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Title of Journal: J Math Sci

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Abbravation: Journal of Mathematical Sciences

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Springer US

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DOI

10.1002/jhet.5570080307

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1573-8795

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Vortex singularities of optimal strategies at the

Authors: M I Zelikin L V Lokutsievskii R A Usachev
Publish Date: 2009/06/20
Volume: 160, Issue: 2, Pages: 197-220
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Abstract

This work studies optimal strategies having unexpected singularities So in particular in the problem of searching for an object E on a segment that has the probability distribution function of location going to infinity to both side of a pursuer P the modulus of whose speed does not exceed some constant the optimal search strategy has no derivative at the initial instant of time During an arbitrary small interval of time the pursuer P changes the direction of his motion infinitely many times in order to be at both sides where the probability of location of E is maximal In the case of a twodimensional manifold with analogous singularities of the probability distribution function of location of E the velocity of P executes countably many turns during an arbitrary small interval of time for the optimal search this is the reason for which such singularities are said to be vortex singularities It is more difficult to imagine optimal strategies arising in the search on manifolds of dimension more than 2 Here during an arbitrary small initial interval of time the player P tends to completely inspect a neighborhood of the boundary of the visibility domain at the initial instant of timeAnother unexpected phenomenon in the search problem on a segment is as follows if the distribution function tends to zero at the endpoints of the segment then the player P changes the direction of motion infinitely many times when approaching the endpoints of the segment Precisely when the probability of finding E near a given endpoint of the segment becomes sufficiently small P runs to another endpoint of the segment and there not arriving at this end he turns backward and this occurs infinitely many times In this case in principle the search can be performed arbitrarily many time despite the fact that the inspection of the whole segment requires a fixed finite time However the expectation of the search time turns out to be minimal This paper finds the formulas for switching points and in the case of infinitely many switching points the asymptotics of turn points is calculated

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