On one problem of search for mobile target

Abstract

At present the problems of control in conditions of conflict and uncertainty are especially relevant. This work deals with the problem of search for a mobile target in condition when the pursuer knows only probability distribution of the target initial state. In this paper we first address the auxiliary problem of controlled object convergence given terminal set. Herewith, its motion is described by a system of nonlinear differential equations and probability distribution of its initial state. We deduce formula for the probability of bringing the trajectory of controlled object to the terminal set at fixed moment of time. In so doing, the Fokker–Planck–Kolmogorov equation is used. In the case of the linear dynamics of the controlled object this formula takes an explicit form. In the paper, we apply this formula to study the problem of search in the case of linear dynamics of the pursuing controlled object (pursuer) and the mobile target. The pursuer starts moving from a given point and strives to get close to a given distance from the target. At the time it occurs the search is considered completed. Therewith, information on current state of the target is not available to the pursuer; however he is aware of probability distribution of its initial state. We derive sufficient conditions, under which at a certain time the pursuer can achieve its goal with certain probability and deduce formula for this probability. To this end, the idea of Pontryagin’s first direct method, based on the condition of the same name, is employed. In so doing, we use Minkowski operation of geometric subtraction, the properties of the set-valued mappings, and the measurable choice theorem. It is shown that, in the case of simple motion of the target and the Gaussian probability distribution of its initial state this probability takes its maximal value at the above mentioned time. On the sake of geometric descriptiveness this is illustra­ted with the example of simple motions on the plain.