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  "body": "\n## Summary\n\nA singular control is an optimal control, or a candidate for one, on a time interval where the maximum principle gives no information.\nThere the maximum condition of [[Pontryagin's maximum principle|7RPV2T29CF]] does not determine the control value.\nThis happens when the [[Hamiltonian|RACVWG5QTK]] is affine in the control.\nThe coefficient of the control, called the switching function, then vanishes on a whole interval.\nOn such an interval the control usually takes interior values, unlike a [[bang–bang control|GK1Z3Q1WRZ]].\nSingular arcs are found by differentiating the switching function in time until the control appears.\n\n## Explanations\n\n### L1 human\n\nFor many problems the best control is at full power in one direction or the other, depending on the sign of a switching function.\nSometimes this switching function stays at zero for a while, and then the rule gives no answer.\nThe control on such a stretch is called singular, and it is usually some intermediate value.\nFor example, to keep a quantity at zero with a control between $-1$ and $1$, the best control after reaching zero is $0$, which is neither extreme.\n",
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