Fixed Duration Pursuit-Evasion Differential Game with Integral Constraints

Journal of Physics: Conference Series OPEN ACCESS Fixed Duration Pursuit-Evasion Differential Game with Integral Constraints To cite this article: Ibragimov G I and Kuchkarov A Sh 213 J. Phys.: Conf. Ser. 435 1217 View the article online for updates and enhancements. Related content - THE PURSUIT PROBLEM IN - PERSON DIFFERENTIAL GAMES N L Grigorenko - Environment-Dependent Payoffs in Finite Populations Xu Wei-Hong, Zhang Yan-Ling, Xie Guang-Ming et al. - LINEAR DIFFERENTIAL GAMES OF PURSUIT L S Pontrjagin Recent citations - Gafurjan Ibragimov et al This content was downloaded from IP address 46.3.193.238 on 27/1/218 at 2:1

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 Fixed Duration Pursuit-Evasion Differential Game with Integral Constraints Ibragimov G.I. 1, and Kuchkarov A.Sh. 2 1 INSPEM & Department of Mathematics, FS, UPM, 434, Serdang, Selangor, Malaysia 2 Institute of Mathematics, Do rman yuli str., 29,1125, Tashkent, Uzbekistan E-mail: gafur@science.upm.edu.my, kuchkarov1@yandex.ru Abstract. We investigate a pursuit-evasion differential game of countably many pursuers and one evader. Integral constraints are imposed on control functions of the players. Duration of the game is fixed and the payoff of the game is infimum of the distances between the evader and pursuers when the game is completed. Purpose of the pursuers is to minimize the payoff and that of the evader is to maximize it. Optimal strategies of the players are constructed, and the value of the game is found. It should be noted that energy resource of any pursuer may be less than that of the evader. 1. Statement of the problem. The study of two person zero-sum differential games was initiated by Isaacs [16]. Berkovitz [3], Fleming [5], Friedman [6], Hajek [8], Krasovskii, [19], Petrosyan [22], Pontryagin [23], Subbotin [26] and others developed mathematical foundations for the theory of differential games. Many investigations were devoted to study the differential games with integral constraints; e.g., [1], [2], [4], [7], [9]-[15], [18], [2], [21], [25], [27]. Constructing the optimal strategies, and finding the value of the game are of interest in differential games, e.g., see [1], [11], [17], [24], [26], [27]. Such problems in the case of many pursuers were studied, for example, in [1] and [11]. In [1], a differential game of optimal approach of countably many pursuers to one evader was studied in Hilbert space with geometric constraints on controls of players. In [11], such differential game was studied for inertial players with integral constraints under the assumption that the control resource of the evader less than that of each pursuer. In the present paper, we also discuss an optimal pursuit problem with countably many pursuers and one evader in Hilbert space l 2, and control resource of the evader σ can be greater than that of any pursuer. In the space l 2 with elements and inner product and norm ı α = α 1, α 2,..., α k,...), αk 2 <, k=1 α, β) = ı ı 1/2 α k β k, α = αk) 2, k=1 Content from this work may be used under the terms of the Creative Commons Attribution 3. licence. Any further distribution of this work must maintain attribution to the authors) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 k=1

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 motions of the pursuers P i and the evader E are described by the equations P i : ẋ i = u i, x i ) = x i, E : ẏ = v, y) = y, 1) where x i, x, u i, y, y, v l 2, u i = u i1, u i2,..., u ik,...) is control parameter of the pursuer P i, and v = v 1, v 2,..., v k,...) is that of the evader E; throughout, i = 1, 2,..., m,... Let be a fixed time, I = {1, 2,..., m,...}, and Hx, r) = {x l 2 : x x r} Sx, r) = {x l 2 : x x = r}. Definition 1. A function u i = u i t), t, with the Borel measurable coordinates u ik : [, ] R 1, k = 1, 2,..., subjected to the condition 1/2 u i t) 2 dt ρ i, is called an admissible control of the pursuer P i, where ρ i are given positive numbers. Definition 2. A function v = vt), t, with the Borel measurable coordinates v k : [, ] R 1, k = 1, 2,..., subjected to the condition 1/2 vt) 2 dt σ, is called an admissible control of the evader E, where σ is given positive number. If it has been chosen admissible controls u i ), v ) of players, then corresponding to them motions x i ), y ) are defined by formulas x i t) = x i1 t), x i2 t),..., x ik t),...), y = y 1 t), y 2 t),..., y k t),...), t x ik t) = x i + t u ik s)ds, y k t) = y + v k s)ds. It is not difficult to verify that x i ), y ) C, ; l 2 ), where C, ; l 2 ) is the space of continuous functions ft) = f 1 t), f 2 t),..., f k t),...) l 2, t, with absolutely continuous components f k t), t. Definition 3. A function U i ξ i, v), U i : [, ρ 2 i ] l 2 l 2, is called a strategy of the pursuer P i if for any admissible control v = vt), t, of evader E, the system of equations ẋ i = U i ξ i, v), x i ) = x i, ξ i = U i ξ i, v) 2, ξ i ) = ρ 2 i, ẏ = v, y) = y, has a unique solution x i ), ξ i ), y )), where x i ), y ) C, ; l 2 ), and ξ i ) is absolutely continuous scalar function on [, ]. A strategy U i is said to be admissible, if every control generated by U i is admissible. 2

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 Definition 4. Strategies U i of the pursuers P i are referred to as the optimal strategies if where inf Γ 1U 1,..., U m,...) = Γ 1 U 1,..., U m,...), U 1,...,U m,... Γ 1 U 1,..., U m,...) = sup v ) inf x i) y), i I U i are admissible strategies of the pursuers P i, and v ) is admissible control of the evader E. Definition 5. A function V x 1,..., x m,..., y), V : l 2... l 2... l 2 l 2, is called a strategy of the evader E, if for any admissible controls u i = u i t), t, of the pursuers P i, the system of equations ẋ k = u k, x k ) = x k, k = 1, 2,..., m,... ẏ = V x 1,..., x m,..., y), y) = y, has a unique solution x 1 ),..., x m ),..., y )), x i ), y ) C,, l 2 ). A strategy V is said to be admissible, if every control generated by V is admissible. Definition 6. Strategy V of the evader E is said to be optimal, if sup Γ 2 V ) = Γ 2 V ), V where Γ 2 V ) = inf inf x i) y), u 1 ),...,u m ),... i I u i ) are admissible controls of pursuers P i, V is admissible control of evader E. If Γ 1 U 1,..., U m,...) = Γ 2 V ) = γ, then we say that the game has the value γ [26]. The problem is to find the optimal strategies U i, V of the players P i and E, respectively, and the value of the game. 2. An auxiliary game. The attainability set of the pursuer P i from the initial position x i up to the time, i.e., the set of all points x) = x i + u i s)ds, where u i ) is an admissible control of the ith pursuer, is the ball Hx i, ρ i ). Indeed, by the Cauchy-Schwartz inequality we have x i ) x i = 1ds u i s)ds u i s) 2 ds 1/2 On the other hand, if x Hx i, ρ i ), then for the control u i s) ds ρ i. u i s) = x x i )/, s, 3

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 of the pursuer we get x i ) = x. Admissibility of this control follows from the relations u i s) 2 ds = 1 2 x x i 2 ds 1 ρ2 i = ρ 2 i. In a similar fashion we can show that the attainability set of the evader E from the initial position y up to the time is the ball Hy, σ ). For the simplicity in this section we will drop index i, i.e., we use designations ρ i = ρ, x i = x, x i = x. Let x y and X = {z l 2 : 2y x, z) ρ 2 σ 2 ) + y 2 x 2 }, e = y x y x, Consider the following game with one pursuer P : ẋ = u, x) = x, E : ẏ = v, y) = y. 2) The goal of the pursuer P is to realize the equality xτ) = yτ) at some τ, τ, and that of the evader E is opposite. Construct the strategy of the pursuer as follows: ut) = 1 y x ) + vt), t. 3) Lemma 1. If y) X, then the strategy 3) of the pursuer P is admissible and ensures the equality x) = y) in the game 2). We show that if y) X, then the strategy 3) is admissible. As then from the inequality y) = y + vs)ds, 2y x, y)) ρ 2 σ 2 ) + y 2 x 2 we have 2 y x, vs))ds ρ 2 σ 2 ) x y 2. Hence from 3) we have us) 2 ds = 1 y x 2 + 2 1 y x 2 + 1 y x, vs))ds + vs) 2 ds ρ 2 σ 2 ) x y 2) + σ 2 = ρ 2, 4

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 and so strategy 3) is admissible. Then x) = x + This proves lemma. 3. Main Result. Now we consider the game 1). Define us)ds = x + y x + vs)ds = y). γ = inf{l : Hy, σ ) Hx i, ρ i + l)}. 4) Theorem. If there exists a non-zero vector p l 2 such that y x i, p ) for all i I, then the number γ defined by the formula 4) is the value of the game 1). Proof. To prove this theorem we need the following statement see, [1], Assertions 4 and 5 in Appendix) Lemma 2. Suppose there exists a non-zero vector p l 2 such that y x i, p ) for all i I. a) If Hy, r) Hx i, R i ) then b) If for any ε > Hy, r) {y : 2y x i, y) Ri 2 r 2 + y 2 x i 2 }. Hy, r) i I Hx i, R i ε) + ) a + = max{, a}), then there exists a point ȳ Sy, r) such that ȳ x i R i for all i I. 1. Construction the strategies of the pursuers. We introduce fictitious pursuers FPs) z i, whose motions are described by the equations ż i = w ε i, z i ) = x i, 1/2 wi ε s) 2 ds ρ i ε) = ρ i + γ + ε k i, 5) where ε is an arbitrary positive number, k i = max{1, ρ i }. It can be shown easily that the attainability set of the FP z i from the initial position x i up to the time is the ball Hx i, ρ i ε) ) = Hx i, ρ i + γ + ε/ki ). We define strategies for FPs z i on the time interval [, ] as follows. w ε i t) = { 1 y x i ) + vt), t τ ε i, τ ε i < t, 6) where τ ε i, τ ε i, is the time for which 5

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 τ ε i w ε i s) 2 ds = ρ 2 i ε), if such a time exists. We define strategies of the pursuers x i by the strategies of the FPs as follows. where ρ i = ρ i ) = ρ i + γ, u i t) = ρ i ρ i w i t), t, 7) w i t) = { 1 y x i ) + vt), t τ i, τ i < t, 8) where τ i, τ i, is the time for which that is, w i t) is obtained from 6) at ε =. As ρ i ε) > ρ i, then w i s) 2 ds = ρ 2 i, that is, ε w ε i s) 2 ds = ρ 2 i ε) > ρ 2 i = w i s) 2 ds, ε y x i + vt) 2 dt > y x i + vt) 2 ds. Hence τ ε i > τ i. 2. Guaranteed result for the pursuers. We shall show that strategies 7) of pursuers guarantee that In accordance with the definition of the number γ we have Denote sup inf y) x i) γ. 9) v ) i I Hy, σ ) Hx i, ρ i + γ + ε/ki ). 1) X i = {z : 2y x i, z) ρ i + γ + ε/ki ) 2 σ ) 2 + y 2 x i 2 }. 6

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 According to the condition of the theorem y x i, p ) for all i I. Then by Lemma 2 it follows from 1) that Hy, σ ) X i. Consequently for the point y) Hy, σ ) at some s I we have 2y x s, y)) ρ s + γ + ε/ks ) 2 σ ) 2 + y 2 x s 2. 11) If x s = y, then according to 6) the strategy of sth FP takes the form w ε st) = vt), t and from 11) we get ρ s + γ + ε/ks σ, and so w ε st) 2 dt = vt) 2 dt σ 2 ρ s + γ + ε ) 2, k s that is, w ε s ) is admissible and, moreover, z s ) = y). If x i y then by Lemma 1 for the strategies 6) of FPs we get z s ) = y). Then taking into account 5) and 7), we get y) x s ) = z s ) x s ) = wst) ε ρ ) s w s t) dt ρ s wst) ε w s t) dt + w s t) ρ s ρ s w s t) dt. 12) We now estimate right-hand side of the inequality 12). Firstly, we show that w ε i t) w i t) dt K ε 13) for all i I and some constant K. Indeed, as we noted above that τi ε > τ i and according to 6) and 8) wi εt) = w it) for t τ i ; w i t) = for t > τ i, wi εt) = for t > τ i ε, then we have + w ε i t) w i t) dt = w ε i t) w i t) dt = τ ε i ε wi ε t) w i t) dt + τ i wi ε t) dt τi ε τ i w ε i t) w i t) dt τ ε i 1/2 wi ε t) 2 dt τ ε i τ i τ i 7

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 ε wi ε t) 2 dt = ε k i 1/2 wi ε t) 2 dt 2ρ i + 2γ + ε k i = ) 1/2 ρ 2 i ε) ρ 2 i )) 1/2 K ε, where K does not depend on i. We note that the sequence ρ 1,..., ρ m,... may be unbounded or their infimum may equal to. Thus, we have shown the inequality 13). For the second integral of 12) we have 1 ρ ) s w s t) ρ s It follows then from 12) that 1 ρ s ρ s ) w s t) dt y) x s ) γ + K ε. 1 ρ s ρ s ) ρs = γ. Thus, if pursuers use strategies 7), then inequality 9) holds. So the result γ is guaranteed for the pursuers. 3.Guaranteed result for the evader. We construct a strategy for the evader, which provides the inequality inf inf y) x i) γ, 14) u 1 ),...,u m ),... i I where u 1 ),..., u m ),... are arbitrary admissible controls of pursuers. If γ =, then for any admissible control of the evader validity of 14) is clear. Let γ >. Then by definition of the number γ, for any ε >, the set Hx i, ρ i + γ ε) does not contain the ball Hy, σ ). Consequently, by Lemma 2 there exists a point ȳ Sy, σ ) such that ȳ x i ρ i + γ. Hence ȳ x i ) ȳ x i x i ) x i ρ i + γ ρi = γ. The control vt) = 1 ȳ y ), t, supplies validity of the inequality 14), since for this control we have y) = y + vs)ds = ȳ. Now, we give an illustrative example. 8

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 Example. Let ρ i = 1, = 9, σ = 4 in the game 1). We consider the following initial positions x i =,...,, 5,,...), y =,,...), of the players, where number 5 is ith coordinate of the point x i. Note that ρ i = 3, σ = 12. We show that the value of the game is γ = 1. It is sufficient to show that 1) for any ε > the inclusion HO, 12) Hx i, 13 + ε), holds, where O is the origin; 2) the ball HO, 12) is not contained in the set Hx i, 13). Indeed, let z = z 1, z 2,...) be an arbitrary point of the ball HO, 12). So zi 2 144. Then either the vector z has a nonnegative coordinate or all the coordinates of the vector z are negative. In the former case, z has a non negative coordinate z k. Then z x k = ) 1/2 z1 2 +... + zk 1 2 + 5 z k ) 2 + zk+1 2 +... Hence, z Hx k, 13 + ε). In the latter case, since ) 1/2 = zi 2 + 25 1z k 169 1z k ) 1/2 13 < 13 + ε. z x k = z 2 i is convergent then z k as k and therefore ) 1/2 zi 2 + 25 1z k 169 1z k ) 1/2 < 13 + ε for an index k. On the other hand, any point z SO, 12) with negative coordinates does not belong to the set Hx i, 13), since for any number i z x i = 169 1z i ) 1/2 > 13. Therefore by the theorem the number γ = inf{l : Hy, σ ) Hx i, ρ i + l)} is the value of the game. = inf{l : HO, 12) Hx i, 3 + l)} = 1 9

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 4. Conclusion We have studied a simple motion pursuit-evasion differential game of fixed duration with countably many pursuers. Control functions of all players are subjected to integral constraints. The value of the game has been found, and the optimal strategies of players have been constructed. The main assumption in the game is existence of a non-zero vector p l 2 such that y x i, p ) for all i I. If the initial positions of the players do not satisfy this condition then the stated problem is open. The advantage of the pursuers strategies in this work with respect to those of the paper [11] is that σ can be greater than ρ i for any i I. Acknowledgements. This research was partially supported by the Research Grant RUGS) of the Universiti Putra Malaysia, No. 5-2-12-1868RU. References [1] Azamov A.A., Samatov B. 2. Π-strategy. An elementary introduction to the theory of differential games. Tashkent, NUU press). [2] Azimov A.Ya. 1974. A linear evasion differential game with integral constraints on the controls. USSR Computational Mathematics and Mathematical Physics 14, 2), 56 65. http://dx.doi.org/1.116/41-555374)9169-4. [3] Berkovitz L. D. 1967. A Survey of Differential Games, Mathematical Theory of Control. New York, Academic Press) 373 385. [4] Chikrii A.A., Belousov A.A. 29. On linear differential games with integral constraints. Memoirs of Institute of Mathematics and Mechanics, Ural Division of RAS. 15, No.4, 29 31. Russian) [5] Fleming W.H. 1961. The convergence problem for differential games. Journal of Mathematical Analysis and Applications.3, 12 116. [6] Friedman A. 1971 Differential Games New York, John Wiley & Sons). [7] Gusiatnikov P.B., Mohon ko E.Z. 198. On l -escape in a linear many-person differential game with integral constraints. Journal of Applied Mathematics and Mechanics. 44, No 4, 436 44. [8] Hajek O. 1975. Pursuit games. New York, Academic Press). [9] Ibragimov G.I. 24 Collective pursuit with integral constraints on the controls of players. Siberian Advances in Mathematics. 14, No 2, 14 26. [1] Ibragimov G.I. 25. Optimal Pursuit with Countable Many Pursuers and One Evader. Differential Equations 41, No. 5, 627 635. [11] Ibragimov G.I, Mehdi Salimi.29. Pursuit-Evasion differential game with many inertial players. Mathematical Problems in Engineering, 29, doi:1.1155/29/653723. [12] Ibragimov G.I, Risman Mat Hasim. 21. An Evasion Differential Game in Hilbert Space. International Game Theory Review IGTR). 12, No.3, 239 251. [13] Ibragimov G.I, Azamov A., Khakestari M. 211 Solution of a Linear Pursuit-Evasion Game with Integral Constraints. ANZIAM J. 52E) 59 75. [14] Ibragimov G.I., Mehdi Salimi, Massoud Amini. 212. Evasion from Many Pursuers in Simple Motion Differential Game with Integral Constraints. European Journal of Operational Research. 218212), No 2, 55 511. doi:1.116/j.ejor.211.11.26. [15] Ibragimov G.I., Satimov N.Yu. 212. A Multi Player Pursuit Differential Game on Closed Convex Set with Integral Constraints. Abstract and Applied Analysis. 212, doi:1.1155/212/46171. [16] Isaacs R. 1967 Differential games. A Mathematical theory with Applications to Warfare and Pursuit, Control and Optimization. New York, John Wiley and Sons) [17] Ivanov R.P., Ledyayev Yu.S. 1981 Optimality of pursuit time in differential game of many objects with simple motion. Trudi MIAN USSR. 158. 87 97. Russian) [18] Krasovskii N.N. 1968 The Theory of Motion Control. Moscow, Nauka.) [19] Krasovskii N.N., Subbotin A.I. 1988. Game-theoretical control problems. N.Y., Springer).1988. [2] Mezentsev A.V. 1974 A sufficient condition for linear evasion games with integral constraints. DAN USSR, 218, No 5, 121 123. [21] Nikolskii M.S. 1969 The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR. 2, 49 59. [22] Petrosyan L.A. 1993. Differential games of pursuit. London, World Scientific). [23] Pontryagin L.S. 1988. Collected works. Moscow, Nauka). Russian) [24] Pshenichnii B.N., and Onopchuk Yu.N. 1968 Linear differential games with integral constraints. Izvestige Akademii Nauk SSSR, Tekhnicheskaya Kibernetika 1, 13 22. [25] Satimov N.Yu., Rikhsiev B.B. 2 Methods of Solving of Evasion Problems in Mathematical Control Theory. Tashkent, Fan). Russian) 1

212 icast: Contemporary Mathematics, Mathematical Physics and their Applications IOP Publishing Journal of Physics: Conference Series 435 213) 1217 doi:1.188/1742-6596/435/1/1217 [26] Subbotin A.I., Chentsov A.G. 1981. Optimization of the guarantee in control problems. Moscow, Nauka) Russian) [27] Ushakov V.N. 1972 Extremal strategies in differential games with integral restrictions. J. Appl. Math.Mech., 36, No 1, 12 19. 11