A Pursuit Problem Described by Infinite System of Differential Equations with Coordinate-Wise Integral Constraints on Control Functions

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1 Malaysan Journal of Mathematcal Scences 9(1): (15) MALAYSIAN JOURNAL OF MAHEMAICAL SCIENCES Journal homepage: A Pursut Problem Descrbed by Infnte System of Dfferental Equatons wth Coordnate-Wse Integral Constrants on Control Functons 1,3* Gafurjan Ibragmov, Mumnjon ukhtasnov, 3 Rsman Mat Hasm and 1,3 Idham Arf Alas 1 Insttute for Mathematcal Research, Unverst Putra Malaysa, 434 UPM Serdang, Selangor, Malaysa Natonal Unversty of Uzbekstan, Unversty Street, 1174, Almazar Dstrct, ashkent, Uzbekstan 3 Department of Mathematcs, Faculty of Scence, Unverst Putra Malaysa, 434 UPM Serdang, Selangor, Malaysa E-mal: bragm1ov@gmal.com *Correspondng author ABSRAC We consder a pursut dfferental game of one pursuer and one evader descrbed by nfnte system of frst order dfferental equatons. he coordnate-wse ntegral constrants are mposed on control functons of players. By defnton pursut s sad to be completed f the state of system equals zero at some tme. A suffcent condton of completon of pursut s obtaned. Strategy for the pursuer s constructed and an explct formula for the guaranteed pursut tme s gven. Keywords: Dfferental game, Hlbert space, nfnte system, pursut, evason, strategy. 1. INRODUCION A consderable amount of lterature has been publshed on dfferental games where the control functons of players are subjected to ntegral constrants, see, for example, Azmov (1975), Azamov and Samatov (), Chkr and Belousov (9), Ibragmov et al (14), Ibragmov and Salleh (1), Ibragmov and Satmov (1), Ibragmov et al (11), Ibragmov (), Krasovsk (1968), Kuchkarov et al (13), Nkolsk

2 Gafurjan Ibragmov et al. (1969), Pshenchn and Onopchuk (1968), Satmov and ukhtasnov (7), Satmov and ukhtasnov (5), ukhtasnov (1995), Ushakov (197). As t s known (see, for example, Avdonn and Ivanov (1989), Butkovsky (1975), Chernous ko (199)) that f we use the decomposton method for the systems wth dstrbuted parameters of evoluton type, then we certanly arrve at an nfnte system of dfferental equatons. he man constrants on control parameters of players n fnte dmensonal dfferental games are geometrc, ntegral, and mxed constrants (see, for example, Nkolsk (1969)). Also these constrants n some respect can be preserved for the control problems descrbed by nfnte systems of dfferental equatons. In the papers by Ibragmov et al (14), Ibragmov (), Satmov and ukhtasnov (7), Satmov and ukhtasnov (5) varous dfferental game problems for parabolc and hyperbolc equatons were studed under dfferent constrants on controls of players. Here, the decomposton method and some approaches from work of Avdonn and Ivanov (1989) were used to obtan varous suffcent condtons under whch the game problems are solvable. he paper of ukhtasnov and Ibragmov (11) s devoted to the problems of keepng the trajectory of system wthn some lmts, wth ntegral constrants beng subjected to controls. In all of these problems, the control parameters are n the rght hand sdes of equatons n addtve form. o solve the stated problems the control methods of nfnte systems of dfferental equatons were appled. herefore, there s an mportant lnk between the control problems descrbed by some partal dfferental equatons and nfnte systems of dfferental equatons. he latter s of ndependent nterest. he papers of Ibragmov (4), Ibragmov et al (14), Ibragmov (), Satmov and ukhtasnov (7), Satmov and ukhtasnov (5), ukhtasnov (1995) were devoted to dfferental game problems descrbed by nfnte system of dfferental equatons. However, nfnte systems requre the exstence and unqueness theorem. herefore, n the paper of Ibragmov (4) such theorem was proved for the frst order nfnte system, and then a game problem was studed. In the present paper, we study a dfferental game problem descrbed by nfnte system of dfferental equatons of frst order. he control functons of players are subjected to coordnate-wse ntegral constrants. It should be noted that for such constrants the exstence and unqueness theorem n the paper of Ibragmov (4), n general, doesn t 68 Malaysan Journal of Mathematcal Scences

3 A Pursut Problem Descrbed by Infnte System of Dfferental Equatons wth Coordnate-wse Integral Constrants on Control Functons work snce the ntegral u ( ) t dt may dverge. We obtan suffcent 1 condtons of completon of pursut from any ponts of the state space.. SAEMEN OF HE PROBLEM We study a two-person zero sum dfferental game descrbed by the followng nfnte system of dfferental equatons z z u v, 1,,... (1) n where z,, u v R, n s a postve nteger, are gven postve numbers whch express the elastcty of medum, u ( u1, u,...) and v ( v1, v,...) are control parameters of the pursuer and the evader respectvely. hese parameters vary dependng on tme, and therefore they become the functons of tme. We denote them by u( t), v( t), t. Let be an arbtrary number. Defnton 1. A vector functon u( t) ( u1( t), u( t),...), t, wth n measurable coordnates u () t, s called admssble control of the -th pursuer f t satsfes the followng ntegral constrant u ( s ) ds, () where s gven nonnegatve number whch we call -control resource of the pursuer. Defnton. A vector functon v( t) ( v1( t), v( t),...), t, wth n measurable coordnates v () t, s called admssble control of the evader f t satsfes the followng ntegral constrant v ( s ) ds, (3) where s gven nonnegatve number whch we call -control resource of the evader. Malaysan Journal of Mathematcal Scences 69

4 Pursut starts from the ntal postons Gafurjan Ibragmov et al. z () z, 1,,... (4) at tme t where z R n, 1,,.... If we replace the parameters u, v n the equaton (1) by some admssble controls u ( t ), v () t, t, then t follows from the theory of dfferental equatons that the ntal value problem (1), (4) has a unque soluton on the tme nterval [, ]. he soluton z( t) ( z ( t), z ( t),...), t, 1 of nfnte system of dfferental equatons (1) s consdered n the space of functons f ( t) ( f1( t), f( t),...) wth absolutely contnuous coordnates f () t defned on the nterval t. Defnton 3. A functon of the form U ( t, v) ( t) v ( ( t) v, ( t) v,...), t, 1 1 where U (, ) ( ) n t v t v R, 1,,..., s called strategy of the pursuer f for any admssble control of the evader v v( t), t, the followng nequaltes hold. U t v t dt, 1,,..., (, ( )) Defnton 4. We say that pursut can be completed for the tme n the dfferental game (1) (4) from the ntal poston z ( z1, z,...) f there exsts a strategy of the pursuer U ( t, v ) such that for any admssble control of the evader v ( t), t, the soluton z( t), t, of the ntal value problem z z U ( t, v( t)) v ( t), t, (5) z () z, 1,,..., equals zero at some tme,,.e. z ( ) at some,. In the sequel, the tme s called guaranteed pursut tme. 7 Malaysan Journal of Mathematcal Scences

5 A Pursut Problem Descrbed by Infnte System of Dfferental Equatons wth Coordnate-wse Integral Constrants on Control Functons Problem. Fnd a guaranteed pursut tme n the game (1) (4), and construct the strategy for pursuer that enables to complete the game for ths tme. 3. MAIN RESUL he followng theorem presents a suffcent condton of completon of the game (1) (4). z heorem. Let, 1,,..., and sup. hen N 1 z ' sup, ln 1, (6) s a guaranteed pursut tme n the game (1) (4). z Proof. he hypothess of the theorem sup mples that ' s N fnte. Let v ( t), t ', 1,,..., be an arbtrary control of the evader. Construct the strategy of the pursuer as follows. s z e ( ) v, s, U ( s, v) z (, ) (7) v, s ', where t s (, t) e ds, t. Verfy admssblty of the strategy of the pursuer (7). Indeed, ' z s ' (, ( )) e ( ) ( ) ( ) U s v s ds v s ds v s ds z (, ) ( ) 1 ' s s (, ( )) ( ). e ds e z v s ds v s ds (, ) z (, ) (8) Malaysan Journal of Mathematcal Scences 71

6 Gafurjan Ibragmov et al. ' Snce ( ) v s, then ' 1 (, ( )) ( ) s (, ( )) U s v s ds e z v s ds. z (, ) Usng the Cauchy-Schwartz nequalty, yelds ' (, ( )) ( ) ( ) U s v s ds, whch mples that ' (, ( )) U s v s ds. hus, the strategy (7) s admssble. Show that z( ). Let v ( s), s ', be any admssble control of the evader. Replacng u n the equaton z z u v, t, z() z. by s z (, ( )) e u s v s ( ) v ( s), z (, ) we obtan that t ( ) ( ) t ts z t e z e ( u ( s, v ( s)) v ( s)) ds t t s e z e ( u ( s, v ( s)) v ( s)) ds. t t z s e z ( ) e ds z (, ). (9) 7 Malaysan Journal of Mathematcal Scences

7 A Pursut Problem Descrbed by Infnte System of Dfferental Equatons wth Coordnate-wse Integral Constrants on Control Functons Hence (, ) z z ( ) e z ( ) z (, ) z e z ( ) (, ). (1) z Snce 1 (, ) s e e ds, z where s defned by (6), we conclude that (, ). hen the expresson n brackets n (1) equals zero, and hence z ( ). Accordng to (7), u ( s, v ( s)) v ( s), s, and therefore the equalty z ( t) keeps to hold on the nterval [, '] as well. hus, we conclude that z ( '), 1,,... hs completes the proof of the theorem. 4. CONCLUSION In ths paper, we have studed a dfferental game descrbed by an nfnte system of dfferental equatons. In the past, such dfferental games were studed n Hlbert spaces (see, for example, Ibragmov (, 4, 14), Satmov and ukhtasnov (5, 7). In the present research, the state zt () doesn t need to be n a Hlbert space, because the seres may be dvergent. he control functons of the players are subjected to coordnate-wse ntegral constrants. We have obtaned suffcent condtons of completon game. Moreover, we have constructed a strategy for the pursuer n explct form. 1 ACKNOWLEDGMEN he present research was partally supported by the Natonal Fundamental Research Grant Scheme FRGS of Malaysa, FR. Malaysan Journal of Mathematcal Scences 73

8 Gafurjan Ibragmov et al. REFERENCES Azmov, A. Ya. (1975). Lnear dfferental pursut game wth ntegral constrants on the control. Dfferental Equatons. 11: Azamov, A. A., Samatov, B. (). -strategy. An elementary ntroducton to the theory of dfferental games. NUU Press. Uzbekstan: ashkent. Avdonn, S. A and Ivanov, S. A. (1989). Controllablty of Systems wth Dstrbuted Parameters and Famly of Exponentals. UMKVO, Kev. Butkovsky, A. G. (1975). Control Methods n Systems wth Dstrbuted Parameters. Nauka, Moscow. Chernous ko, F. L. (199). Bounded controls n dstrbuted-parameter systems. Prkl. Math. Mekh. 56(5): [J.Appl.Math.Mech. 56(5): (199)]. Chkr, A. A. and Belousov, A. A. (9). On lnear dfferental games wth ntegral constrants. Memors of Insttute of Mathematcs and Mechancs, Ural Dvson of RAS. Ekaternburg, 15(4): Ibragmov, G. I. (4). On possblty of evason n a dfferental game, descrbed by countable number dfferental equatons. Uzbek Math. Journal. 1: 5-55 Ibragmov, G. I., Allahab, F. and Kuchkarov, A. Sh. (14). A pursut problem n an nfnte system of dfferental equatons of second order. Ukranan Mathematcal Journal. 65(8): Ibragmov, G. I. and Yusra Salleh. (1). Smple Moton Evason Dfferental Game of Many Pursuers and One Evader wth Integral Constrants on Control Functons of Players. Journal of Appled Mathematcs. Artcle ID 74896, 1 p., do:1.1155/1/ Ibragmov, G. I. and Satmov, N. Yu. (1). A Mult Player Pursut Dfferental Game on Closed Convex Set wth Integral Constrants. Abstract and Appled Analyss. Artcle ID 46171, 1p., do:1.1155/1/ Malaysan Journal of Mathematcal Scences

9 A Pursut Problem Descrbed by Infnte System of Dfferental Equatons wth Coordnate-wse Integral Constrants on Control Functons Ibragmov, G. I, Azamov, A. and Khakestar, M. (11). Soluton of a Lnear Pursut-Evason Game wth Integral Constrants. ANZIAM J. 5(E): E59-E75. Ibragmov, G. I. (). An optmal pursut problem n systems wth dstrbuted parameters. Prkladnaya Matematka Mekhanka. 66(5): Krasovsk, N. N. (1968). he heory of Moton Control. Nauka, Moscow. Kuchkarov, A. Sh, Ibragmov, G. I. and Khakestar, M. (13). On a Lnear Dfferental Game of Optmal Approach of Many Pursuers wth One Evader. Journal of Dynamcal and Control Systems. 19(1): 1-15, DOI: 1.17/s z. Nkolsk, M. S. (1969). he drect method n lnear dfferental games wth ntegral constrants. Controlled systems, IM, IK, SO AN SSSR. : Pshenchn, B. N. and Onopchuk, Yu. N. (1968). Lnear dfferental games wth ntegral constrants. Izvestya Akadem Nauk SSSR, ekhncheskaya Kbernetka. 1: 13-, Satmov, N.Yu. and ukhtasnov, M. (7). On Game Problems for Second- Order Evoluton Equatons. Russan Mathematcs (Iz. VUZ). 51(1): Satmov, N.Yu. and ukhtasnov, M. (5). On some game problems for frst-order controlled evoluton equatons. Dfferentsal nye Uravnenya. 41(8): [Dfferental Equatons 41(8): (5]. ukhtasnov, M. and Ibragmov, U. M. (11). Sets Invarant under an Integral Constrant on Controls. Russan Mathematcs (Iz.VUZ). 55(8): ukhtasnov, M. (1995). Some problems n the theory of dfferental pursut games n systems wth dstrbuted parameters. Prkl. Math. Mekh. 59(6): [J. Appl. Math.Mech.59(6): ]. Malaysan Journal of Mathematcal Scences 75

10 Gafurjan Ibragmov et al. Ushakov, V. N. (197). Extremal strateges n dfferental games wth ntegral constrants. Prkl. Math. Mekh. 36(1): Malaysan Journal of Mathematcal Scences