Simple Motion Pursuit and Evasion Differential Games with Many Pursuers on Manifolds with Euclidean Metric

We consider pursuit and evasion differential games of a group ofm pursuers and one evader on manifolds with Euclidean metric. The motions of all players are simple, and maximal speeds of all players are equal. If the state of a pursuer coincides with that of the evader at some time, we say that pursuit is completed. We establish that each of the differential games (pursuit or evasion) is equivalent to a differential game of m groups of countably many pursuers and one group of countably many evaders in Euclidean space. All the players in any of these groups are controlled by one controlled parameter. We find a condition under which pursuit can be completed, and if this condition is not satisfied, then evasion is possible. We construct strategies for the pursuers in pursuit game which ensure completion the game for a finite time and give a formula for this time. In the case of evasion game, we construct a strategy for the evader.


Introduction
Multiplayer differential games are natural extension of twoperson differential games.New methods of solution for the games of many players were proposed in many works such as [1][2][3][4][5][6][7].The main problem for such games is to find conditions of evasion or completion of pursuit and construct strategies of players as well.
Pursuit and evasion differential games of many players attract attention of many researchers.The case when all players have equal dynamic possibilities [8] proved to be an alternative: if the initial position of the evader belongs to the interior of convex hull of the initial positions of the pursuers, then pursuit problem is solvable; else evasion problem is solvable.This work was extended by many researchers such as [2,3,6,9].It should be noted that one of the main methods for solving differential games of many pursuers and one evader is the method of resolving functions developed by Pshenichnii et al. [6].
In the evasion game with many pursuers studied by Chernous'ko [1], it was shown that an evader whose speed is bounded by 1 can, by remaining in a neighborhood of a given motion, avoid an exact contact with any finite number of pursuers whose speeds are bounded by a number less than 1.Later on, this result was extended by Zak [10].The work [11] presented a sufficient condition of existence of evasion strategy in the game of many pursuers in the plane.
In the real control systems, the state constraints occur and they are given usually in the form of equalities and inequalities.For example, the works [17,18] study control problems with state constraints of equality type and [19] does properties of shortest curves in the domain described by equality and inequality type of constraints.
In differential games, mainly two types of state constraints are considered.According to the first constraint a player moves within a given set, and according to the second one it moves along a submanifold of the state space.
In pursuit-evasion differential games, the state constraint can be imposed on the states of pursuers or evaders separately.In the papers [20,21], the Isaacs method was applied to study the simple motion pursuit differential games of one pursuer and one evader on two-dimensional Riemannian manifolds.In [22], the problem of evasion of one evader from many pursuers was studied on a hypersurface in R  .
In the works [23][24][25] devoted to generalized "Lion and Man" differential game of Rado, the evader moves along a given absolutely continuous curve and the pursuer moves in the space.In the case where the evader's speed is greater than that of the pursuer, an evasion strategy was constructed, which enables us to estimate from below the distance between players.If the maximal speeds of the players are equal and the curve does not intersect itself, necessary and sufficient conditions of completion of pursuit from a given initial position as well as from all the initial positions [25] were obtained.
Simple motion pursuit and evasion differential games studied in [26] occur in a ball of the Riemannian manifold, and all players have the same dynamic possibilities.The pursuers move in the ball, and with respect to the motion of the evader two cases were considered: in the first case, the evader moves on the sphere of the ball, and in the second case, the evader also moves in the ball.For each case, sufficient conditions of both completion of pursuit and evasion were obtained.
In the games of many pursuers and many evaders studied in [5,7,27], all of the evaders are controlled by one controlled parameter, but pursuers have different control parameters.By definition, game is said to be completed if the state of a pursuer coincides with that of an evader at some time.In these papers, sufficient conditions of completion of pursuit were obtained.In the work [28], a simple motion evasion game of many pursuers and many evaders was studied under integral constraints on controls of players.
In the present paper, we deal with the manifolds, each point of which has a neighborhood isometric to a region of Euclidean space; in other words, the Riemannian metric at each point of the manifold, in a local coordinate system, is expressed as Euclidean one.Such manifolds are called ones with Euclidean metric or ones locally isometric to Euclidean space [29,30].For example, two-dimensional manifolds locally isometric to Euclidean space are only plane, cylinder, torus, Mobius strip, and the Klein bottle.Note that a differential game of optimal approach [31] was studied on such manifolds, where duration of the game is fixed, and payoff of the game is distance between the players when the game is terminated.We will study, however, pursuit and evasion differential games of many pursuers and one evader on manifolds with Euclidean metric.Maximal speeds of all players are assumed to be identical.We find a condition on initial positions of players under which pursuit problem is solvable.If this condition is not satisfied, then we show that evasion problem is solvable.
To solve the pursuit or evasion differential game of  pursuers and one evader on manifold  with Euclidean metric, we first reduce the game to an equivalent one in Euclidean space R  , where a group of countably many pursuers  1  ,  2  , . . .∈ R  ,  = 1, 2, . . ., , and group of countably many evaders  1 ,  2 , . . .∈ R  correspond to each pursuer   ∈  and evader  ∈ , respectively.Each group of players is controlled by one control parameter.In the new game in R  , if    () =   () at some  ∈ {1, 2, . . ., }, ,  ∈ N and  > 0, then game is completed; otherwise evasion is possible.At first glance, the condition   0 ∈ int conv{  0 ,  = 1, . . ., ;  = 1, 2, . ..} seems to be the condition for completing pursuit.It turns out that this condition does not guarantee that pursuit can be completed in the game on .We obtain another condition (see Theorem 10), which is necessary and sufficient condition of completion of pursuit on manifolds with local Euclidean metric.

Statement of Problem
Let  be ,  ≥ 2, dimensional manifold with positive defined symmetric bilinear form ⟨⋅, ⋅⟩  on the tangent space   at the point  ∈ , and let ‖ ⋅ ‖  be the norm defined by ⟨, ⟩  = ⟨ () , ⟩ , where ⟨⋅, ⋅⟩ is inner product in R  ; the matrix (⋅), called metric tensor, defines the Riemannian metric on the manifold , which becomes identity matrix at each point of the manifold in a local coordinate system; that is,  is a manifold with local Euclidean metric.
) is called a strategy of the evader .
(2) The trajectory (⋅) is defined as the solution of initial value problem Thus, strategy of the group of pursuers is defined as a contrstrategy, and corresponding motion is the solution of the initial value problem, whereas strategy of the evader is defined as a positional (closed loop) strategy, and corresponding to it trajectory of the evader is defined as a stepwise motion [32].
The problems of the players are as follows.
Dynamics of the groups of pursuers and evaders are described by the following equations: where =   0 for all  ∈ {1, 2, . . ., }, ,  ∈ {1, 2, . ..};  1 ,  2 , . . .,   and V are the control parameters of the groups of pursuers and group of evaders, respectively, which satisfy the constraints Controls, strategies of the groups of pursuers and group of evaders, trajectories generated by the initial position, strategies of pursuers, and control of evaders (or vice versa) are defined as in Definitions 2 and 3.

Properties of the Multivalued Mapping
The multivalued mapping  −1 :  → R  has the following properties [29,30]: (1)  −1 maps the straight lines (i.e., geodesics) on manifold  to a family of parallel straight lines in R  . ( (3) There exists a positive number ℎ such that Let Then the properties of mapping  −1 mentioned above imply the following statement.
(3) If  is a compact set, then there is a bounded set  0 ⊂ R  of nonempty interior such that  :  0 →  is one-to-one [29].Since R  can be represented as a union of  0 and the bounded sets,  1 ,  2 , . .., which can be obtained by parallel translation of  0 and whose interiors are disjoint [29], therefore, for any  ∈ , each of the compact sets   ,  = 0, 1, . .., contains only one element of  −1 ().This implies that dim conv  −1 () = , which is the desired conclusion.

Main Result
We now formulate the main result of the present paper.
It should be noted that condition (21) implies that  >  and  > 0. Proof.
Remark 11.Obviously, if  ≤ , inclusion (21) fails to hold.In this case, by Theorem 10, evasion is possible from any initial points.In addition, in the proof of this statement, we have used only inequality (15), and other properties of multivalued mapping  −1 :  → R  have not been used.Therefore, if  ≤ , then evasion is possible in the game of type ( 11)-( 12) from any initial points that satisfy (15).
It should be noted that condition (21) of Theorem 10 is fairly easy to check.In the following example, we give conditions which are equivalent to condition (21).

Conclusion
We have studied pursuit and evasion differential games of  pursuers and one evader on manifolds with Euclidean metric.
The following are our main contributions: (1) We have proposed a method of reduction of simple motion differential game with many pursuers and one evader on a class of manifolds (e.g., cylinder and torus) to an equivalent differential game in R  , which contains  groups of countably many pursuers,  1  ,  2  , . ..,  = 1, . . ., , and one group of countably many evaders,  1 ,  2 , . ... (2) We have obtained necessary and sufficient condition of evasion in the equivalent game.It should be noted that even though the condition   0 ∈ {  0 ,  = 1, . . ., ;  = 1, 2, . ..}, for some positive integer , is more similar to the condition of Pshenichnii [8] than condition (21), it is not sufficient for completion of pursuit in the game.
(3) If condition (21) is not satisfied, we have constructed an evasion strategy and proved that evasion is possible.Note that many researchers (see, e.g., Pshenichnii [8] and Grigorenko [3]) suggested to the evader a constant velocity depending on initial positions of players, which guarantees the evasion in R  .In the game we studied, evasion using constant velocity does not work.In general, the evader will be captured if it moves with constant velocity.Evasion strategy used in the present paper depends on current positions of players and requires a specific construction.