research article linear discrete pursuit game problem with...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 840925 5 pageshttpdxdoiorg1011552013840925

Research ArticleLinear Discrete Pursuit Game Problem with Total Constraints

Atamurat Kuchkarov1 Gafurjan Ibragimov2 and Akmal Sotvoldiev1

1 Institute of Mathematics of Uzbekistan 29 Dorman yuli Street 100125 Tashkent Uzbekistan2 Institute for Mathematical Research and Department of Mathematics Faculty of Science (FS) Universiti Putra Malaysia43400 Serdang Selangor Malaysia

Correspondence should be addressed to Gafurjan Ibragimov gafurscienceupmedumy

Received 8 March 2013 Revised 28 May 2013 Accepted 28 May 2013

Academic Editor Valery Y Glizer

Copyright copy 2013 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 study a linear discrete pursuit game problem of one pursuer and one evader Control vectors of the players are subjected to totalconstraints which are discrete analogs of the integral constraints By definition pursuit can be completed in the game if there existsa strategy of the pursuer such that for any control of the evader the state of system 119911(119905) reaches the origin at some time We obtainsufficient conditions of completion of the game from any initial position of the state space Strategy of the pursuer is defined as afunction of the current state of system and value of control parameter of the evader

1 Introduction

Pursuit and evasion differential game problems with integralconstraints for continuous time dynamical systems wereinvestigated in many works (see for example [1ndash14]) Suchconstraints represent the constraints on resource energy fueland so forth Discrete analog of integral constraint is calledtotal constraint

There are many works on discrete controlled systemswhere controls are subjected to total or geometric constraintsSirotin [15] studied some conditions of 0-controllability andasymptotical controllability of discrete time linear systemsIn this work the control is bounded in Holder norm Inparticular he obtained necessary and sufficient conditionsof 0-controllability and asymptotical controllability of thesystem

Sazanova [16] studied the problem of steering the state ofa linear discrete system from the given point at a given timeto another given point at a given time by a control whichminimizes a given functional She constructed an optimalprogram control

However discrete linear pursuit games with total con-straints on control vectors of players were studied in a fewworks for example [17ndash20] Satimov et al [19] examinedlinear discrete pursuit games of many objects under total

constraints on controls of players Game is described by thefollowing systems

119911119894 (119899 + 1) = 119862119894119911119894 (119899) + 119906119894 (119899) minus V (119899)

119911119894 (0) = 1199111198940 119894 = 1 119898

(1)

It is assumed that the eigenvalues of matrices 119862119894are real

Sufficient conditions of completion of pursuit were obtainedThe paper of Satimov and Ibragimov [20] was also

devoted to discrete time pursuit games of several playersdescribed by the same linear systemsThey obtained sufficientconditions of completion of the game for any matrices 119862

119894 It

should be noted that in case of one pursuer these conditionsare also necessary In particular they obtained necessary andsufficient conditions of 0-controllability when the coefficientmatrix of control is an identity matrix

Azamov and Kuchkarov [17] obtained necessary and suf-ficient conditions of solvability of both the 0-controllabilityproblem and pursuit problem for a linear discrete systemwhen the control parameter of pursuer is subjected togeometric constraint and that of evader is subjected to totalconstraint

In the monograph of Azamov [21] the fundamentals oftheory of discrete time pursuit and evasion dynamic games

2 Abstract and Applied Analysis

are given systematically The main notions of the theory ofdiscrete games were explained by interesting examples

In the present paper we study a linear discrete pursuitgame problem described by the linear equation with constantcoefficients of general form Controls of both pursuer andevader are subjected to total constraints We give sufficientconditions of completion of pursuit from any initial position

2 Statement of the Problem

We consider a discrete pursuit game described by the follow-ing equation

119911 (119905 + 1) = 119860119911 (119905) + 119861119906 (119905) + 119862V (119905) (2)

where 119905 is a step number which belongs to the set ofnonnegative integers 119911(119905) isin R119899 119860 119861 and 119862 are 119899 times 119899 119899 times119898and 119899times119897 constantmatrices respectively 119906 isin R119898 is the controlparameter of pursuer and V isin R119897 is that of evader

Definition 1 Sequences 119906(sdot) 119873 rarr R119898 and V(sdot) 119873 rarr R119897subjected to the following total constraints

119906 (sdot)119897119901= (

infin

sum

119905=0

|119906(119905)|119901)

1119901

le 120588 (3)

V(sdot)119897119901 = (infin

sum

119905=0

|V(119905)|119901)1119901

le 120590 (4)

are referred to as the controls of the pursuer and evaderrespectively where119873 is the set of nonnegative integers 120588 gt 0and 120590 ge 0 are given numbers and 119901 gt 1

The conditions (3) and (4) are the discrete analogs ofintegral constraints and they are called total constraints

Definition 2 A function 119880(119911 V) 119880 R119899 times R119897 rarr R119898 iscalled strategy of the pursuerThe triple (119911

0 119880 V(sdot)) consisting

of a given initial point 1199110 the strategy 119880 and a control of the

evader V(sdot) generates the unique trajectory 119911(119905) defined by thefollowing formula

119911 (119905 + 1) = 119860119911 (119905) + 119861119880 (119911 (119905) V (119905)) + 119862V (119905)

119911 (0) = 1199110

(5)

The sequence 119906(119905) = 119880(119911(119905) V(119905)) 119905 isin 119873 is referred to as therealization of the strategy 119880 The strategy of the pursuer 119880 iscalled admissible if all its realizations with fixed 119911

0satisfy the

condition (3)

Definition 3 Pursuit is said to be completed in the game (2)ndash(4) from the initial position 119911

0if there exists a strategy of

the pursuer 119880 such that for any control of the evader V(sdot) thetrajectory generated by 119911

0 119880 and V(sdot) satisfies the condition

119911(119905) = 0 at some 119905 119905 isin 119873

The aim of the pursuer is to realize the equality 119911(119905) =0 as earlier as possible Thus we deal with pursuit problem

The pursuer uses a strategy and the evader uses any controlV(sdot) By Definition 2 at any time 119905 the value of the strategy isconstructed based on the state 119911(119905) and value of the controlparameter of the evader V(119905)

Problem 1 Find the conditions under which pursuit can becompleted in the game (2)ndash(4) from any initial position

3 Main Result

To prove the main result we first prove two lemmas Let1205821 1205822 120582

119899be the eigenvalues of the matrix 119860 Denote

120573 = max1le119894le119899

1003816100381610038161003816120582119894

1003816100381610038161003816 119862 = max

|119911|=1

|119862119911|

119902 =

119901

119901 minus 1

(6)

Let119881 be the set of all sequences V(sdot) that satisfy the constraint(4)

Lemma 1 If 120573 lt 1 then

(a) for all 119905 = 1 2 the set

119883(119905) =

119905

sum

119894=0

119860119894119862V (119905 minus 119894) | V (sdot) isin 119881 (7)

is contained in a ball 119878119899119903= 119909 isin R119899 | |119909| le 119903

(b) for any triple consisting of V(sdot) isin 119881 120576 gt 0 and 1199050isin

0 1 2 there exists a number 119905lowast 119905lowast gt 1199050 such that

10038161003816100381610038161003816100381610038161003816100381610038161003816

119905lowast

sum

119894=0

119860119894119862V (119905lowast minus 119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576 (8)

Proof Since 120573 lt 1 then 119860119905 rarr 0 as 119905 rarr infin thereforethere exists a positive integer ℎ such that 120583 def

= 119860ℎ lt 1

Then according to the property of norm we have

10038171003817100381710038171003817119860ℎ11989610038171003817100381710038171003817le

10038171003817100381710038171003817119860ℎ10038171003817100381710038171003817

119896

lt 120583119896 119896 = 1 2 (9)

Clearly

1003816100381610038161003816100381610038161003816100381610038161003816

119896ℎ

sum

119894=0

119860119894119862V (119896ℎ minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119896

sum

119894=0

119860ℎ119894

ℎminus1

sum

119895=0

119860119895119862V ((119896 minus 119894) ℎ minus 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119896

sum

119894=0

10038171003817100381710038171003817119860ℎ11989410038171003817100381710038171003817

ℎminus1

sum

119895=0

(

1003817100381710038171003817100381711986011989510038171003817100381710038171003817sdot 119862 sdot

1003816100381610038161003816V ((119896 minus 119894) ℎ minus 119895)100381610038161003816

1003816)

119896 = 1 2

(10)

Abstract and Applied Analysis 3

Using the Holder inequality we obtain for any V(sdot) isin 119881 that

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817sdot1003816100381610038161003816V ((119896 minus 119894) ℎ minus 119895)100381610038161003816

1003816

le (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(

ℎminus1

sum

119895=0

1003816100381610038161003816V((119896 minus 119894)ℎ minus 119895)100381610038161003816

1003816

119901)

1119901

le 119886120590 119896 = 1 2 119886 = (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(11)

It follows from (9)ndash(11) that1003816100381610038161003816100381610038161003816100381610038161003816

119896ℎ

sum

119894=0

119860119894119862V (119896ℎ minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119886120590 119862

119896

sum

119894=0

120583119894= 119886120590 119862

1 minus 120583119896+1

1 minus 120583

le

119886120590 119862

1 minus 120583

(12)

for all 119896 = 1 2 Let 119903 = 119886120590119862(1 minus 120583)minus1 It is easy to

check that if 0 lt 1199051lt 1199052 then 119883(119905

1) sub 119883(119905

2) Therefore

the inequality (12) implies that1003816100381610038161003816100381610038161003816100381610038161003816

119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119903 119905 = 1 119896ℎ (13)

Since (12) holds for any positive integer 119896 then the statement(a) of Lemma 1 follows

We now turn to the statement (b) of the lemma Assumethe contrary there exist a control V(sdot) isin 119881 and positivenumbers 120576 119905

0such that1003816100381610038161003816100381610038161003816100381610038161003816

119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

gt 120576 forall119905 ge 1199050 (14)

It follows from (9) that

max119906isin119878119899119903

10038161003816100381610038161003816119860ℎ119905119906

10038161003816100381610038161003816le max119906isin119878119899119903

(

10038171003817100381710038171003817119860ℎ11990510038171003817100381710038171003817sdot |119906|) le 120583

119905119903

119905 = 0 1 2

(15)

Since 0 le 120583 lt 1 then we can conclude that there exists apositive integer 119879 119879 ge 119905

0+ 1 such that

max119906isin119878119899119903

10038161003816100381610038161003816119860119905119906

10038161003816100381610038161003816le

120576

2

forall 119905 ge 119879 (16)

Clearly (14) implies that

120576 lt

1003816100381610038161003816100381610038161003816100381610038161003816

(119896+1)119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198681+ 1198682

1198681=

1003816100381610038161003816100381610038161003816100381610038161003816

119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

1198682=

1003816100381610038161003816100381610038161003816100381610038161003816

119860119879

(119896+1)119879minus1

sum

119894=119879

119860119894minus119879119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

(17)

Setting 119895 = 119894 minus 119879 we get

(119896+1)119879minus1

sum

119894=119879


=

119896119879minus1

sum

119894=0

119860119895119862V (119896119879 minus 1 minus 119895)

(18)

According to the part (a) of Lemma 1 this vector belongs to119878119899

119903 Then by (16) 119868

2le 1205762 Hence from (17) we obtain that

120576 lt 1198681+ (1205762) Therefore using the Holder inequality yields

120576

2

lt 1198681le 119887(

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901)

1119901

(19)

where 119887 = (sum119879minus1119894=0

119860119894119862

119902

)

1119902

This gives

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901 gt (120576

2119887

)

119901

(20)

Substituting 119895 = (119896 + 1)119879 minus 1 minus 119894 and then replacing 119895 by 119894again we obtain

(119896+1)119879minus1

sum

119894=119896119879

|V (119894)|119901 gt (120576

2119887

)

119901

forall119896 = 1 2 (21)

We conclude from these inequalities that

(119896+1)119879minus1

sum

119894=119879

|V(119894)|119901 gt 119896(120576

2119887

)

119901

119896 = 1 2 (22)

According to (4) for any 119896 the left hand side of this inequalityis bounded by120590119901 while the right hand side rarr infin as 119896 rarr infinContradiction The proof of Lemma 1 is complete

Lemma 2 If 120573 lt 1 then for any V(sdot) isin 119881 the solution 119911(sdot) =119911(sdot V(sdot) 119911

0) of the system

119911 (119905 + 1) = 119860119911 (119905) + 119862V (119905) 119911 (0) = 1199110 (23)

satisfies the equality

120574 (119911 (sdot V (sdot) 1199110)) = inf119905ge0

1003816100381610038161003816119911 (119905 V (sdot) 1199110)

1003816100381610038161003816= 0 (24)

Proof Let V(sdot) isin 119881 and 120576 be an arbitrary positive numberSince120573 lt 1 then the system 119911(119905+1) = 119860119911(119905) is asymptoticallystable Therefore for any 119911

0there exists a number 119905

1such that

100381610038161003816100381610038161198601199051199110

10038161003816100381610038161003816lt

120576

2

119905 ge 1199051 (25)

Next by Lemma 1 there exists a number 1199052(1199052ge 1199051) such

that1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

120576

2

(26)


Using (25) and (26) we have

120574 (119911 (119905 V (sdot) 1199110))

le1003816100381610038161003816119911 (1199052 V (sdot) 1199110)

1003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

11986011990521199110+

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381611986011990521199110

10038161003816100381610038161003816+

1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

lt

120576

2

+

120576

2

= 120576

(27)

As the number 120576 is arbitrary then 120574(119911(119905 V(sdot) 1199110)) = 0 and

Lemma 2 follows

Theorem 3 If 120573 lt 1 and dim1198611198781198981= 119899 then pursuit can be

completed in the game (2)ndash(4) from any initial position 1199110isin

R119899

Proof Define the strategy of the pursuer as follows

119861119880 (119911 V) = 0 if 119860119911 + 119862V notin 119861119878119898

120588

minus119860119911 minus 119862V if 119860119911 + 119862V isin 119861119878119898120588

(28)

Since dim1198611198781198981= 119899 the set119861119878119898

120588contains a ball 119878119899

120575of radius

120575 gt 0 of the spaceR119899 It follows from Lemma 2 that for everycontrol V(sdot) isin 119881 there exists a step 119905 isin 119873 such that |119860119911(119905) +119862V(119905)| le 120575 Consequently for some 120591

119860119911 (120591) + 119862V (120591) isin 119861119878119898120588 (29)

Then by (28) we have 119861119880(119911(120591) V(120591)) = minus119860119911(120591) minus 119862V(120591) andtherefore

119911 (120591 + 1) = 119860119911 (120591) + 119861119880 (119911 (120591) V (120591)) + 119862V (120591) = 0 (30)

This proves the theorem

We now give an illustrative example

Example 4 Consider the following discrete system in R2

119911 (119905 + 1) = [03 minus04

04 03] 119911 (119905)

+ [1 2 0

minus1 1 3] 119906 (119905) + [

1 2 minus1 3

3 1 4 5] V (119905)

(31)

where 119911(119905) isin R2 119906(119905) = (1199061(119905) 1199062(119905) 1199063(119905)) and V(119905) =

(V1(119905) V2(119905) V3(119905) V4(119905)) Eigenvalues of the matrix

119860 = [03 minus04

04 03] (32)

are 12058212= 03 plusmn 04119894 and |120582

12| = 05 lt 1 Next it is not

difficult to verify that dim11986111987831= 2 where

1198611198783

1= 119861120585 | 120585 = (120585

1 1205852 1205853) 1205852

1+ 1205852

2+ 1205852

3le 1 (33)

Thus all hypotheses ofTheorem 3 are satisfied and thereforepursuit can be completed from any initial position 119911

0isin R2

4 Conclusion

In the present paper we have studied a linear discrete pursuitgame problem under total constraints We have obtainedconditions under which pursuit starting from any initialposition can be completed in a finite number of steps Wehave described a strategy for the pursuer which is constructedbased on the information about the position 119911(119905) and valueof the control parameter of the evader V(119905) at each step 119905 Itshould be noted that the resource of the pursuer 120588 does notneed to be greater than that of the evader 120590

Acknowledgment

The present research was partially supported by the NationalFundamental Research Grant Scheme (FRGS) of Malaysia01-01-13-1228FR

References

[1] A Ya Azimov ldquoLinear differential pursuit game with integralconstraints on the controlrdquo Differential Equations vol 11 pp1283ndash1289 1975

[2] A A Chikrii and A A Belousov ldquoOn linear differential gameswith integral constraintsrdquo Memoirs of Institute of Mathematicsand Mechanics Ural Division of RAS Ekaterinburg vol 15 no4 pp 290ndash301 2009 (Russian)

[3] P B Gusiatnikov and E Z Mohonrsquoko ldquoOn 119897infin-escape in a

linear many-person differential game with integral constraintsrdquoJournal of AppliedMathematics andMechanics vol 44 no 4 pp436ndash440

[4] G I Ibragimov A A Azamov and M Khakestari ldquoSolutionof a linear pursuit-evasion game with integral constraintsrdquoANZIAM Journal Electronic Supplement vol 52 pp E59ndashE752010

[5] G I Ibragimov M Salimi and M Amini ldquoEvasion frommany pursuers in simple motion differential game with integralconstraintsrdquo European Journal of Operational Research vol 218no 2 pp 505ndash511 2012

[6] G I Ibragimov and N Yu Satimov ldquoA multiplayer pursuit dif-ferential game on a closed convex set with integral constraintsrdquoAbstract and Applied Analysis vol 2012 Article ID 460171 12pages 2012

[7] R Isaacs Differential Games A Mathematical Theory withApplications to Warfare and Pursuit Control and OptimizationNew York NY USA 1967

[8] NNKrasovskiiTheTheory ofMotionControl NaukaMoscowRussia 1968

[9] A V Mesencev ldquoSufficient conditions for evasion in lineargames with integral constraintsrdquoDoklady Akademii Nauk SSSRvol 218 pp 1021ndash1023 1974

[10] M S Nikolskii ldquoThe direct method in linear differential gameswith integral constraintsrdquo Controlled Systems IM IK SO ANSSSR no 2 pp 49ndash59 1969

[11] L S Pontryagin Selected Scientific Papers vol 2 NaukaMoscow Russia 1988

[12] B N Pshenichnii and V V Ostapenko Differential GamesNaukova Dumka Kiev Russia 1992

[13] N Yu SatimovMethods for Solving a Pursuit Problem in theThe-ory of Differential Games NUUz Press Tashkent Uzbekistan2003 in Russian


[14] V N Ushakov ldquoExtremal strategies in differential games withintegral constraintsrdquo Prikladna Matematika I Mehanika vol36 no 1 pp 15ndash23 1972

[15] A N Sirotin ldquoOn null-controllable and asymptotically null-controllable finitedimensional linear systems with controlsbounded in the Holder norms of controlrdquo Automation andRemote Control vol 60 no 11 part 1 pp 1729ndash1738 1999

[16] L A Sazanova ldquoOptimal control of linear discrete systemsrdquoProceedings of the Steklov Institute of Mathematics supplement2 pp S141ndashS157 2000

[17] A A Azamov and A Sh Kuchkarov ldquoOn controllabilityand pursuit problems in linear discrete systemsrdquo Journal ofComputer and Systems Sciences International vol 49 no 3 pp360ndash365 2010

[18] G I Ibragimov ldquoProblems of linear discrete games of pursuitrdquoMathematical Notes vol 77 no 5 pp 653ndash662 2005

[19] N Yu Satimov B B Rikhsiev and A A Khamdamov ldquoOna pursuit problem for linear differential and discrete n-persongames with integral constraintsrdquo Matematicheskii Sbornik vol46 no 4 pp 459ndash469 1982

[20] N Yu Satimov and G I Ibragimov ldquoOn a pursuit problemfor discrete games with several participantsrdquo Izvestiya VysshikhUchebnykh Zavedeniı Matematika Kazanskiı GosudarstvennyıUniversitet no 12 pp 46ndash57 2004

[21] A A Azamov Fundamentals of Theory of Discrete Games NisoPoligraf Tashkent Uzbekistan 2011 in Russian

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Stochastic AnalysisInternational Journal of


are given systematically The main notions of the theory ofdiscrete games were explained by interesting examples

In the present paper we study a linear discrete pursuitgame problem described by the linear equation with constantcoefficients of general form Controls of both pursuer andevader are subjected to total constraints We give sufficientconditions of completion of pursuit from any initial position

2 Statement of the Problem

We consider a discrete pursuit game described by the follow-ing equation

119911 (119905 + 1) = 119860119911 (119905) + 119861119906 (119905) + 119862V (119905) (2)

where 119905 is a step number which belongs to the set ofnonnegative integers 119911(119905) isin R119899 119860 119861 and 119862 are 119899 times 119899 119899 times119898and 119899times119897 constantmatrices respectively 119906 isin R119898 is the controlparameter of pursuer and V isin R119897 is that of evader

Definition 1 Sequences 119906(sdot) 119873 rarr R119898 and V(sdot) 119873 rarr R119897subjected to the following total constraints

119906 (sdot)119897119901= (

infin

sum

119905=0

|119906(119905)|119901)

1119901

le 120588 (3)

V(sdot)119897119901 = (infin

sum

119905=0

|V(119905)|119901)1119901

le 120590 (4)

are referred to as the controls of the pursuer and evaderrespectively where119873 is the set of nonnegative integers 120588 gt 0and 120590 ge 0 are given numbers and 119901 gt 1

The conditions (3) and (4) are the discrete analogs ofintegral constraints and they are called total constraints

Definition 2 A function 119880(119911 V) 119880 R119899 times R119897 rarr R119898 iscalled strategy of the pursuerThe triple (119911

0 119880 V(sdot)) consisting

of a given initial point 1199110 the strategy 119880 and a control of the

evader V(sdot) generates the unique trajectory 119911(119905) defined by thefollowing formula

119911 (119905 + 1) = 119860119911 (119905) + 119861119880 (119911 (119905) V (119905)) + 119862V (119905)

119911 (0) = 1199110

(5)

The sequence 119906(119905) = 119880(119911(119905) V(119905)) 119905 isin 119873 is referred to as therealization of the strategy 119880 The strategy of the pursuer 119880 iscalled admissible if all its realizations with fixed 119911

0satisfy the

condition (3)

Definition 3 Pursuit is said to be completed in the game (2)ndash(4) from the initial position 119911

0if there exists a strategy of

the pursuer 119880 such that for any control of the evader V(sdot) thetrajectory generated by 119911

0 119880 and V(sdot) satisfies the condition

119911(119905) = 0 at some 119905 119905 isin 119873

The aim of the pursuer is to realize the equality 119911(119905) =0 as earlier as possible Thus we deal with pursuit problem

The pursuer uses a strategy and the evader uses any controlV(sdot) By Definition 2 at any time 119905 the value of the strategy isconstructed based on the state 119911(119905) and value of the controlparameter of the evader V(119905)

Problem 1 Find the conditions under which pursuit can becompleted in the game (2)ndash(4) from any initial position

3 Main Result

To prove the main result we first prove two lemmas Let1205821 1205822 120582

119899be the eigenvalues of the matrix 119860 Denote

120573 = max1le119894le119899

1003816100381610038161003816120582119894

1003816100381610038161003816 119862 = max

|119911|=1

|119862119911|

119902 =

119901

119901 minus 1

(6)

Let119881 be the set of all sequences V(sdot) that satisfy the constraint(4)

Lemma 1 If 120573 lt 1 then

(a) for all 119905 = 1 2 the set

119883(119905) =

119905

sum

119894=0

119860119894119862V (119905 minus 119894) | V (sdot) isin 119881 (7)

is contained in a ball 119878119899119903= 119909 isin R119899 | |119909| le 119903

(b) for any triple consisting of V(sdot) isin 119881 120576 gt 0 and 1199050isin

0 1 2 there exists a number 119905lowast 119905lowast gt 1199050 such that

10038161003816100381610038161003816100381610038161003816100381610038161003816

119905lowast

sum

119894=0

119860119894119862V (119905lowast minus 119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576 (8)

Proof Since 120573 lt 1 then 119860119905 rarr 0 as 119905 rarr infin thereforethere exists a positive integer ℎ such that 120583 def

= 119860ℎ lt 1

Then according to the property of norm we have

10038171003817100381710038171003817119860ℎ11989610038171003817100381710038171003817le

10038171003817100381710038171003817119860ℎ10038171003817100381710038171003817

119896

lt 120583119896 119896 = 1 2 (9)

Clearly

1003816100381610038161003816100381610038161003816100381610038161003816

119896ℎ

sum

119894=0

119860119894119862V (119896ℎ minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119896

sum

119894=0

119860ℎ119894

ℎminus1

sum

119895=0

119860119895119862V ((119896 minus 119894) ℎ minus 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119896

sum

119894=0

10038171003817100381710038171003817119860ℎ11989410038171003817100381710038171003817

ℎminus1

sum

119895=0

(

1003817100381710038171003817100381711986011989510038171003817100381710038171003817sdot 119862 sdot

1003816100381610038161003816V ((119896 minus 119894) ℎ minus 119895)100381610038161003816

1003816)

119896 = 1 2

(10)



ℎminus1

sum

119895=0


1003816

le (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(

ℎminus1

sum

119895=0

1003816100381610038161003816V((119896 minus 119894)ℎ minus 119895)100381610038161003816

1003816

119901)

1119901

le 119886120590 119896 = 1 2 119886 = (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(11)


119896ℎ

sum

119894=0

119860119894119862V (119896ℎ minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119886120590 119862

119896

sum

119894=0

120583119894= 119886120590 119862

1 minus 120583119896+1

1 minus 120583

le

119886120590 119862

1 minus 120583

(12)



1) sub 119883(119905

2) Therefore


119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119903 119905 = 1 119896ℎ (13)



0such that1003816100381610038161003816100381610038161003816100381610038161003816

119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

gt 120576 forall119905 ge 1199050 (14)


max119906isin119878119899119903

10038161003816100381610038161003816119860ℎ119905119906

10038161003816100381610038161003816le max119906isin119878119899119903

(

10038171003817100381710038171003817119860ℎ11990510038171003817100381710038171003817sdot |119906|) le 120583

119905119903

119905 = 0 1 2

(15)


0+ 1 such that

max119906isin119878119899119903

10038161003816100381610038161003816119860119905119906

10038161003816100381610038161003816le

120576

2

forall 119905 ge 119879 (16)


120576 lt

1003816100381610038161003816100381610038161003816100381610038161003816

(119896+1)119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198681+ 1198682

1198681=

1003816100381610038161003816100381610038161003816100381610038161003816

119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

1198682=

1003816100381610038161003816100381610038161003816100381610038161003816

119860119879

(119896+1)119879minus1

sum

119894=119879


1003816100381610038161003816100381610038161003816100381610038161003816

(17)


(119896+1)119879minus1

sum

119894=119879


=

119896119879minus1

sum

119894=0

119860119895119862V (119896119879 minus 1 minus 119895)

(18)


119903 Then by (16) 119868



120576

2

lt 1198681le 119887(

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901)

1119901

(19)

where 119887 = (sum119879minus1119894=0

119860119894119862

119902

)

1119902

This gives

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901 gt (120576

2119887

)

119901

(20)


(119896+1)119879minus1

sum

119894=119896119879

|V (119894)|119901 gt (120576

2119887

)

119901

forall119896 = 1 2 (21)


(119896+1)119879minus1

sum

119894=119879

|V(119894)|119901 gt 119896(120576

2119887

)

119901

119896 = 1 2 (22)



0) of the system

119911 (119905 + 1) = 119860119911 (119905) + 119862V (119905) 119911 (0) = 1199110 (23)


120574 (119911 (sdot V (sdot) 1199110)) = inf119905ge0

1003816100381610038161003816119911 (119905 V (sdot) 1199110)

1003816100381610038161003816= 0 (24)



1such that

100381610038161003816100381610038161198601199051199110

10038161003816100381610038161003816lt

120576

2

119905 ge 1199051 (25)


that1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

120576

2

(26)



120574 (119911 (119905 V (sdot) 1199110))

le1003816100381610038161003816119911 (1199052 V (sdot) 1199110)

1003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

11986011990521199110+

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381611986011990521199110

10038161003816100381610038161003816+

1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

lt

120576

2

+

120576

2

= 120576

(27)


Lemma 2 follows



R119899


119861119880 (119911 V) = 0 if 119860119911 + 119862V notin 119861119878119898

120588


(28)

Since dim1198611198781198981= 119899 the set119861119878119898

120588contains a ball 119878119899

120575of radius


119860119911 (120591) + 119862V (120591) isin 119861119878119898120588 (29)


119911 (120591 + 1) = 119860119911 (120591) + 119861119880 (119911 (120591) V (120591)) + 119862V (120591) = 0 (30)




119911 (119905 + 1) = [03 minus04

04 03] 119911 (119905)

+ [1 2 0

minus1 1 3] 119906 (119905) + [

1 2 minus1 3

3 1 4 5] V (119905)

(31)

where 119911(119905) isin R2 119906(119905) = (1199061(119905) 1199062(119905) 1199063(119905)) and V(119905) =


119860 = [03 minus04

04 03] (32)

are 12058212= 03 plusmn 04119894 and |120582



1198611198783

1= 119861120585 | 120585 = (120585

1 1205852 1205853) 1205852

1+ 1205852

2+ 1205852

3le 1 (33)


0isin R2

4 Conclusion


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎminus1

sum

119895=0


1003816

le (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(

ℎminus1

sum

119895=0

1003816100381610038161003816V((119896 minus 119894)ℎ minus 119895)100381610038161003816

1003816

119901)

1119901

le 119886120590 119896 = 1 2 119886 = (

ℎminus1

sum

119895=0

1003817100381710038171003817100381711986011989510038171003817100381710038171003817

119902

)

1119902

(11)


119896ℎ

sum

119894=0

119860119894119862V (119896ℎ minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119886120590 119862

119896

sum

119894=0

120583119894= 119886120590 119862

1 minus 120583119896+1

1 minus 120583

le

119886120590 119862

1 minus 120583

(12)



1) sub 119883(119905

2) Therefore


119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 119903 119905 = 1 119896ℎ (13)



0such that1003816100381610038161003816100381610038161003816100381610038161003816

119905

sum

119894=0

119860119894119862V (119905 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

gt 120576 forall119905 ge 1199050 (14)


max119906isin119878119899119903

10038161003816100381610038161003816119860ℎ119905119906

10038161003816100381610038161003816le max119906isin119878119899119903

(

10038171003817100381710038171003817119860ℎ11990510038171003817100381710038171003817sdot |119906|) le 120583

119905119903

119905 = 0 1 2

(15)


0+ 1 such that

max119906isin119878119899119903

10038161003816100381610038161003816119860119905119906

10038161003816100381610038161003816le

120576

2

forall 119905 ge 119879 (16)


120576 lt

1003816100381610038161003816100381610038161003816100381610038161003816

(119896+1)119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198681+ 1198682

1198681=

1003816100381610038161003816100381610038161003816100381610038161003816

119879minus1

sum

119894=0

119860119894119862V ((119896 + 1) 119879 minus 1 minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

1198682=

1003816100381610038161003816100381610038161003816100381610038161003816

119860119879

(119896+1)119879minus1

sum

119894=119879


1003816100381610038161003816100381610038161003816100381610038161003816

(17)


(119896+1)119879minus1

sum

119894=119879


=

119896119879minus1

sum

119894=0

119860119895119862V (119896119879 minus 1 minus 119895)

(18)


119903 Then by (16) 119868



120576

2

lt 1198681le 119887(

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901)

1119901

(19)

where 119887 = (sum119879minus1119894=0

119860119894119862

119902

)

1119902

This gives

119879minus1

sum

119894=0

|V((119896 + 1)119879 minus 1 minus 119894)|119901 gt (120576

2119887

)

119901

(20)


(119896+1)119879minus1

sum

119894=119896119879

|V (119894)|119901 gt (120576

2119887

)

119901

forall119896 = 1 2 (21)


(119896+1)119879minus1

sum

119894=119879

|V(119894)|119901 gt 119896(120576

2119887

)

119901

119896 = 1 2 (22)



0) of the system

119911 (119905 + 1) = 119860119911 (119905) + 119862V (119905) 119911 (0) = 1199110 (23)


120574 (119911 (sdot V (sdot) 1199110)) = inf119905ge0

1003816100381610038161003816119911 (119905 V (sdot) 1199110)

1003816100381610038161003816= 0 (24)



1such that

100381610038161003816100381610038161198601199051199110

10038161003816100381610038161003816lt

120576

2

119905 ge 1199051 (25)


that1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

120576

2

(26)



120574 (119911 (119905 V (sdot) 1199110))

le1003816100381610038161003816119911 (1199052 V (sdot) 1199110)

1003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

11986011990521199110+

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381611986011990521199110

10038161003816100381610038161003816+

1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

lt

120576

2

+

120576

2

= 120576

(27)


Lemma 2 follows



R119899


119861119880 (119911 V) = 0 if 119860119911 + 119862V notin 119861119878119898

120588


(28)

Since dim1198611198781198981= 119899 the set119861119878119898

120588contains a ball 119878119899

120575of radius


119860119911 (120591) + 119862V (120591) isin 119861119878119898120588 (29)


119911 (120591 + 1) = 119860119911 (120591) + 119861119880 (119911 (120591) V (120591)) + 119862V (120591) = 0 (30)




119911 (119905 + 1) = [03 minus04

04 03] 119911 (119905)

+ [1 2 0

minus1 1 3] 119906 (119905) + [

1 2 minus1 3

3 1 4 5] V (119905)

(31)

where 119911(119905) isin R2 119906(119905) = (1199061(119905) 1199062(119905) 1199063(119905)) and V(119905) =


119860 = [03 minus04

04 03] (32)

are 12058212= 03 plusmn 04119894 and |120582



1198611198783

1= 119861120585 | 120585 = (120585

1 1205852 1205853) 1205852

1+ 1205852

2+ 1205852

3le 1 (33)


0isin R2

4 Conclusion


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574 (119911 (119905 V (sdot) 1199110))

le1003816100381610038161003816119911 (1199052 V (sdot) 1199110)

1003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

11986011990521199110+

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381611986011990521199110

10038161003816100381610038161003816+

1003816100381610038161003816100381610038161003816100381610038161003816

1199052

sum

119894=0

119860119894119862V (1199052minus 119894)

1003816100381610038161003816100381610038161003816100381610038161003816

lt

120576

2

+

120576

2

= 120576

(27)


Lemma 2 follows



R119899


119861119880 (119911 V) = 0 if 119860119911 + 119862V notin 119861119878119898

120588


(28)

Since dim1198611198781198981= 119899 the set119861119878119898

120588contains a ball 119878119899

120575of radius


119860119911 (120591) + 119862V (120591) isin 119861119878119898120588 (29)


119911 (120591 + 1) = 119860119911 (120591) + 119861119880 (119911 (120591) V (120591)) + 119862V (120591) = 0 (30)




119911 (119905 + 1) = [03 minus04

04 03] 119911 (119905)

+ [1 2 0

minus1 1 3] 119906 (119905) + [

1 2 minus1 3

3 1 4 5] V (119905)

(31)

where 119911(119905) isin R2 119906(119905) = (1199061(119905) 1199062(119905) 1199063(119905)) and V(119905) =


119860 = [03 minus04

04 03] (32)

are 12058212= 03 plusmn 04119894 and |120582



1198611198783

1= 119861120585 | 120585 = (120585

1 1205852 1205853) 1205852

1+ 1205852

2+ 1205852

3le 1 (33)


0isin R2

4 Conclusion


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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