A Question of Bellman

ISSN： 1072-3374

Source： Journal of Mathematical Sciences, Vol.131, Iss.1, 2005-11, pp. : 5286-5306

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Abstract

From a random point O in an infinite strip of width 1, we move in a randomly chosen direction along a curve Г. What shape of Г gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments suggesting that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval OA of length a, its smooth continuation, an arc AB of radius 1 and small length &phis;, an interval BD that is smooth continuation of the arc, and an interval DF (with a corner at the point D). If O is the origin and OA is the x axis of a coordinate system, then the coordinates of the above-mentioned points are as follows: A(a, 0), B(a + sin &phis;, 1 − cos &phis;), F(a, 1), and $$D\left( {a + \frac{{\cos \varphi \sqrt {1 + a^2 } - a}}{{\cos \varphi - a\sin \varphi }},\;1 + \frac{{\sin \varphi \sqrt {1 + a^2 } }}{{\cos \varphi - a\sin \varphi }}} \right).$$ For the best curve, we have: a ≍ 0.814 and &phis; ≍ 0.032. Related questions are discussed. Bibliography: 9 titles.