Malaysan Journal of Mathematcal Scences 1(S) August: 267 277 (216) Specal Issue: The 7 th Internatonal Conference on Research and Educaton n Mathematcs (ICREM7) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://enspem.upm.edu.my/journal Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons Gafurjan Ibragmov 1, Rsman Mat Hasm 2, Askar Rakhmanov 3, and Idham Arf Alas 4 1,2,4 Insttute for Mathematcal Research, Unverst Putra Malaysa, Malaysa 3 Department of Informatcs, Tashkent Unversty of Informaton Technologes, Uzbekstan E-mal: bragmov@upm.edu.my Correspondng author ABSTRACT We consder a dfferental game of one pursuer and one evader. The game s descrbed by an nfnte system of frst order dfferental equatons. Control functons of the players are subject to coordnate-wse ntegral constrants. Game s sad to be completed f each component of state vector equal to zero at some unspecfed tme. The pursuer tres to complete the game and the evader pursues the opposte goal. A formula for optmal pursut tme s found and optmal strateges of players are constructed. Keywords: Infnte system of dfferental equatons, pursuer, evader, strategy, optmal pursut tme.
Gafurjan, I. et al. 1. Introducton There are many works devoted to dfferental games wth ntegral constrants n fnte dmensonal spaces such as Krasovsk (1968), Mezentsev (1971), Ushakov (1972), Azmov (1974), Subbotn and Ushakov (1975), Okhezn (1977), Gusatnkov et al. (1979), Satmov et al. (1983), Solomatn (1984), Konovalov (1987), Ukhobotov (1987), Lokshn (199), Ibragmov (1998), Gusenov et al. (1999), Azamov and Samatov (2), Ibragmov (25), Chkr and Belousov (29), Belousov (211), Ibragmov and Satmov (212), Chkr and Belousov (213) and Kuchkarov (213). However, there a few works devoted to dfferental game problems descrbed by nfnte system of dfferental equatons. For example, the followng papers are devoted to such game problems L (1986), Tukhtasnov (1995), Ibragmov (22), Satmov and Tukhtasnov (25), Satmov and Tukhtasnov (25), Tukhtasnov (25), Satmov and Tukhtasnov (26), Satmov and Tukhtasnov (27), Ibragmov and Hussn (21), Azamov and Ruzboyev (213), Ibragmov (213), Ibragmov et al. (214) and Ibragmov et al. (215), Salm et al. (215). In the paper Ibragmov et al. (215), a pursut game problem s studed for an nfnte system of dfferental equatons, where control functons of players are subjected to coordnate-wse ntegral constrants. In the present paper, we prove that guaranteed pursut tme found n Ibragmov et al. (215) s optmal pursut tme as well. 2. Statement of problem Consder dfferental game descrbed by the followng frst order nfnte system of dfferental equatons ż + λ z = u v, z () = z = 1, 2,..., (1) where z, u, v R n, n s a postve nteger, λ are gven postve numbers, u = (u 1, u 2,...) and v = (v 1, v 2,...) are control parameters of the pursuer and evader respectvely. Let T > be an arbtrary number. Defnton 2.1. A functon u(t) = (u 1 (t), u 2 (t),...), t T, wth measurable coordnates u (t) R n, s called admssble control of the -th pursuer f t satsfes the followng ntegral constrant T u (s) 2 ds ρ 2, (2) 268 Malaysan Journal of Mathematcal Scences
Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons where ρ s a gven postve number. Defnton 2.2. A functon v(t) = (v 1 (t), v 2 (t),...), t T, wth measurable coordnates v (t) R n, s called admssble control of the evader f t satsfes the followng ntegral constrant T where σ s a gven postve number. v (s) 2 ds σ 2, (3) Pursut starts from the ntal postons z () = z, = 1, 2,... at tme t = where z R n, = 1, 2,... Next, we defne soluton of the system (1). If we replace the parameters u and v n the equaton (1) by some admssble controls u (t) and v (t), t T, then t follows from the theory of dfferental equatons that the ntal value problem (1) has a unque soluton on the tme nterval [, T ]. The soluton z(t) = (z 1 (t), z 2 (t),...), t T, of nfnte system of dfferental equatons (1) s consdered n the space of functons f(t) = (f 1 (t), f 2 (t),...) wth absolutely contnuous coordnates f (t) defned on the nterval t T. Defne strategy of the pursuer. Defnton 2.3. A functon of the form U(t, v) = ψ(t) + v = (ψ 1 (t) + v 1, ψ 2 (t) + v 2,...), t T, where U (t, v ) = ψ (t) + v R n, = 1, 2,..., s called strategy of the pursuer f for any admssble control of the evader v = v(t), t T, the followng nequaltes T U (t, v (t)) 2 dt ρ 2, = 1, 2,..., hold, where ψ(t) = (ψ 1 (t), ψ 2 (t),...), t T, s a functon wth measurable coordnates ψ (t) R n. Defnton 2.4. We say that pursut can be completed for the tme θ > n the dfferental game (1) (3) from the ntal poston z = (z 1, z 2,...) f there exsts a strategy of the pursuer U(t, v) such that for any admssble control of the evader v (t), t θ, the soluton z(t), t θ, of the ntal value problem ż + λ z = U (t, v(t)) v (t), t θ, z () = z, = 1, 2,..., equals zero at some tme τ, τ θ,.e. z(τ) =. Malaysan Journal of Mathematcal Scences 269
Gafurjan, I. et al. In the sequel, such a tme θ s called guaranteed pursut tme. To defne strategy of the evader, we need the varables p(t) = (p 1 (t), p 2 (t),...), and q(t) = (q 1 (t), q 2 (t),...), whch are defned as the solutons of the followng equatons Clearly, and ṗ = u 2 (t), p () = ρ 2, q = v 2 (t), q () = σ 2, = 1, 2,... p (t) = ρ 2 q (t) = σ 2 u 2 (s)ds, v 2 (s)ds. The quantty u2 (s)ds expresses the amount of energy spent by the pursuer n the -th component of control u. The quantty p (t) s the amount of energy remaned for the -th component of control u whch the pursuer can use startng from tme t. The quanttes v2 (s)ds and q (t) also have smlar meanngs for the evader. Defnton 2.5. A functon V (t) = (V 1 (t), V 2 (t),...), t, wth the coordnates of the form V (t) t τ, q (τ ), V (t) = τ < t τ + ε, u (t ε ) t > τ + ε, s called strategy of evader, where V (t), t, s a measurable functon, q (t) = σ 2 V 2 (s)ds, t = τ s the frst tme for whch p (t) = q (t), u(t) = (u 1 (t), u 2 (t),...) s any admssble control of pursuer, ε s a postve number. Defnton 2.6. A guaranteed pursut tme θ s called optmal pursut tme f there exsts a strategy of the evader V such that for any admssble control of the pursuer z(t) for all t [, θ). Problem 1. Fnd optmal pursut tme n the game (1) (3), and construct the strategy for pursuer that enables to complete the game for ths tme. 27 Malaysan Journal of Mathematcal Scences
Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons 3. Man result The followng theorem presents a formula for optmal pursut tme n the game (1) (3). Theorem 3.1. Let ρ > σ, = 1, 2,..., and sup N z ρ σ <. Then ( T = sup T, T = 1 ( ) ) 2 z ln 1 + 2λ 2λ ρ σ (4) s optmal pursut tme n the game (1) (3). Proof. Earler by Ibragmov et al. (215) was shown that T s guaranteed pursut tme n the game (1) (3). Therefore t s suffcent to show that on the nterval [, T ) evason s possble. 3.1 Constructon the strategy of the evader. We construct t n two steps. The frst part of strategy s as follows: v (s) = z z e λs ϕ (, T ) σ, s τ, = 1, 2,..., (5) where τ, τ < T, s some tme at whch q (τ ) = p (τ ), b ϕ (a, b) = a e 2λs ds. Note that at the tme t =, p () = ρ 2 > σ 2 = q (). The evader uses (1) unless q (τ ) = p (τ ) at some tme τ. Startng from the tme τ the evader uses the second part of the strategy. It s as follows: v (t) = {, τ t τ + ε, u (t ε ), t > τ + ε, = 1, 2,..., (6) where ε s a postve number. 3.2 Proof that evason s possble. Show that constructed strategy (5) (6) guarantees the evason on the tme nterval [, T ). Let the evader use the frst part of ts strategy (5). Show that Malaysan Journal of Mathematcal Scences 271
Gafurjan, I. et al. z (t) as long as q (t) p (t), or more precsely q (t) < p (t). Assume the contrary. Let there exst a control of the pursuer u (t) such that z (τ ) = at some τ τ, where q (τ ) p (τ ). Then substtutng (5) nto the formula ) z (t) = e (z λt + e λs (u (s) v (s))ds yelds We have z + e λs u (s)ds + z z σ τ ϕ (, T ) e 2λs ds =. ( e λs u (s) ds z 1 + σ ϕ 2 (, τ ) ). (7) z ϕ (, T ) Use the Cauchy-Schwartz nequalty to estmate left hand sde of (7) Then from ths and (7) we obtan Accordng to (5) ( 1/2 e λs u (s) ds ϕ (, τ ) u (s) ds) 2. u (s) 2 ds z 2 Combnng (8) and (9), we obtan ϕ 2 (, τ ) ( 1 + σ ϕ 2 (, τ ) ) 2 (8) z ϕ (, T ) v (s) 2 ds = σ2 ϕ2 (, τ ) ϕ 2 (, T ). (9) u (s) 2 ds v (s) 2 ds = z 2 ϕ 2 (, τ ) + 2 z σ ϕ (, T ) > z 2 ϕ 2 (, T ) + 2σ z ϕ (, T ) = (ρ σ ) 2 + 2σ (ρ σ ) = ρ 2 σ 2. Hence, q 2(τ ) > p2 (τ ), whch s n contradcton wth defnton of τ. Thus, on the nterval [, τ ], z (t). In partcular, z (τ ). Startng from the tme τ the evader uses the strategy (6) v (t) = {, τ t τ + ε, u (t ε ), t > τ + ε, = 1, 2,..., 272 Malaysan Journal of Mathematcal Scences
Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons where ε s a postve number that wll be chosen below. Observe that ) z (t) = e (z λt + e λs (u (s) v (s))ds = f and only f y (t) = z + e λs (u (s) v (s))ds =. Therefore t suffces to show the nequalty y (t), t [τ, ). Let τ t τ + ε. Then We have y (t) = z + = y (τ ) + = y (τ ) + e λs (u (s) v (s))ds τ e λs (u (s) v (s))ds τ e λs u (s)ds. ( ) 1/2 e λs u (s) ds ϕ (τ, t) u (s) 2 ds ρ ϕ (τ, τ + ε ). τ τ If b a = ε, a [, T ], then b 1 ϕ (a, b) = e 2λs ds = (e 2λ 2λb e 2λa ) a = e λa 1 2λ ( e 2λ (b a) 1 ) e λt 1 2λ (e 2λε 1). Then and so Let the number ε be chosen such that y (t) y (τ ) ρ e λt 1 2λ (e 2λε 1) y (τ ) /2. (1) ρ ϕ (τ, τ + ε ) y (τ ) /2, τ e λs u (s) ds y (τ ) /2, τ t τ + ε. Malaysan Journal of Mathematcal Scences 273
Gafurjan, I. et al. Let now t > τ + ε. Then Snce y (t) = y (τ ) + = y (τ ) + +ε τ e λs u (s)ds + therefore (11) mples that τ e λs u (s)ds τ +ε e λs u (s ε)ds = τ +ε e λs (u (s) v (s))ds τ +ε e λs u (s ε )ds. (11) ε y (t) = y (τ ) + e λs u (s)ds τ = y (τ ) + Hence, t follows from (1) that τ t ε e λs u (s)ds. e λs u (s)ds, ε y (t) y (τ ) e λs u (s) ds t ε τ e λs u (s)ds y (τ ) ρ ϕ (t ε, t) y (τ ) /2, for all t τ + ε. Thus, y (t) y (τ ) /2 > for all t τ. In other words, evason s possble on the nterval [, T ). Show that evason s possble on the nterval [, T ), T = sup =1,2,... T. Let t [, T ) be any tme. Then by defnton of supremum t [, T j ) at some j N. As proved above y j (t) for t [, T j ), and hence z j (t). In ts turn ths nequalty mples that z(t) meanng that evason s possble on the nterval [, T ). Thus, T defned by (4) s optmal pursut tme. Proof s complete. 4. Concluson We have studed a dfferental game of one pursuer and one evader descrbed by nfnte system of frst order dfferental equatons. Control functons of pursuer and evader are subject to coordnate-wse ntegral constrants. Earler we showed that the pursut tme defned by formula (4) s guaranteed pursut tme. In the present paper, we proved that ths tme s optmal pursut tme. Moreover, we have constructed optmal strateges of players. The dfferental 274 Malaysan Journal of Mathematcal Scences
Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons game can be extended by studyng dfferental games 1) descrbed hgher order dfferental equatons, or 2) wth many pursuers. Acknowledgements Ths research was partally supported by the Natonal Fundamental Research Grant Scheme (FRGS) of Malaysa, No. 1-1-13-1228FR. References Azamov, A. A. and Ruzboyev, M. B. (213). The tme-optmal problem for evolutonary partal dfferental equatons. Journal of Appled Mathematcs and Mechancs, 77(2):22 224. Azamov, A. A. and Samatov, B. T. (2). P-strategy. En elementary ntroducton to the Theory of Dfferental Games. Natonal Unversty of Uzbekstan, Tashkent, Uzbekstan. Azmov, A. Y. (1974). Lnear evason dfferental game wth ntegral constrants on the control. Journal of Computatonal Mathematcs and Mathematcal Physcs, 14(6):1427 1436. Belousov, A. A. (211). Dfferental games wth ntegral constrants on controls n the norm of l 1. Theory of Optmal Solutons, 1:3 8. Chkr, A. A. and Belousov, A. A. (29). On lnear dfferental games wth ntegral constrants. Trudy Inst. Mat. I Mekh. UrO RAN, 15(4):29 31. Chkr, A. A. and Belousov, A. A. (213). On lnear dfferental games wth convex ntegral constrants. Trudy Inst. Mat. I Mekh. UrO RAN, 19(4):38 319. Gusenov, K. G., Neznakhn, A. A., and Ushakov, V. N. (1999). Approxmate constructon of attanablty sets of control systems wth ntegral constrants on the controls. Journal of Appled Mathematcs and Mechancs, 63(4):557 567. Gusatnkov, P. B., Ledaev, I. S., and Mezentsev, A. V. (1979). Dfferental escape games wth nformaton lag. J. Appl. Maths Mekhs, 43(2):213 221. Ibragmov, G. (1998). On the optmal pursut game of several pursuers and one evader. Prkladnaya Matematka Mekhanka, 62(2):199 25. Ibragmov, G. (22). An optmal pursut problem n systems wth dstrbuted parameters. Prkladnaya Matematka Mekhanka, 66(5):753 759. Malaysan Journal of Mathematcal Scences 275
Gafurjan, I. et al. Ibragmov, G. and Hussn, N. (21). A pursut-evason dfferental game wth many pursuers and one evader. MJMS, 4(2):183 194. Ibragmov, G., Tukhtasnov, M., Rsman, M. H., and Arf Alas, I. (215). A pursut problem descrbed by nfnte system of dfferental equatons wth coordnate-wse ntegral constrants on control functons. Malaysan Journal of Mathematcal Scences, 9(1):67 76. Ibragmov, G. I. (25). A group pursut game. Automaton and Remote Control, 66(8):1214 1223. Ibragmov, G. I. (213). The optmal pursut problem reduced to an nfnte system of dfferental equatons. Journal of Appled Mathematcs and Mechancs, 77(5):47 476. Ibragmov, G. I., Allahab, F., and Kuchkarov, A. (214). A pursut problem n an nfnte system of second-order dfferental equatons. Ukranan Mathematcal Journal, 65(8):123 1216. Ibragmov, G. I. and Satmov, N. Y. (212). A multplayer pursut dfferental game on a closed convex set wth ntegral constrants. Abstract and Appled Analyss, pages 1 12. Konovalov, A. P. (1987). Lnear dfferental evason games wth lag and ntegral constrants. Cybernetcs, (4):41 45. Krasovsk, N. N. (1968). Theory of control of moton: Lnear systems (n Russan). Nauka, Moscow. Kuchkarov, A. S. (213). On a dfferental game wth ntegral and phase constrants. Automaton and Remote Control, 74(1):12 25. L, X. J. (1986). N-person dfferental games governed by nfnte-dmensonal systems. Journal of Optmzaton Theory and Applcatons, 5(3):431 45. Lokshn, M. D. (199). Dfferental games wth ntegral constrants on dsturbances. Journal of Appled Mathematcs and Mechancs, 54(3):331 337. Mezentsev, A. V. (1971). A drect method n lnear dfferental games wth dfferent constrants. Zh. Vchsl. Mat. I Mat. Fz, 11(2):365 374. Okhezn, S. P. (1977). Dfferental encounter-evason game for parabolc system under ntegral constrants on the player s controls. Prkl Mat I Mekh, 41(2):22 29. Salm, M., Ibragmov, G., Segmund, S., and Sharf, S. (215). On a fxed duraton pursut dfferental game wth geometrc and ntegral constrants. Dynamc Games and Applcatons, 1.17/s13235-15-161-3. 276 Malaysan Journal of Mathematcal Scences
Optmal Pursut Tme n Dfferental Game for an Infnte System of Dfferental Equatons Satmov, N. Y., Rkhsev, B. B., and Khamdamov, A. A. (1983). On a pursut problem for n-person lnear dfferental and dscrete games wth ntegral constrants. Math. USSR Sbornk, 46(4):459 471. Satmov, N. Y. and Tukhtasnov, M. (26). On game problems wth fxed duraton for frst order controlled evolutonary equatons. (Matematcheske Zametk) Mathematcal Notes, 8(4):613 626. Satmov, N. Y. and Tukhtasnov, M. (27). On game problems for secondorder evoluton equatons. Russan Mathematcs, 51(1):49 57. Satmov, N. Y. and Tukhtasnov, M. T. (25). Some game problems n dstrbuted controlled systems. Journal of Appled Mathematcs and Mechancs, 69(6):885 89. Solomatn, A. M. (1984). A game theoretc approach evason problem for a lnear system wth ntegral constrants mposed on the player control. Journal of Appled Mathematcs and Mechancs, 48(4):41 45. Subbotn, A. and Ushakov, V. N. (1975). Alternatve for an encounter-evason dfferental game wth ntegral constrants on the players contols. J. Appl. Maths Mekhs, 39(3):367 375. Tukhtasnov, M. (1995). Some problems n the theory of dfferental pursut games n systems wth dstrbuted parameters. J. Appl. Maths Mekhs, 59(6):935 94. Tukhtasnov, M. (25). On some game problems for frst-order controlled evoluton equatons. 41(8):1169 1177. Ukhobotov, V. I. (1987). A type of lnear game wth mxed constrants on the controls. Journal of Appled Mathematcs and Mechancs, 51(2):139 144. Ushakov, V. N. (1972). Extremal strateges n dfferental games wth ntegral constrant. J. Appl. Maths Mekhs, 36(1):12 19. Malaysan Journal of Mathematcal Scences 277