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  "body": "\n## Problem\n\nA boat moves in the plane with constant speed $V \u003e 0$ relative to the water.\nThe water moves with a velocity field $w = (w_1, w_2) \\colon \\R^2 \\to \\R^2$ of class $C^1$ with $\\abs{w} \u003c V$ everywhere.\nThe captain controls the heading $\\theta(t)$, so that the position $(x(t), y(t))$ obeys the [[ordinary differential equations|0NT64JV96A]]\n$$x' = V \\cos \\theta + w_1(x, y), \\qquad y' = V \\sin \\theta + w_2(x, y).$$\nGiven two points $P$ and $Q$, find a heading $t \\mapsto \\theta(t)$ that brings the boat from $P$ to $Q$ in the least time.\n\n## Remarks\n\nIt is a [[time-optimal control|YTR959CYMX]] problem, and the [[Pontryagin maximum principle|7RPV2T29CF]] gives necessary conditions on optimal headings.\nZermelo derived a differential equation for the optimal heading along the path, known as Zermelo's navigation formula.\nFor a constant current $w$, the optimal heading is constant and the boat moves along the straight segment from $P$ to $Q$.\nOne aims so that the sum of $V(\\cos\\theta, \\sin\\theta)$ and $w$ points from $P$ to $Q$.\nThe condition $\\abs{w} \u003c V$ ensures that the boat can make headway in every direction.\n\n## Explanations\n\n### L1 human\n\nA boat crosses a river or a sea with currents, and the captain wants to arrive as soon as possible.\nPointing straight at the target is not best, because the current pushes the boat off course.\nThe problem asks how to steer at each moment, and it is one of the first problems of optimal control.\n",
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      "first_stated": "1931",
      "posed_by": "Ernst Zermelo"
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