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, β

1

,β

2

, …,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

M

Ф

N

of Ф

N

depending on the index of evaders caught there. Then, we study Ф

2

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 Ф

1

2

(Ф

2

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 Ф

2

is strategically equivalent to the associated game with the minimal value.