EFFECTIVE PURSUIT STRATEGIES IN SIMPLE MOTION DIFFERENTIAL GAME OF THREE PURSUERS AND ONE EVADER

Abstract: We consider a simple motion pursuit differential game of three pursuers and one evader with equal dynamic possibilities in the plane. Geometric constraints are imposed on the controls of players and the control set of each player is unit circle centered at the origin. Pursuit is said to be completed if the state of at least one of the pursuers coincides with that of evader. In this paper, we construct a new effective strategy for the pursuers. We prove that new strategy has advantage over the P-strategy.

In the present paper, we consider a simple motion pursuit differential game of three pursuers and one evader in the plane.Control parameters of players are subjected to geometric constraints.More precisely, the control sets of players are unit circle centered at the origin.We construct new effective strategies for the pursuers and show its advantage over the P-strategy.

Statement of the Problem
Consider a differential game of three pursuers and one evader described by the following equations (2.1) where x i , u i , y, v, ∈ R 2 , u i is the control parameter of the pursuer x i , v is that of the evader y, x i0 = y 0 , i = 1, 2, 3.
has a unique solution (x i (t), y(t)) with absolutely continuous components x i (t) and y(t).
If x i (τ ) = y(τ ) for some i ∈ {1, 2, 3} at some τ > 0, then pursuit is said to be completed.Purpose of the pursuers is to complete the pursuit, that of the evader is to avoid from capturing.
Problem 1. Construct more effective strategy for the pursuers than Pstrategy to complete the pursuit.

Construction Strategies for the Pursuers
It is known that if y 0 / ∈ int conv{x 10 , x 20 , x 30 }, then evasion is possible (see, [20]).Therefore, we assume that y 0 ∈ int conv{x 10 , x 20 , x 30 }.In this case, pursuit can be completed [20].Note that P-strategy was used by pursuers to complete the game in that paper.In the present paper, we suggest another new effective strategy for pursuers that is more effective than P-strategy to complete the game.Let x i = x i (t), y = y(t) be current positions of the players, and let Clearly, m i is midpoint of segment with endpoints x i and y.Let ), and y i = (y 1 , y 2 ).Coordinates of the point m i are Any point z = (ξ, η) of the straight line that passes through the point m i and orthogonal to e i satisfies the equation Let a 12 , a 23 , a 31 be intersection points of straight lines (2.3 1 ) and (2.3 2 ), (2.3 2 ) and (2.3 3 ), (2.3 3 ) and (2.3 1 ), respectively (Diag.1).Next, find coordinates of the points a 12 , a 23 , a 31 .Since a ij , (i, j) = (1, 2), (2, 3), (3,1), is the intersection point of the lines (2.3i) and (2.3j), then for the coordinates (ξ ij , η ij ) of the point a ij , we have .
Note that e i ⊥p i,i−1 , e i ⊥p i,i+1 .
Next, construct strategies of pursuers.Set, ) Constructed strategy needs some comments.To imagine when the inequality (v, p i,i−1 ) > (e i,i−1 , p i,i−1 ) in (3.1) may occur, consider the case i = 1.Then this inequality takes the form (v, p 13 ) > (e 13 , p 13 ).
It may occur only if v goes out from the angle between the rays [ya 31 ) and [yb 31 ) shown in the Diag.3, where the ray [yb 31 ) parallel to [x 1 a 31 ).It should be noted, however, the fact that v belongs to this angle doesn't guarantee that the inequality (v, p 13 ) > (e 13 , p 13 ) is satisfied.In case this inequality fails to hold, according to (3.1), the pursuer x 1 applies P-strategy.If v doesn't belong to these angles shown in the Diag.3, the pursuer x 1 applies P-strategy as well.

Advantage of Constructed Strategy
We compare the constructed strategy with the P-strategy and show advantage of the new strategy.Consider the rate of decrease of the expression .
Compare the value of the quantity (v, e i ) − (u i , e i ) corresponding to the strategy (3.1) and P-strategy.Let u i (t) be defined by (3.1).If u i (t) is defined by formula (see the third line of (3.1)) then, clearly, (v, e i ) − (u i , e i ) is the same as the value corresponding to Pstrategy.
When the pursuer x i uses P-strategy We compare (3.2) and (3.3), and show that right hand side of (3.2) is greater than that of (3.3).To this end it suffices to show that Indeed, square both sides of (3.4) to obtain or equivalently (e i,i−1 , p i,i−1 ) < (v, p i,i−1 ), which is true according to the inequality in the first line of (3.1).
Thus, when the pursuer x i uses (3.1), the expression α(t) decreases faster than x i uses P-strategy.
In a similar fashion, it can be shown that when x i uses u i (t) = e i,i+1 , (v, p i,i+1 ) > (e i,i+1 , p i,i+1 ), the expression α(t) decreases faster than x i uses P-strategy.Therefore, the strategy (3.1) has an advantage over the P-strategy.
It is important to point out that when pursuers use P-strategy, the best control for the evader is as follows.Pass a straight line through the point y parallel the longest side of triangle with vertices a 12 , a 23 , a 31 .Let [sw] be a segment of this line between the sides of the triangle.In Diag.4

Conclusion
A pursuit differential game of three pursuers and one evader in the plane has been studied when all the players have equal dynamic possibilities.New strategies for pursuers have been constructed.We have compared the new strategy with the familiar P-strategy and shown that new strategy has an advantage over the P-strategy.The present work can be extended by considering many pursuers and one evader.

Diag. 2 .
Strategy for x 1 .From now on, denote the straight line passing through the given two points a, b ∈ R 2 , a = b, by (ab), and the ray going out of the point a and passing through the point b by [ab).

|a 12 −
a 23 | ≤ |a 12 − a 31 | ≤ |a 23 − a 31 |.First, the evader moves along the ray [ys) to the distance |s − y| − ǫ/2, where ǫ is a given sufficiently small positive number.Then, turns back and moves in the direction [sw) to the distance |w − y| − ǫ/2.Until this time the pursuer x 3 moves parallel to the side [a 23 a 31 ] of the triangle.Finally, the evader moves parallel to the side of middle length, that is, [a 12 a 31 ] until he meets the pursuer x 3 .In this way, the evader can move not being captured until the time |s − w| + |w − a 31 | − ǫ.Therefore, T = |s − w| + |w − a 31 | is a guaranteed evasion time for the evader provided that pursuers use P-strategy.For any ǫ > 0, the evader can move not being captured on the time interval [0, T − ǫ].If the pursuers use the strategy (3.1), the evader is captured earlier than the time T since if the evader moves in the direction [ys) or [yw) pursuer x 3 moves towards [x 3 a 23 ) or [x 3 a 31 ), respectively, and as shown above, the distance |y − x 3 | reduces faster.This shows advantage of the strategy (3.1).