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  "body": "\n## Example\n\n*Fuller's problem* is the optimal control problem\n$$\\text{minimize } \\int_0^\\infty x_1(t)^2 \\, dt \\quad \\text{subject to } x_1' = x_2, \\; x_2' = u, \\; \\abs{u(t)} \\le 1, \\; x(0) = \\xi,$$\nwhere $\\xi \\in \\R^2$ is a given nonzero initial state and $u$ is a measurable control.\nFor every $\\xi \\ne 0$ the optimal control is [[bang–bang|GK1Z3Q1WRZ]] with $u = \\pm 1$.\nIt switches infinitely many times, and the switching times accumulate at a finite time $t^*$, at which the state reaches the origin.\nAfter $t^*$ the optimal control is $u = 0$ and the state stays at the origin.\nSo the optimal control is a [[chattering control|6K8Q2YERTF]].\nThis behaviour is called *Fuller's phenomenon*.\n\n## Remarks\n\nA scaling argument explains the pattern.\nIf $(x_1(t), x_2(t), u(t))$ is admissible, then so is $(\\lambda^2 x_1(t/\\lambda), \\lambda x_2(t/\\lambda), u(t/\\lambda))$ for every $\\lambda \u003e 0$, and its cost is multiplied by $\\lambda^5$.\nSo the optimal feedback is invariant under this scaling, and the time between two switches shrinks by a fixed factor at each switch.\nThe intervals then form a geometric series with finite sum, which gives the finite time $t^*$.\nThe optimal feedback has the form $u = -\\sgn(x_1 + C x_2 \\abs{x_2})$ for a certain constant $C \u003e 0$.\nNecessary conditions of [[Pontryagin's maximum principle|7RPV2T29CF]] together with this scaling are the main tools of the analysis.\n\n## Explanations\n\n### L1 human\n\nA cart on a rail is pushed by a force that can be at most one in either direction.\nThe goal is to keep the cart's position close to zero in total, measured by the integral of the squared position.\nThe best strategy pushes at full force, alternating direction, with shorter and shorter pushes.\nInfinitely many switches happen before the cart comes to rest at zero, after a finite time.\nThis shows that optimal controls can be surprisingly wild even for very simple systems.\n",
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