eu ABOUT THE PROBLEM OF GROUP PERSECUTION IN LINEAR DIFFERENTIAL GAMES WITH A SIMPLE MATRIX AND STATE CONSTRAINTS

Consider two dependent problems of evasion one evader from many pursuers. In the first problem assumed, that the all players have a simple motion, among the pursuers are both participants the maximum speed of which coincide with the maximum speed of evader and parties who maximum speeds strictly less than the maximum speed of evader while evader is not leaves the convex compact set with non-empty interior. In the second problem consider linear dipendent problem of persecution one evader by group pursuers, provided that the matrix of the system is the product of a function and identity matrix, among the pursuers are both participants, which the set of admissible controls, is a sphere with center at the origin, coincides with the set of admissible controls the evader, and the pursuers with fewer opportunities and evader does not leave the confines of a convex cone with vertex at the origin. We prove that if the number of pursuers, opportunities coincide with opportunities the evader, less than the dimension space, then the pursuers with fewer features do not affect the solvability evasion problem. AMS Subject Classification: 49N70, 49N75, 49N90


Introduction
Differential games of two players, first considered in the book of Isaacs [1], now present wide field of research [2]- [11].Methods were developed for solving various classes of game problems: Isaaks' method, based on the analysis of a certain partial differential equation and its characteristics, Krasovskii's method of extremal guidans, Pontryagin's method and others.A natural generalization of differential games pursuit-evasion of two persons are games with a group of pursuers and one or several evaders [12]- [15].These games are interesting from the theoretical point of view, they cannot be treated by theory for twoperson games.One reason for this is, that the union sets of the reachability of all pursuers and the union of all target sets are sets, is non-convex and, furthermore, is not connected.On the other hand, there are some applications of these games to the problems of motion vehicles, collisions of avoidance for ships and others.Among a large number of papers, devoted to the pursuitevasion games with a group of pursuers, mention the works [16]- [33].Works [24], [26]- [31] is devoted to model problems, where all players have a simple motion.In work [24] proved the possibility of evasion evader from any number of pursuers, provided, that maximum at a rate speed of the evader greater than the maximum at a rate of speed of any pursuer.The generalization of this work to a wide class of problems is work [25].The task of pursuit-evasion with equal opportunities to all the participants considered in [26], [27], [29].In [28] proved the possibility of evasion the evader from a group of pursuers, if all participants have equal opportunities, number of pursuers less than the dimension of space, evader does not leave the confines of a convex compact set with non-empty interior.In [32], [33] proved the possibility of evasion evader from group of pursuers in linear differential game with a simple matrix if the number of pursuers less then dimension of space and evader does not leave the confines of a convex cone.Problem of evasion group of evaders from a group of pursuers in various productions considered in [14], [16], [30].This work is devoted to two non-stationary problems of evasion of one the evader from a group of pursuers.In the first problem assumed, that the all players have a simple motion, among the pursuers are both participants the maximum speed of which coincide with the maximum speed of evader and parties who maximum speeds strictly less than the maximum speed of evader while evader is not leaves the convex compact set with non-empty interior.In the second problem consider linear dependent problem of persecution one evader by group pursuers , provided that the matrix of the system is the product of a function and identity matrix, among the pursuers are both participants, which the set of admissible controls, is a sphere with center at the origin, coincides with the set of admissible controls the evader, and the pursuers with fewer opportunities and evader does not leave the confines of a convex cone with vertex at the origin.
We prove that if the number of pursuers, opportunities coincide with opportunities the evader, less than the dimension space, then the pursuers with fewer features do not affect the solvability evasion problem.

Statement of the Problem
In space R k (k ≥ 2) we consider differential game n + 1 objects: n pursuers P 1 , . . ., P n and evader E.
The law of motion of each of the pursuers P i has the form where α j = 1 for all j = 1, . . ., m < n α j < 1 for all j = m + 1, . . ., n.
The law of motion of evader E has the form At t = t 0 set the initial position of pursuers x 0 1 , . . ., x 0 n and the initial position of evader y 0 , and x 0 i = y 0 , i = 1, . . ., n.
We denote this game by Γ(n).
Definition 2. We say, in game Γ(n) evasion occurs from meeting, if there partitioning σ of interval [t 0 , ∞) which has no finite condensation points, strategy V of evader E, which corresponds to partitioning σ such that for all trajectories x 1 (t), . . ., x n (t) of pursuers P 1 , . . ., P n takes place where y(t) -implemented in this situation, the trajectory of the evader E.
We denote by IntD interior of the set D.

The Theorem about Evasion
Theorem 1.Let y 0 ∈ IntD, b -is a function, which is bounded on any compact and m < k.Then in game Γ(n) occurs evasion of meeting from any initial positions.
Proof.Inasmuch as y 0 ∈ IntD, there is D r (q) -a sphere of radius r with center q such that y 0 ∈ IntD r (q) ⊂ D. Assume further that ε -distance from Note, that if t ∈ Ω(τ ) τ, t ∈ I l for some l, that .
Note, that due to (1) numbers m l exist for all l.As a partition σ of interval [t 0 , ∞) take such a partition, that constriction σ to any segment We introduce the following notation Note, that ∆ j > 0, δ j > 0 if x i (t j ) = y(t j ).
We set the strategy V of evader E to [t j , t j+1 ) follows.Let v j -is vector, which satisfies the system (v j , x l (t j ) − y(t j )) = 0, l = 1, . . ., m, (v j , y(t j ) − q) ≤ 0, v j = 1.inasmuch as m < k, that the system has a solution.
Assume further v j (t, t j , v j , γ j ) -is the control of the evader E, which guarantees to it evading his pursuers P m+1 , . . ., P n [t j , t j+1 ) and such that y(t) − y(t) < γ j for all t ∈ [t j , t j+1 ), where y(t) = y(t j ) + v j t t j |b(s)|ds, y(t) -is trajectory of the evader E, control which meets v j (t, t j , v j , γ j ). by virtue of [25], [26] such control E exists.We assume control of the evader E in game Γ(n) to [t j , t j+1 ) equal to v 0 j (t) = v j (t, t j , v j , γ j ).
We show that V is a strategy of evasion.Consider the segment [t j , t j+1 ].Let u i (t) -arbitrary control pursuer P i , Then ûi (t) ≤ 1 for all t ∈ [t j , t j+1 ).
By the triangle inequality, we have Estimate the first term. x From the last inequality implies that if capture does not occur until the time t j , then it does not happen on [t j , t j+1 ).Since x 0 i = y 0 for all i, then y(t) = x i (t) for all i, t ≥ t 0 .
We show that the evader E does not leave the confines of the set D. Let us prove that if the inequality y(t j ) − q ≤ r − ε j , then Indeed, Since y(t) − y(t) < γ j and Thus proved that y(t) ∈ D r (q) ⊂ D for all t ≥ t 0 .The theorem is proved.
Remark.Note that the case k = 2, m = 1, D -is circle, b(t) = 1 for all t considered in [29], where proposed a different approach to the selection of parameters.

Statement of the Problem
In space R k (k ≥ 2) we consider differential game n + 1 objects: n pursuers P 1 , . . ., P n and evader E.
The law of motion of each of the pursuers P i has the form ẋi = a(t) where α j = 1 for all j = 1, . . ., m < n and α j < 1 for all j = m + 1, . . ., n.
The law of motion of evader E has the form At t = t 0 set the initial position of pursuers x 0 1 , . . ., x 0 n and the initial position of evader y 0 , and It is further assumed, that evader E in the game does not leave limits of convex cone where p 1 , . . .p r -the unit vectors R k such that IntD = ∅.
The evader uses a piecewise-program strategies.

The Theorem about Evasion
Theorem 2. Let y 0 ∈ IntD, a -is a function, which is bounded on any compact and m < k.Then in game Γ(n) occurs evasion of meeting from any initial positions.
Proof.Consider the segment I l = [t 0 +l−1, t 0 +l], t l = t 0 +l−1, l = 1, 2, . . ., In systems (2), (3) we make the change variables Get systems Note that x i (τ ) = y(τ ) under some i, τ ∈ I l if and only if, when t l a(s)ds , D r (q) -a sphere of radius r with center q such that y 0 ∈ D r (q) ⊂ D, ε -distance from y 0 to border D r (q), q For segment I 1 by ε 1 , and function b 1 define number m 1 and partition σ 1 by the scheme previous section.For segment I 2 by ε 2 = K 2 ε 1 m 1 +2 and function b 2 define number m 2 and partition σ 2 and so on.For segment I l by ε l = and function b l define number m l and partition σ l .As a partition σ of interval [t 0 , ∞) take such a partition σ, whose restriction to any segment I l coincides with σ l .
Consider the segment I l and the corresponding partition Since m < k, v l j is always exists.Assume further v l j (t, τ l j , v l j , γ j ) is the control of the evader E to segment [τ l j , τ l j+1 ), which guarantee him evasion in this segment from the pursuers P m+1 , . . ., P n in the game, described (4) and is such that where z l j (t) = z l j (t l j )+v l j • t t l j b(s)ds, z l j (t) -the trajectory of the evader E, which is meets a control v l j (t, τ j , v l j , γ j ).By virtue of [24], [25] such control exists.We put control of the evader E in the game Γ(n) to [τ l j , τ l j+1 ) equals v 0 j (t) = v l i (t, τ l j , v l j , γ j ).Let us show that for all l and all t ∈ I l we have the inequalities Consider the segment I 1 = [t 0 , t 1 ] and partition σ 1 of this segment.Then Further proof of (5) for I 1 is similar proof of the corresponding inequality in the previous section.Assume that the inequality ( 5) proved for all I l , l ≤ s.We prove this inequality for the interval I s+1 .By assumption we have the inequality Therefore the proof of (5) for I s+1 similar to the proof of inequality (5) for I 1 .Therefore, z l (t) ∈ D r l (q l ) for t ∈ I l and for all l.
Let us prove that D r l (q l ) ⊂ D. Note that t 0 a(s)ds q.
Similarly, r l = e Therefore, ĉ ∈ D r (q) ⊂ D. In that D is a cone, that c ∈ D. Then D r l (q l ) ⊂ D for all l.
We show that strategy V guarantees evasion.Let us show that z l (t) = w l i (t) for all i, t ∈ I l .Let τ l j , τ l j+1 ∈ σ l .From the system (4) we have z l (t) = z l (τ l j ) + Therefore for all t ∈ [τ l j , τ l j+1 ) z l (t) − w l i (t) ≥ z l (τ l j ) + M l j (t)v l j − w l i (τ l j ) − M l j (t) = = (a l i (τ l j )) 2 + 2M l j (t)(v l j , a l i (τ l j )) + (M l j (t)) 2 − M l j (t) = = (a l i (τ l j )) 2 − (M l j (t)) 2 − M l j (t) > 0 if a l i (τ l j ) = 0.
Therefore, if the capture did not occur until τ l j , it does not happen on a segment [τ l j , τ l j+1 ].In that z 0 = w 0 i for all i, it is proved that z l (t) = w l i (t) for all i, l, t ∈ I l .Therefore, the strategy V is a strategy of evasion.The theorem is proved.
Let c ∈ D r l (q l ).Represent c in the form c = ĉe s)u i (s)ds.