Vortex singularities of optimal strategies at the beginning of motion in search problems on n-dimensional Riemannian manifolds

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 two-dimensional 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 time.

Another 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.