Approaching Coalitions of Evaders on the Average

Abstract

In the game Φ _N with simple motions, the pursuer P and the coalition E _N = E ₁, E ₂, …, E _N of evaders move in a plane with constant speeds 1, β₁,β₂, …,β _N . The average distance from a point to a set of points is defined as a weighted sum of the corresponding Euclidean distances with given positive constant weights. P strives to minimize the distance to E _N and terminates the game when the distance shortening is not guaranteed.

First, we describe several conditions that are met by the states on the terminal manifold MФ _N of ФN depending on the index of evaders caught there. Then, we study Ф₂ in detail. This game is a game of alternative pursuit since there are three different terminal sub-manifolds: P catches E ₁ (E ₂) on M ¹ _Ф2 (M ^{2 Ф2}) and all players are apart on M ^θ _Ф2. We set up and study associated games Ф¹ ₂ (Ф² ₂) and Ф^θ ₂ with the payoffs equal to the average distance to E ² at instants when the state reaches M ¹ _Ф2 (M ² _Ф2) and M ^θ _Ф2 correspondingly. It is shown that Ф₂ is strategically equivalent to the associated game with the minimal value.