switches before a finite time is called chattering.\n",
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