Approaching Coalitions of Evaders on the Average
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 Фθ 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.
KeywordsAverage Distance Playing Space Optimal Trajectory Differential Game Optimal Direction
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