eu GROUP PURSUIT WITH PHASE CONSTRAINTS IN RECURRENT PONTRYAGIN ’ S EXAMPLE

We consider a generalized non-stationary Pontryagin’s example under the same dynamic and inertial capabilities players and phase constraints on the state of the runaway. Boundary of phase constraints is not a ”line of death” for the evader. Sufficient conditions for the capture of a group of pursuers one evader are obtained in this article. 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]- [8].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 [9]- [12].These games are interesting from the theoretical point of view, they cannot be solved by theory for two-person 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.In this case, the problem becomes much more complicated if the players on the state of the phase constraints are imposed.Among the large number of papers devoted to the problem of group pursuit with phase constraints, we mention the works [13]- [20].In [13]- [17] there are the various tasks simple pursuit with phase constraints.The general problem of evasion with state constraints is considered in [18].Sufficient mustache capture in stationary Pontryagin's example obtained in [19].Terms of capture in an unsteady Pontryagin's example with phase constraints under other assumptions are presented in [20].
In this paper we consider the problem of persecution of a group of pursuers one evader at equal opportunities and inertial dynamic players.It is assumed that the evader in the game did not leave a convex polyhedral set, terminal sets -convex kompaty.Provided that some of the functions defined by the initial conditions and parameters of the game are recurrent, we obtain sufficient conditions for the solvability of persecution.

Statement of the Problem
In space R k (k ≥ 2) we consider differential game Γ(n, D) 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 The law of motion of evader E has the form where x i , y j , u i , v j ∈ R k , the functions a 1 (t), a 2 (t), . . ., a l (t) are continuous [t 0 , ∞), V -convex compact.At t = t 0 the initial conditions are set where It is further assumed, that evader E in the course of the game does not leave a convex set with non-empty interior, where (a, b) is the scalar product of vectors a and b, p 1 , . . ., p r are the unit vectors R k , µ 1 , . . ., µ r are real numbers.
Suppose further Assume that ξ i (t) / ∈ M i for all i, t ≥ t 0 , for if ξ i (τ ) ∈ M i for some i, τ, then the pursuer P i catches the evader E, supposing u i (t) = v(t).Definition 1.We will say that a quasi-strategy U i of pursuer P i , is given if a mapping U i (t, z 0 , v t (•)) is defined that assigns to the initial state z 0 , the point in time t, and an arbitrary past history of control v t (•) of the evader E such that y(t) ∈ D for all t ≥ t 0 , a measurable function u i (t) with values in V.

Definition 2.
There is a capture in the game Γ(n, D) if there is a time T (z 0 ) and quasi-strategies U 1 (t, z 0 , v t (•)), . . ., U n (t, z 0 , v t (•)) of pursuers P 1 , . . ., P n such that, for any measurable function v(•), v(t) ∈ V , y(t) ∈ D, t ∈ [t 0 , T (z 0 )] there are numbers s ∈ I and a time τ ∈ [t 0 , T (z 0 )], such that z s (τ ) ∈ M s .Definition 3. (see [21]) The function f : R 1 → R k is called recurrent if, for any ε > 0 , there exists T (ε) > 0 such that for any a, t ∈ R 1 , there exists τ (t) ∈ [a, a + T (ε)] for which it holds that We define the functions Assumptions 2. There are times τ 0 i ≥ t 0 , are positive numbers ε, δ such that: 1) for all i and for all Lemma 1. Suppose that assumptions 1, 2, r = 1.Then there time T ≥ t 0 such that for any admissible control v(•) evader E, any h ∈ S there exists a number m ∈ I for which Proof.Since control v(t) of the evader E is admissible, then for all t ≥ t 0 We define the set The last two relations imply that Further, we have Since F (t) → ∞ at t → ∞ and µ(t) is bounded, then we obtain the desired result.The lemma is proved.
Define the number T 0 Assumptions 3.There are times τ i ≥ T 0 such that: Remarks.a) the existence of τ i in paragraph 1 of assumptions 3 guaranteed assumption that recurrence functions ξ i (t); b) if the assumption 3 of all τ i = τ, paragraph 2 of this assumption is made automatically by Lemma 1.
Proof.By the Cauchy's formula, the solution of problem (4), ( 5) for all t ≥ t 0 for any admissible control has the form Let τ i -points satisfying assumption 3, v(s), s ∈ [t 0 , T 1 ] be an arbitrary admissible control of the evader E, where T 1 = max i τ i .Consider the function We denote by τ 0 ≥ t 0 is the first root of this function.Note that, by the assumption 2, the time τ 0 exists , wherein τ 0 ≤ τ i for at least one of i.Furthermore, there is an index m such that 1 − For j = m is also denoted t j -times for which the condition (7), if such times exist.By Filippov's lemma [22] for each i there are measurable functions m i (s), u i (s), s ∈ [t 0 , T 1 ], which for each fixed s, the solution of the equation We specify the control of pursuers P i , in the following way: Then The theorem is proved.

Capture Conditions in the Case
We denote IntX, coX respectively the interior and the convex hull of the set X, Proof.Note that (see [9], p. 46) Suppose that δ + = 0. Then there exists an v 0 , v 0 = 1 such that λ(Q i , v 0 ) = 0 for all i, (p j , v 0 ) ≤ 0 for all j.
Assumptions 4.There τ 0 i ≥ t 0 such that Lemma 3. Suppose that assumptions 1, 4. Then there are positive numbers ε, T (ε) for which the following propositions: 1) for all h i ∈ S ε (ξ i (τ 0 i )) the following inclusion 2) for each t ≥ t 0 there are moments The first proposition follows from the properties of the strict separation of convex sets, and the second assertion follows from the properties of recurrent functions.
We fix ε > 0 and T (ε) > 0, so that there have been propositions of Lemma 4.
Lemma 5. Let Q i , i ∈ I be convex compact sets R k , 0 / ∈ Q i for all i and the following conditions: 2) the number of elements Proof.According to the conditions of Lemma there are q 1 , . . ., q s ∈ n i=1 Q i such that 0 ∈ Intco{q 1 , . . ., q s , p 1 , . . ., p r }.

Consider the set
The lemma is proved.
Assumptions 5.For any h ∈ S in the set n i=1 (h i − M i ) there is a k linear independent vectors.Corollary 2. Suppose that assumptions 1, 4, 5.Then, for any h ∈ S exist a vector p(h) ∈ R k and the number of µ(h) ∈ R 1 such that: Lemma 6.Let V = S 1 (0) and assumptions 1, 4, 5 be fulfiled.Then there is the time T ≥ t 0 such that for any admissible control v(•) of the evader E in the game Γ(n, D 1 ), any h ∈ S there exists a number m ∈ I such that where D 1 is defined in corollary fact 2.
Proof.Suppose h ∈ S. By Lemma 5, we have δ > 0. Therefore, the assumption that the conditions 2. Therefore, we can apply Lemma 1, which implies the assertion.The lemma is proved.Theorem 3. Let the assumptions of 1,4,5, and there are τ i ≥ T 0 such that: Then in the game Γ(n, D 1 ) capture occurs.
The validity of this assertion follows from theorem 1.
Corollary 3. Suppose that all the conditions of theorem 3. Then in the game Γ(n, D) capture occurs.
For this example, assumption 1.
Then in the game Γ(n, D) capture occurs.
Function ϕ 0 (t) is recurrent, but is not almost periodic (see [21]).Assumption 1 is satisfied.Recurrence functions ξ i (t) from the results of (see [21]).Then in the game Γ(n, D) capture occurs.Note that for this example are not fulfilled the conditions of [20].