Research Article Simple Motion Pursuit and Evasion Differential Games ... · Simple Motion Pursuit and Evasion Differential Games with Many Pursuers on Manifolds with Euclidean Metric

Research ArticleSimple Motion Pursuit and Evasion Differential Games withMany Pursuers on Manifolds with Euclidean Metric

Atamurat Kuchkarov1 Gafurjan Ibragimov2 and Massimiliano Ferrara34

1 Institute of Mathematics National University of Uzbekistan 100125 Tashkent Uzbekistan2Department of Mathematics amp Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Malaysia3Department of Law and Economics Mediterranea University of Reggio Calabria 89127 Reggio Calabria Italy4ICRIOS-Bocconi University 20123 Milan Italy

Correspondence should be addressed to Massimiliano Ferrara massimilianoferraraunircit

Received 17 May 2016 Accepted 4 July 2016

Academic Editor Filippo Cacace

Copyright copy 2016 Atamurat Kuchkarov et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider pursuit and evasion differential games of a group of119898 pursuers and one evader on manifolds with Euclidean metricThe motions of all players are simple and maximal speeds of all players are equal If the state of a pursuer coincides with that ofthe evader at some time we say that pursuit is completed We establish that each of the differential games (pursuit or evasion) isequivalent to a differential game of 119898 groups of countably many pursuers and one group of countably many evaders in Euclideanspace All the players in any of these groups are controlled by one controlled parameter We find a condition under which pursuitcan be completed and if this condition is not satisfied then evasion is possible We construct strategies for the pursuers in pursuitgame which ensure completion the game for a finite time and give a formula for this time In the case of evasion game we constructa strategy for the evader

1 Introduction

Multiplayer differential games are natural extension of two-person differential games New methods of solution for thegames of many players were proposed in many works such as[1ndash7] The main problem for such games is to find conditionsof evasion or completion of pursuit and construct strategiesof players as well

Pursuit and evasion differential games of many playersattract attention of many researchers The case when allplayers have equal dynamic possibilities [8] proved to be analternative if the initial position of the evader belongs to theinterior of convex hull of the initial positions of the pursuersthen pursuit problem is solvable else evasion problem issolvableThisworkwas extended bymany researchers such as[2 3 6 9] It should be noted that one of themainmethods forsolving differential games ofmany pursuers and one evader isthe method of resolving functions developed by Pshenichniiet al [6]

In the evasion game with many pursuers studied byChernousrsquoko [1] it was shown that an evader whose speed isbounded by 1 can by remaining in a neighborhood of a givenmotion avoid an exact contact with any finite number ofpursuers whose speeds are bounded by a number less than 1Later on this result was extended by Zak [10] The work[11] presented a sufficient condition of existence of evasionstrategy in the game of many pursuers in the plane

In differential games of many pursuers studied by Ivanovand Ledyaev [12] Ibragimov [13] Ibragimov and Rikhsiev[14] Ibragimov and Salimi [15] and Ibragimov et al [16]optimal strategies of players were constructed

In the real control systems the state constraints occurand they are given usually in the form of equalities andinequalities For example the works [17 18] study controlproblems with state constraints of equality type and [19] doesproperties of shortest curves in the domain described byequality and inequality type of constraints

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 1386242 8 pageshttpdxdoiorg10115520161386242

2 Discrete Dynamics in Nature and Society

In differential gamesmainly two types of state constraintsare considered According to the first constraint a playermoves within a given set and according to the second oneit moves along a submanifold of the state space

In pursuit-evasion differential games the state constraintcan be imposed on the states of pursuers or evaders separatelyIn the papers [20 21] the Isaacs method was applied to studythe simple motion pursuit differential games of one pursuerand one evader on two-dimensional Riemannian manifoldsIn [22] the problem of evasion of one evader from manypursuers was studied on a hypersurface in R119899

In the works [23ndash25] devoted to generalized ldquoLion andManrdquo differential game of Rado the evader moves along agiven absolutely continuous curve and the pursuer movesin the space In the case where the evaderrsquos speed is greaterthan that of the pursuer an evasion strategy was constructedwhich enables us to estimate frombelow the distance betweenplayers If the maximal speeds of the players are equal andthe curve does not intersect itself necessary and sufficientconditions of completion of pursuit from a given initialposition as well as from all the initial positions [25] wereobtained

Simple motion pursuit and evasion differential gamesstudied in [26] occur in a ball of the Riemannian manifoldand all players have the same dynamic possibilities Thepursuers move in the ball and with respect to the motion ofthe evader two cases were considered in the first case theevadermoves on the sphere of the ball and in the second casethe evader also moves in the ball For each case sufficientconditions of both completion of pursuit and evasion wereobtained

In the games of many pursuers andmany evaders studiedin [5 7 27] all of the evaders are controlled by one controlledparameter but pursuers have different control parametersBy definition game is said to be completed if the state of apursuer coincides with that of an evader at some time Inthese papers sufficient conditions of completion of pursuitwere obtained In the work [28] a simple motion evasiongame of many pursuers and many evaders was studied underintegral constraints on controls of players

In the present paper we deal with the manifolds eachpoint of which has a neighborhood isometric to a regionof Euclidean space in other words the Riemannian metricat each point of the manifold in a local coordinate systemis expressed as Euclidean one Such manifolds are calledones with Euclidean metric or ones locally isometric toEuclidean space [29 30] For example two-dimensionalmanifolds locally isometric to Euclidean space are only planecylinder torus Mobius strip and the Klein bottle Note thata differential game of optimal approach [31] was studied onsuch manifolds where duration of the game is fixed andpayoff of the game is distance between the players whenthe game is terminated We will study however pursuit andevasion differential games of many pursuers and one evaderon manifolds with Euclidean metric Maximal speeds of allplayers are assumed to be identical We find a condition oninitial positions of players under which pursuit problem issolvable If this condition is not satisfied then we show thatevasion problem is solvable

To solve the pursuit or evasion differential game of 119898pursuers and one evader on manifold 119872 with Euclideanmetric we first reduce the game to an equivalent one inEuclidean space R119899 where a group of countably many pur-suers 1199091

119894 1199092

119894 isin R119899 119894 = 1 2 119898 and group of countably

many evaders 1199101 1199102 isin R119899 correspond to each pursuer119909119894isin 119872 and evader 119910 isin 119872 respectively Each group of players

is controlled by one control parameter In the newgame inR119899if 119909119895119894(120591) = 119910

119896

(120591) at some 119894 isin 1 2 119898 119895 119896 isin N and 120591 gt 0then game is completed otherwise evasion is possible At firstglance the condition 119910

119896

0isin int conv119909119895

1198940 119894 = 1 119898 119895 =

1 2 seems to be the condition for completing pursuit Itturns out that this condition does not guarantee that pursuitcan be completed in the game on 119872 We obtain anothercondition (seeTheorem 10) which is necessary and sufficientcondition of completion of pursuit on manifolds with localEuclidean metric

2 Statement of Problem

Let 119872 be 119899 119899 ge 2 dimensional manifold with positivedefined symmetric bilinear form ⟨sdot sdot⟩

119909on the tangent space

119879119872119909at the point 119909 isin 119872 and let sdot

119909be the norm defined by

⟨119886 119887⟩119909= ⟨119866 (119909) 119886 119887⟩

119886119909= ⟨119886 119886⟩

119909

119909 isin 119872 119886 119887 isin 119879119872119909

(1)

where ⟨sdot sdot⟩ is inner product in R119899 the matrix 119866(sdot) calledmetric tensor defines theRiemannianmetric on themanifold119872 which becomes identity matrix at each point of themanifold in a local coordinate system that is119872 is amanifoldwith local Euclidean metric

The motion of a group of pursuers 1198751 1198752 119875

119898 and

one evader 119864 are described on the manifold 119872 by thefollowing equations

119875119894 119894= 119906119894

119909119894(0) = 119909

1198940

119864 = V

119910 (0) = 1199100

1199100

= 1199091198940

(2)

where 119909119894 119910 isin 119872 119906

119894isin 119879119872

119909119894 V isin 119879119872

119910 and 119906

119894 V are control

parameters 119894 = 1 2 119898

Definition 1 Borel measurable functions V(119905) and 119906119894(119905) 119905 ge

0 are called controls of the evader 119864 and pursuer 119875119894 119894 =

1 2 119898 respectively if for the solutions 119910(119905) and 119909119894(119905) of

initial value problems

= V (119905) 119910 (0) = 1199100

119894= 119906119894(119905) 119909

119894(0) = 119909

1198940

(3)

Discrete Dynamics in Nature and Society 3

the following inequalities

V (119905)119910(119905)

le 1

1003817100381710038171003817119906119894(119905)1003817100381710038171003817119909119894(119905)

le 1

(4)

and inclusions

V (119905) isin 119872119910(119905)

119906119894(119905) isin 119872

119909119894(119905)

(5)

hold true for almost all 119905 119905 ge 0

In the tangent space 119879119872119911 denote the ball 119908 isin 119879119872

119911|

|119911 minus 119908| le 120588 of radius 120588 centered at 119911 by 119861lowast(119911 120588)

Definition 2 A function 119880 = (1198801 1198802 119880

119898) 119880119894= 119880119894(119905 1199091

1199092 119909

119898 119910 V)

119880119894 [0infin) times119872

119898+1

times 119861lowast(119910 1) 997888rarr 119861

lowast(119909119894 1) (6)

is called a strategy of the group of the pursuers 1198751 1198752 119875

119898

if for any control of the evader the following initial valueproblem

= V (119905) 119910 (0) = 1199100

119894(119905) = 119880

119894(119905 1199091 1199092 119909

119898 119910 V (119905))

119909119894(0) = 119909

1198940 119894 = 1 2 119898

(7)

has a unique absolutely continuous solution (1199091(sdot) 1199092(sdot)

119909119898(sdot) 119910(sdot)) on [0infin) The function 119909

119894(sdot) is referred to as

the trajectory of the pursuer 119875119894generated by the triple the

strategy119880 initial position (11990910 11990920 119909

1198980 1199100) and control

V(sdot)

Definition 3 A function 119881(119905 1199091 1199092 119909

119898 119910) 119881 [0infin) times

119872119898+1

rarr 119861lowast(119910 1) is called a strategy of the evader 119864

Let Δ = 1199050= 0 119905

1 be a partition of the interval

[0infin) The trajectory 119910(sdot) of the evader 119864 generated by thecontrols of pursuers 119906

1(sdot) 1199062(sdot) 119906

119898(sdot) a strategy of the

evader119881 initial position (11990910 11990920 119909

1198980 1199100) and partition

Δ is defined as follows

(1) At each time 119905119896 we define a unit vector

V119896= 119881 (119905 119909

1(119905119896) 1199092(119905119896) 119909

119898(119905119896) 119910 (119905

119896))

V119896isin 119861lowast(119910 (119905119896) 1)

(8)

and construct a geodesic 120574119896 [0 119905119896+1

minus 119905119896] rarr 119872 such

that

1205741015840

119896(0) = V

119896

120574119896(0) = 119910 (119905

119894)

100381610038161003816100381610038161205741015840

119894(119904)

10038161003816100381610038161003816= 1

119904 isin [0 119905119896+1

minus 119905119896] 119896 = 1 2

(9)

(note that such a geodesic exists and is unique [29])

(2) The trajectory 119910(sdot) is defined as the solution of initialvalue problem

= 1205741015840

119894(119905 minus 119905119894)

119905 isin [119905119894 119905119894+1) 119894 = 1 2 119910 (0) = 119910

0

(10)

Thus strategy of the group of pursuers is defined as acontrstrategy and corresponding motion is the solution ofthe initial value problem whereas strategy of the evader isdefined as a positional (closed loop) strategy and corre-sponding to it trajectory of the evader is defined as a stepwisemotion [32]

The problems of the players are as follows

Problem 4 (pursuit problem) For each initial position (11990910

11990920 119909

1198980 1199100) find a number 119879 and function (strategy) 119880

such that pursuit can be completed for time 119879 that is for anycontrol V(sdot) of the evader 119864 solution of system (7) satisfies theequation 119909

119894(119905) = 119910(119905) at some 119905 isin [0 119879] and 119894 isin 1 2 119898

Problem 5 (evasion problem) For any initial position (11990910

11990920 119909

1198980 1199100) 1199100

= 1199091198940 construct a function (strategy)

119881 and find a partition Δ such that evasion is possiblethat is for any controls 119906

1(sdot) 1199062(sdot) 119906

119898(sdot) of the pur-

suers1198751 1198752 119875

119898 the trajectories 119909

1(sdot) 1199092(sdot) 119909

119898(sdot) 119910(sdot)

generated by the controls 1199061(sdot) 1199062(sdot) 119906

119898(sdot) strategy 119881

partition Δ and initial position (11990910 11990920 119909

1198980 1199100) satisfy

the inequality 119909119894(119905) = 119910(119905) for all 119905 ge 0 and 119894 isin 1 2 119898

3 An Equivalent Game in Euclidean Space

Wefirst reduce game (2) (4) (5) (7) and (10) onmanifold119872to a specific one in Euclidean spaceR119899 by unfolding119872 inR119899

[29 30] Such unfolding can be conducted by themultivaluedmapping which is inverse to the universal covering 120587 R119899 rarr119872 that is local isometry If119872 itself is not Euclidean space and119911 isin 119872 then the set of its preimages120587minus1(119911) consists of the classof denumerable points equivalent to each other 1199111 1199112 isinR119899

Thus game (2) (4) (5) (7) and (10) is reduced to agame in Euclidean space R119899 in which 119898 groups of pursuers120587minus1

(119909119894) = 119909

1

119894 1199092

119894 119894 = 1 2 119898 consisting of countably

many pursuers try to capture the group of the evaders120587minus1

(119910) = 1199101

1199102

All members of each group have thesame control parameter

Dynamics of the groups of pursuers and evaders aredescribed by the following equations

119895

119894= 119906119894 119909119895

119894(0) = 119909

119895

1198940 119894 = 1 2 119898

119895

= V 119910119895

(0) = 119910119895

0 119895 = 1 2

(11)

where 119909119895119894 119910119895

119906119894 V isin R119899 119909119895

1198940= 119910119896

0for all 119894 isin 1 2 119898 119895 119896 isin

1 2 1199061 1199062 119906

119898and V are the control parameters of

the groups of pursuers and group of evaders respectivelywhich satisfy the constraints

1003816100381610038161003816119906119894

1003816100381610038161003816le 1

|V| le 1(12)

4 Discrete Dynamics in Nature and Society

Controls strategies of the groups of pursuers and groupof evaders trajectories generated by the initial positionstrategies of pursuers and control of evaders (or vice versa)are defined as in Definitions 2 and 3

We say that pursuit can be completed if

119909119895

119894(119905) = 119910

119896

(119905)

at some 119894 isin 1 2 119898 119895 119896 isin 1 2 119905 gt 0(13)

In the differential game (11)-(12) the task of groups ofpursuers is to assure equality (13) and that of the group ofevaders is the opposite

Show that games (2) (4) (5) (7) (10) and (11)-(12) areequivalent Let 119910(119905) 119905 ge 0 be a trajectory of the evader on119872 corresponding to some control function V(119905) 119905 ge 0 Then120587minus1

(119910(119905)) 119905 ge 0 is a family of trajectories in R119899 Since 120587 is alocal isometry and |(119905)| = |V(119905)| le 1 then |119889119910119894(119905)119889119905| le 1 119894 =1 2 Also note that 1198891199101(119905)119889119905 = 119889119910

2

(119905)119889119905 = 1198891199103

(119905)119889119905 =

sdot sdot sdot Hence for given V(119905) isin 119872119910(119905)

|V(119905)| le 1 there is a uniquevector V(119905) isin R119899 |V(119905)| le 1 such that 119889119910119894(119905)119889119905 = V(119905) 119894 =1 2 The same conclusion can be drawn for the pursuers

Note that (13) is equivalent to the equation 120587minus1(119909119894(119905)) =

120587minus1

(119910(119905)) in R119899 and hence it is equivalent to the equation119909119894(119905) = 119910(119905) on manifold 119872 Thus differential game (2)

(4) (5) (7) and (10) of pursuers 1199091 119909

119898and evader 119910

on manifold 119872 is equivalent to game (11)-(12) of groupsof countably many pursuers 120587minus1(119909

119894) = 119909

1

119894 1199092

119894 119894 =

1 2 119898 and a group of countably many evaders 120587minus1(119910) =1199101

1199102

in R119899 where each group is controlled by onecontrol parameter

4 Properties of the Multivalued Mapping

The multivalued mapping 120587minus1 119872 rarr R119899 has the followingproperties [29 30]

(1) 120587minus1 maps the straight lines (ie geodesics) on mani-fold119872 to a family of parallel straight lines in R119899

(2) If 120587minus1(119911) = 1199111 1199112 119911 isin 119872 then

119911119894

+ (119911119895

minus 119911119896

) isin 120587minus1

(119911) forall119894 119895 119896 = 1 2 (14)

(3) There exists a positive number ℎ such that

10038161003816100381610038161003816119911119894

minus 11991111989510038161003816100381610038161003816gt ℎ 119894 = 119895 (15)

Let

119873(119872) = conv 119911119895 minus 119911119896 | 119911119895 isin 120587minus1 (119911) 119911119896

isin 120587minus1

(119911) 119911 isin 119872

(16)

Then the properties of mapping 120587minus1 mentioned above implythe following statement

Lemma 6 The following statements hold true

(1) For any 119911 isin 119872 and 119911lowastisin 120587minus1

(119911) inclusion 120587minus1(119911) sub119911lowast+ 119873(119872) holds

(2) There exists a number 119903 gt 0 such that for any 119886 isin

119873(119872) the ball 119909 | |119909 minus 119886| le 119903 119909 isin 119873(119872) containsat least one point of the set 120587minus1(119911) 119911 isin 119872

(3) 119873(119872) is a subspace ofR119899 Moreover if119872 is a compactset then119873(119872) = R119899

Proof (1) Let 119911 isin 119872 and 119911lowastisin 120587minus1

(119911) Then for any 119911lowast isin120587minus1

(119911) we obtain from (16) that 119911lowast minus 119911lowastisin 119873(119872) that is

119911lowast

isin 119911lowast+ 119873(119872) Therefore 120587minus1(119911) sub 119911

lowast+ 119873(119872)

(2) Without loss of generality we assume that 1199111 = 0 anddim conv120587minus1(119911) = 119896 Then 120587minus1(119911) contains a linearly inde-pendent set of 119896 vectors say 1199112 1199113 119911119896+1 Denote the 119896-dimensional prism constructed on the vectors 1199112 1199113 119911119896+1by 119875 Since 1199111 = 0 therefore by (14) and (16) we obtainconv120587minus1(119911) = 119873(119872) Cover the set conv120587minus1(119911) by 119875 andprisms that can be obtained by parallel translation of 119875 Inview of (16) vertices of these prisms belong to 120587minus1(119911) Thisimplies in particular that119873(119872) is a 119896-dimensional subspaceofR119899 To prove part (2) of lemma it suffices to put 119903 = diam119875

(3) If 119872 is a compact set then there is a bounded set1198720sub R119899 of nonempty interior such that 120587 119872

0rarr 119872 is

one-to-one [29] Since R119899 can be represented as a union of1198720and the bounded sets119872

11198722 which can be obtained

by parallel translation of1198720and whose interiors are disjoint

[29] therefore for any 119911 isin 119872 each of the compact sets119872119894 119894 =

0 1 contains only one element of120587minus1(119911)This implies thatdim conv120587minus1(119911) = 119899 which is the desired conclusion

We now study some examples to illustrate the set119873(119872)

Example 7 Let119872 be a plane Then clearly119873(119872) = 0

Example 8 Let

119872

= 119911 = (119911(1)

119911(2)

119911(3)

) isin R3

| (119911(1)

)

2

+ (119911(2)

)

2

= 1

(17)

that is119872 is two-dimensional cylinder in R3 Then [30] forsome 119886 119886 isin [0 2120587]

120587minus1

(119911) = 119909 = (119909(1)

119909(2)

) isin R2

| 119909(2)

= 119911(3)

119909(1)

= 119886

+ 2120587119899 119899 isin Z 119911 isin 119872

(18)

whereZ is the set of integers It is obvious that the set119873(119872) =

119911 = (119909(1)

119909(2)

) isin R2 | 119909(2) = 0 is a straight line Here thenumber 119903mentioned in Lemma 6 can be any number not lessthan 2120587

Example 9 Let now119872 be two-dimensional torus119872 = 1198781

times

1198781 where 1198781 is a unit circle then [30]

120587minus1

(119911) = 119909 = (119909(1)

119909(2)

) isin R2

119909(1)

= 119886 + 2120587119899 119909(2)

= 119887 + 2120587119896 119899 119896 isin Z 119911 isin 119872

(19)

Discrete Dynamics in Nature and Society 5

where 119886 and 119887 are some numbers in [0 2120587] Therefore119873(119872)

is a plane and the number 119903 mentioned in Lemma 6 can befor example 120587radic2

5 Main Result

We now formulate the main result of the present paper

Theorem 10 (i) If there exist points

1199111 1199112 119911

119898isin R119899

119908119894119905ℎ 119911119894isin 120587minus1

(1199091198940) = 119909

1

1198940 1199092

1198940 119894 = 1 2 119898

(20)

and number 119895 isin 1 2 such that

119910119895

0isin int conv 119911

1 1199112 119911

119898 (21)

then pursuit can be completed in game (11)-(12) (and hence ingame (2) (4) (5) (7) and (10)) for the time

119879 =

1

119860

119898

sum

119894=1

10038161003816100381610038161003816119911119894minus 119910119895

0

10038161003816100381610038161003816

119860 = min|V|=1

119898

sum

119894=1

(1003816100381610038161003816⟨119890119894 V⟩100381610038161003816

1003816minus ⟨119890119894 V⟩)

(22)

where

1199101

0 1199102

0 = 120587

minus1

(1199100)

119890119894=

(119910119895

0minus 119911119894)

10038161003816100381610038161003816119910119895

0minus 119911119894

10038161003816100381610038161003816

119894 = 1 2 119898

(23)

(ii) If for any set 1199111 1199112 119911

119898 with 119911

119894isin 120587minus1

(1199091198940) 119894 =

1 2 119898 and for any number 119895 isin 1 2 inclusion (21)fails to hold then evasion is possible in game (11)-(12) (andhence in game (2) (4) (5) (7) and (10))

It should be noted that condition (21) implies that 119898 gt 119899

and 119860 gt 0

Proof

Part I It follows from condition (14) that if for a number119895 isin 1 2 there exists a set 119911

1 1199112 119911

119898 to satisfy (21)

then for each 119895 isin 1 2 such a set exists as well Thereforefrom now on one writes 119910

0and 119910(119905) instead of 119910119895

0and 119910119895(119905)

respectively This means that the group of evaders consists ofonly one evader

Assume that there exists a set 1199111 1199112 119911

119898 with 119911

119894isin

120587minus1

(1199091198940) 119894 = 1 2 119898 satisfying inclusion (21) Without

loss of generality we may assume that 119911119894= 1199091

1198940 119894 = 1 119898

Let V(sdot) be an arbitrary control of the evader Let thepursuers use the strategy of parallel approach [33]

119880119894(V) = V + 120582

119894(V) 119890119894

120582119894(V) = minus ⟨V 119890

119894⟩ + radic1 minus |V|2 + ⟨V 119890

119894⟩2

(24)

Analysis similar to [8] shows that pursuit can be com-pleted for time 119879

Part II We now turn to the case where there is no set1199111 1199112 119911

119898 with 119911

119894isin 120587minus1

(1199091198940) 119894 = 1 2 119898 to satisfy

(21) We need to consider the following two cases

Case 1 (119898 le 119899) In this case observe that int conv1199111 1199112

119911119898 = 0 and inclusion (21) fails to hold for any set 119911

1

1199112 119911

119898 119911119894isin 120587minus1

(1199091198940) 119894 = 1 2 119898

For simplicity of notation we denote the (119898 + 1)-tuple(120587minus1

(1199091) 120587minus1

(1199092) 120587

minus1

(119909119898) 119910) by 119908 Set

119882(119908) = 119909119895

119894|

10038161003816100381610038161003816119909119895

119894minus 119910

10038161003816100381610038161003816le

ℎ

2

at some 119894

isin 1 2 119898 119895 isin 1 2

(25)

From now on we assume that the number ℎ gt 0 satisfiesinequalities (15) for all 119911 = 119911

119894 119894 = 1 119898 It follows

from inequality (15) that for each 119894 isin 1 119898 the set119882(119908) contains not more than one element of the set 120587minus1(119909

119894)

Therefore for any119908 the number of elements of the set119882(119908)

does not exceed119898Let the partition Δ be defined by numbers 119905

119896= 119896ℎ4 119896 =

0 1 Define the strategy 119881(119908) of the evader by requiringconditions

min|V|=1

max119909119895

119894isin119882(119908)

⟨

119909119895

119894minus 119910

10038161003816100381610038161003816119909119895

119894minus 119910

10038161003816100381610038161003816

V⟩

= max119909119895

119894isin119882(119908)

⟨

119909119895

119894minus 119910

10038161003816100381610038161003816119909119895

119894minus 119910

10038161003816100381610038161003816

119881 (119908)⟩ |119881 (119908)| = 1

(26)

Since the set 119882(119908) is finite max in (26) attains In view of119898 le 119899 we conclude that 119881(119908)must satisfy inequality

max119909119895

i isin119866(119908)

⟨

119909119895

119894minus 119910

10038161003816100381610038161003816119909119895

119894minus 119910

10038161003816100381610038161003816

119881 (119908)⟩ le 0 (27)

Let 1199061(sdot) 1199062(sdot) 119906

119898(sdot) be any controls of the groups of

pursuers and let 119910(sdot) 119909119895119894(sdot) 119895 = 1 2 119894 = 1 2 119898 be

the trajectories generated by these controls initial positionsstrategy 119881(119908) and partition Δ We shall have establishedthe theorem if we prove the following statement if theinequalities 119909119895

119894(119905) = 119910(119905) 119894 = 1 2 119898 119895 = 1 2 hold

for all 119905 isin [119905119896minus1 119905119896] then they hold true for all 119905 isin [119905

119896 119905119896+1]

119896 = 1 2 as well (recall 1199050= 0) Then clearly 119909119895

119894(119905) = 119910(119905)

for all 119894 = 1 2 119898 119895 = 1 2 and 119905 ge 0Let 11990911(119905119896) isin 119882(119908(119905

119896)) Then (27) implies that

⟨

1199091

1(119905119896) minus 119910 (119905

119896)

10038161003816100381610038161199091

1(119905119896) minus 119910 (119905

119896)1003816100381610038161003816

119881 (119908 (119905119896))⟩ le 0 (28)

6 Discrete Dynamics in Nature and Society

and so the greatest (nonacute) angle of the triangle withvertices 119910(119905) = 119910(119905

119896) + (119905 minus 119905

119896)119881(119908(119905

119896)) 11990911(119905119896) and 119910(119905

119896) is at

the vertex 119910(119905119896)Therefore

100381610038161003816100381610038161199091

1(119905119896) minus 119910 (119905)

10038161003816100381610038161003816gt1003816100381610038161003816(119905 minus 119905119896) 119881 (119908 (119905

119896))1003816100381610038161003816= 119905 minus 119905

119896

119905 isin [119905119896 119905119896+1]

(29)

This inequality and (12) now give100381610038161003816100381610038161199091

1(119905) minus 119910 (119905)

10038161003816100381610038161003816ge

100381610038161003816100381610038161199091

1(119905119896) minus 119910 (119905)

10038161003816100381610038161003816minus

100381610038161003816100381610038161199091

1(119905) minus 119909

1

1(119905119896)

10038161003816100381610038161003816

gt 119905 minus 119905119896minus

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119896

1199061(120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

ge 119905 minus 119905119896minus int

119905

119905119896

10038161003816100381610038161199061(120591)1003816100381610038161003816119889120591 ge 0

119905119896le 119905 le 119905

119896+1

(30)

We can now turn to the case 11990911(119905119896) notin 119882(119908(119905

119896)) For 119905

119896le

119905 le 119905119896+1

first using the triangle inequality and then (12) (25)and (26) we get100381610038161003816100381610038161199091

1(119905) minus 119910 (119905)

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1199091

1(119905119896) + int

119905

119905119896

1199061(120591) 119889120591 minus 119910 (119905

119896) minus int

119905

119905119896

119881 (119908 (119905119896)) 119889120591

100381610038161003816100381610038161003816100381610038161003816

ge

100381610038161003816100381610038161199091

1(119905119896) minus 119910 (119905

119896)

10038161003816100381610038161003816minus

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119896

1199061(120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

minus

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119896

119881 (119908 (119905119896)) 119889120591

100381610038161003816100381610038161003816100381610038161003816

gt

ℎ

2

minus int

119905

119905119896

10038161003816100381610038161199061(120591)1003816100381610038161003816119889120591 minus (119905 minus 119905

119896) ge

ℎ

2

minus (119905 minus 119905119896) minus

ℎ

4

ge

ℎ

2

minus

ℎ

4

minus

ℎ

4

= 0

(31)

Inequalities (30) and (31) show that for any location of119908(119905119896) the inequality 1199091

1(119905) = 119910(119905) 119905 isin [119905

119896 119905119896+1] holds In a

similar way one can prove that 119909119895119894(119905) = 119910(119905) 119894 = 1 2

119898 119895 = 1 2 for all 119905 isin [119905119896 119905119896+1] This is our assertion in the

case119898 le 119899

Case 2 (119898 gt 119899) Let 119873 = 119873(119872) (see (16)) If 119898 gt 119899 anddim119873 = 119899 then it is not difficult to verify using the secondstatement of Lemma 6 that there exists a set

1199111 1199112 119911

119898 119911

119894isin 1199091

1198940 1199092

1198940 119894 = 1 2 119898 (32)

to satisfy inclusion (21) Therefore if 119898 gt 119899 and 1199100notin

int conv1199111 1199112 119911

119898 for any set 119911

1 1199112 119911

119898 with 119911

119894isin

1199091

1198940 1199092

1198940 119894 = 1 2 119898 then necessarily dim119873 lt 119899

Thus it remains to study the case where119898 gt 119899 and dim119873 lt

119899Denote by 119873perp the orthogonal complement of the sub-

space 119873 and by 119865 119865 R119899 rarr 119873perp the orthogonal projection

operator Then clearly 119865 119873 rarr 0 and it follows from(14) that the set 119865120587minus1(119911) 119911 isin 119872 consists of a unique pointTherefore we denote

119865120587minus1

(1199100) = 119887

119865120587minus1

(1199091198940) = 119886119894 119894 = 1 2 119898

(33)

Show that the following inclusion

119887 isin int conv 1198861 1198862 119886

119898 (34)

fails to hold in the subspace119873perp where the operations int andconv act on the subspace 119873perp Assume the contrary let (34)hold

It follows from condition (14) that if for a number 119895 isin

1 2 there exists a set 1199111 1199112 119911

119898 to satisfy (21) then

for each 119895 isin 1 2 also such a set exists Therefore fromnow on one writes 119910

0= 119887 = 0 and 119910(119905) instead of 119910119895

0and

119910119895

(119905) respectively This means that group of evaders consistsof only one evader

Since 119898 gt 119899 and dim119873perp

lt 119899 then in view of (34) thepoint 119887 = 0 belongs to convex hull of some 119899 points from1198861 1198862 119886

119898 Without loss of generality we assume that

0 isin conv 1198861 1198862 119886

119899 (35)

in the subspace119873perpNext let 119901 isin 119873 be a unit vector and let 119860 be a positive

number Choose linearly independent set of vectors 119911119894isin

120587minus1

(1199091198940) 119894 = 1 2 119899 so that ⟨119911

119894 119901⟩ gt 119860 Construct the

cone

119870 = 119911 isin R119899

119911 = 12057211199111+ 12057221199112+ sdot sdot sdot + 120572

119899119911119899 120572119894ge 0 119894

= 1 2 119899

(36)

and its conjugate

119870lowast

= 119911 isin R119899

119911 = 12057211199111+ 12057221199112+ sdot sdot sdot + 120572

119899119911119899 120572119894

le 0 119894 = 1 2 119899

(37)

Since the set of vectors 1199111 1199112 119911

119899 is linearly independent

therefore these cones have nonempty interiors Observe that119901 isin 119870 and minus119901 isin 119870lowast It follows from inclusions minus119901 isin 119870lowast 119901 isin119873 and statement (2) of Lemma 6 that the set

int119870lowast cap 120587minus1 (119909119899+1) (38)

has infinitely many elements Choose a vector 119911119899+1

from thisset Then by construction of the cones 119870 and 119870lowast in view of(35) we have

0 isin int conv 1199111 1199112 119911

119899+1 (39)

By hypothesis of the theorem there is no set 1199111 1199112 119911

119898

with 119911119894

isin 120587minus1

(1199091198940) 119894 = 1 2 119898 to satisfy (21) a

contradictionThus (34) does not hold and so

0 int conv 1198861 1198862 119886

119898 (40)

Discrete Dynamics in Nature and Society 7

From this it follows that there exists a unit vector 1198810isin 119873perp

such that

⟨1198810 119909119895

1198940minus 1199100⟩ = ⟨119881

0 119865120587minus1

(1199091198940) minus 119865120587

minus1

(1199100)⟩

= ⟨1198810 119886119894⟩ le 0

119894 = 1 2 119898 119895 = 1 2

(41)

Let the evader use the constant control 119881(119908) = 1198810 119905 ge 0

We conclude from (41) that for any 119894 isin 1 2 119898 119895 isin

1 2 and 119905 ge 0 the greatest angle of the triangle withvertices 119909119895

1198940 1199100 and 119910(119905) = 119910

0+ 1198810119905 is at the vertex 119910

0 This

gives10038161003816100381610038161003816119910 (119905) minus 119909

119895

1198940

10038161003816100381610038161003816gt1003816100381610038161003816119910 (119905) minus 119910

0

1003816100381610038161003816= 119905 (42)

Next by the restriction |119906119894(sdot)| le 1 we obtain

10038161003816100381610038161003816119909119895

119894(119905) minus 119909

119895

1198940

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

int

119905

0

119906119894(119904) 119889119904

10038161003816100381610038161003816100381610038161003816

le int

119905

0

1003816100381610038161003816119906119894(119904)1003816100381610038161003816119889119904 le 119905 (43)

From the last two relations we obtain 119910(119905) = 119909119895

119894(119905) for all 119894 =

1 2 119898 119895 = 1 2 and 119905 ge 0 and the proof is complete

Remark 11 Obviously if119898 le 119899 inclusion (21) fails to hold Inthis case by Theorem 10 evasion is possible from any initialpoints In addition in the proof of this statement we haveused only inequality (15) and other properties of multivaluedmapping 120587minus1 119872 rarr R119899 have not been used Therefore if119898 le 119899 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 10is fairly easy to check In the following example we giveconditions which are equivalent to condition (21)

Example 12 Let119872 be the cylinder given in Example 8 Thenin view of (18) one can verify that condition (21) is equivalentto the following inequalities

min1le119894le119898

119909(3)

1198940lt 119910(3)

0lt max1le119894le119898

119909(3)

1198940 119898 ge 3 (44)

Next in the case where the set 119872 is compactdim119873(119872) = 119899 by Lemma 6 and there exists a number119903 such that each ball of radius 119903 in R119899 contains at leastone point of the set 120587minus1(119909) 119909 isin 119872 Therefore if 119898 gt 119899inclusion (21) holds According to this property we obtainthe following corollary of Theorem 10

Corollary 13 If 119872 is a compact set and 119898 gt 119899 thenfor any initial position of the players (119909

10 11990920 119909

1198980 1199100)

1199091198940

= 1199100 119894 = 1 119898 there exists a positive number 119879 =

119879(11990910 11990920 119909

1198980 1199100) such that pursuit can be completed for

time 119879

Remark 14 Condition (21) is equivalent to the following onethere exist geodesics 120574

119894 [0 120572

119894] rarr 119872 such that 120574

119894(0) =

1199100 120574119894(120572119894) = 1199091198940 119894 = 1 2 119898 and the inclusion

0 isin int conv 12057410158401(0) 1205741015840

2(0) 120574

1015840

119898(0) (45)

holds in the tangent space 1198791198721199100 Note that such geodesics in

general are not unique

6 Conclusion

We have studied pursuit and evasion differential games of 119898pursuers and one evader onmanifolds with EuclideanmetricThe following are our main contributions

(1) We have proposed a method of reduction of simplemotion differential game with many pursuers andone evader on a class of manifolds (eg cylinderand torus) to an equivalent differential game in R119899which contains119898 groups of countablymany pursuers1199091

119894 1199092

119894 119894 = 1 119898 and one group of countably

many evaders 1199101 1199102 (2) We have obtained necessary and sufficient condition

of evasion in the equivalent game It should be notedthat even though the condition 119910

119897

0isin 119909

119895

1198940 119894 =

1 119898 119895 = 1 2 for some positive integer 119897 ismore similar to the condition of Pshenichnii [8] thancondition (21) it is not sufficient for completion ofpursuit in the game

(3) If condition (21) is not satisfied we have constructedan evasion strategy and proved that evasion is possi-ble Note that many researchers (see eg Pshenichnii[8] and Grigorenko [3]) suggested to the evader aconstant velocity depending on initial positions ofplayers which guarantees the evasion in R119899 In thegamewe studied evasion using constant velocity doesnot work In general the evader will be captured if itmoves with constant velocity Evasion strategy usedin the present paper depends on current positions ofplayers and requires a specific construction

Competing Interests

The authors declare that they have no competing interests

References

[1] F L Chernousrsquoko ldquoA problem of evasion of several pursuersrdquoJournal of Applied Mathematics and Mechanics vol 40 no 1pp 14ndash24 1976

[2] A Chikrii Conflict-Controlled Processes vol 405 ofMathemat-ics and Its Applications Springer Science amp Business MediaDordrecht The Netherlands 1997

[3] N L Grigorenko ldquoOn the problem of group pursuitrdquo Proceed-ings of the Steklov Institute ofMathematics vol 2 pp 73ndash81 1990

[4] E F Mishchenko M S Nikolrsquoskii and N Yu Satimov ldquoEvoid-ance encounter problem in differential games of many personsrdquoTrudy MIAN USSR vol 143 pp 105ndash128 1977

[5] N N Petrov ldquoSimple pursuit of rigidly linked evadersrdquoAvtomatika i Telemekhanika no 12 pp 89ndash96 1997 EnglishTranslation in Automation and Remote Control vol 58 no 12pp 1914ndash1919 1998

[6] B N Pshenichnii A A Chikrii and I S Rappoport ldquoEffectivemethod of solving differential games with many pursuersrdquoDoklady Akademii Nauk SSSR vol 256 no 3 pp 530ndash535 1981

8 Discrete Dynamics in Nature and Society

[7] N Yu Satimov and M Sh Mamatov ldquoThe pursuit-evasionproblem in differential games between the groups of pursuersand evadersrdquo Differentsialnie Uravneniya vol 26 no 9 pp1541ndash1551 1990

[8] B N Pshenichnii ldquoSimple pursuit by several objectsrdquo Kiber-netika vol 3 pp 145ndash146 1976

[9] P V Prokopovich and A A Chikrii ldquoA linear evasion problemfor interacting groups of objectsrdquo Journal of Applied Mathemat-ics and Mechanics vol 58 no 4 pp 583ndash591 1994

[10] V L Zak ldquoOn a problem of evading many pursuersrdquo Journal ofApplied Mathematics and Mechanics vol 43 no 3 pp 492ndash5011978

[11] P Borowko andW Rzymowski ldquoAvoidance ofmany pursuers inthe simple motion caserdquo Journal of Mathematical Analysis andApplications vol 111 no 2 pp 535ndash546 1985

[12] R P Ivanov and Y S Ledyaev ldquoOptimality of pursuit timein simple motion differential game of many objectsrdquo TrudyMatematicheskogo Instituta imeni VA Steklova vol 158 pp 87ndash97 1981

[13] G I Ibragimov ldquoOptimal pursuit with countably many pur-suers and one evaderrdquo Differential Equations vol 41 no 5 pp627ndash635 2005

[14] G I Ibragimov and B B Rikhsiev ldquoOn some sufficientconditions for optimality of the pursuit time in the differentialgame with multiple pursuersrdquo Automation and Remote Controlvol 67 no 4 pp 529ndash537 2006

[15] G I Ibragimov and M Salimi ldquoPursuit-evasion differentialgame with many inertial playersrdquo Mathematical Problems inEngineering vol 2009 Article ID 653723 15 pages 2009

[16] G Ibragimov N Abd Rasid A Kuchkarov and F IsmailldquoMulti pursuer differential game of optimal approach withintegral constraints on controls of playersrdquo Taiwanese Journalof Mathematics vol 19 no 3 pp 963ndash976 2015

[17] A V Arutyunov and D Y Karamzin ldquoMaximum principle inan optimal control problem with equality state constraintsrdquoDifferential Equations vol 51 no 1 pp 33ndash46 2015

[18] A V Arutyunov and D Y Karamzin ldquoNon-degenerate neces-sary optimality conditions for the optimal control problemwithequality-type state constraintsrdquo Journal of Global Optimizationvol 64 no 4 pp 623ndash647 2016

[19] A V Davydova andD Y Karamzin ldquoOn some properties of theshortest curve in a compound domainrdquo Differential Equationsvol 51 no 12 pp 1626ndash1636 2015

[20] A A Melikyan and N V Ovakimyan ldquoA differential game ofsimple approach on manifoldsrdquo Journal of Applied Mathematicsand Mechanics vol 57 no 1 pp 41ndash51 1993

[21] A A Melikyan N V Ovakimyan and L L HarutiunianldquoGames of simple pursuit and approach on a two-dimensionalconerdquo Journal of Optimization Theory and Applications vol 98no 3 pp 515ndash543 1998

[22] N Yu Satimov and A Sh Kuchkarov ldquoDeviation fromencounterwith several pursuers on a surfacerdquoUzbekskiiMatem-aticheskii Zhurnal vol 1 pp 51ndash55 2001

[23] A A Azamov ldquoOn a problem of escape along a prescribedcurverdquo Journal of Applied Mathematics and Mechanics vol 46no 4 pp 553ndash555 1982

[24] A A Azamov and A S Kuchkarov ldquoGeneralized lsquoLion ampManrsquo Game of R Radordquo Contributions to Game Theory andManagement vol 2 pp 8ndash20 2009

[25] A S Kuchkarov ldquoSolution of simple pursuit-evasion problemwhen evader moves on a given curverdquo International GameTheory Review vol 12 no 3 pp 223ndash238 2010

[26] A Sh Kuchkarov ldquoA simple pursuit-evasion problem on a ballof a Riemannian manifoldrdquo Mathematical Notes vol 85 no 2pp 190ndash197 2009

[27] D Vagin and N Petrov ldquoThe problem of the pursuit of agroup of rigidly coordinated evadersrdquo Journal of Computer andSystems Sciences International vol 40 no 5 pp 749ndash753 2001

[28] I A Alias G Ibragimov and A Rakhmanov ldquoEvasion differ-ential game of infinitely many evaders from infinitely manypursuers in Hilbert spacerdquo Dynamic Games and Applicationsvol 6 no 2 pp 1ndash13 2016

[29] E Cartan ldquoSur une classe remarkable despaces de RiemannrdquoBulletin de la Societe Mathematique de France vol 54 pp 214ndash216 1926

[30] V VNikulin and I R ShafarevichGeometriya i Gruppy NaukaMoscow Russia 1983

[31] A Sh Kuchkarov ldquoThe problem of optimal approach in locallyEuclidean spacesrdquo Automation and Remote Control vol 68 no6 pp 974ndash978 2007

[32] A I Subbotin and A G Chentsov Guaranteed Optimization inControl Problems Nauka Moscow Russia 1981 (Russian)

[33] L A Petrosjan Differential Games of Pursuit World ScientificNew York NY USA 1993

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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