Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations

Transcription

1 Malaysian Journal of Mathematical Sciences 11(2): (217) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations Ibragimov, G. 1,2, Akhmedov, A. 3, Puteri Nur Izzati 2, and Abdul Manaf, N. 4 1 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Malaysia 2 Institute for Mathematical Research, Universiti Putra Malaysia, Malaysia 3 Universiti Malaysia Pahang, Malaysia 4 Malaysia-Japan International Institute of Technology, Universiti Teknologi Malaysia, Malaysia ibragimov@upm.edu.my Corresponding author Received: 5th November 216 Accepted: 13th March 217 ABSTRACT We study a pursuit differential game problem for infinite first order 2- systems of differential equations in the Hilbert space l 2. Geometric constraints are imposed on controls of players. If the state of system coincides with the origin, then we say that pursuit is completed. In the game, pursuer tries to complete the game, while the aim of evader is opposite. The problem is to find a formula for guaranteed pursuit time. In the present paper, an equation for guaranteed pursuit time is obtained. Moreover, a strategy for the pursuer is constructed in explicit form. To prove the main result, we use solution of a control problem. Keywords: Differential game, infinite system, pursuer, evader, geometric constraint, control, strategy.

2 Ibragimov, G. et al. 1. Introduction Differential games, first considered in the book of Isaacs (1965). Since then lots of works with various approaches have been published in developing the theory of differential games for ordinary differential equations. Control and differential games described by partial differential equations are increasing interest (see, for example, Avdonin and Ivanov (1995), Butkovsky (1969), Chernous ko (1992), David (1978), Zuazua (26), Ibragimov ( 759), Ilin (21), Lions (1968), Osipov (1975), Satimov and Tukhtasinov (27), Satimov and Tukhtasinov (26), Satimov and Tukhtasinov (25), Tukhtasinov (1995) Tukhtasinov and Mamatov (28)). Using decomposition method, some of the control and differential game problems described by parabolic and hyperbolic partial differential equations can be reduced to the ones described by infinite system of ordinary differential equations of first and second order respectively. For example, using this method, one can reduce some control problems for parabolic equation (see, e.g. Avdonin and Ivanov (1995), Butkovsky (1969), Chernous ko (1992), Ibragimov ( 759), Satimov and Tukhtasinov (26), Satimov and Tukhtasinov (25), Tukhtasinov (1995), Tukhtasinov and Mamatov (28)) z t + Az = w to the following infinite system of differential equations ż k + λ k z k = w k, k = 1, 2,..., < λ 1 λ 2..., where w k, k = 1, 2,... are control parameters z k, w k R 1, and λ k are eigenvalues of the elliptic operator n ( A = a ij (x) ). x i x j i,j=1 Because of this significant relationship between control problems described by partial differential equations and those described by infinite system of differential equations, the latter can be investigated within one theoretical framework assuming that the coefficients are any real numbers. This opens up new prospects for the differential games in Hilbert spaces. The works Ibragimov et al. (215), Ibragimov et al. (214), Ibragimov (213a), Ibragimov (213b), Ibragimov and Hasim (21), Ibragimov (25), 182 Malaysian Journal of Mathematical Sciences

3 Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations Alias et al. (216), Salimi et al. (216) relate to such differential games, where various differential game problems for infinite system of differential equations were examined. In the present paper, we study a pursuit differential game with geometric constraints on the control functions of players in the Hilbert space l 2 with elements ξ = (ξ 1, ξ 2,..., ξ k,...), ξk 2 <, where inner product and norm are defined by formulas (ξ, η) = ξ k η k, ξ = (ξ, ξ) 1/2. Differential game is described by the following infinite system of differential equations ẋ k = α k x k β k y k + u k1 v k1, x k () = x k, ẏ k = β k x k α k y k + u k2 v k2, y k () = y k,, k = 1, 2,..., (1) where α k, β k are real numbers, α k, x = (x 1, x 2,... ) l 2, y = (y 1, y 2,... ) l 2, u = (u 11, u 12, u 21, u 22,... ) and v = (v 11, v 12, v 21, v 22,... ) are control parameters of pursuer and evader, respectively. The aim of the pursuer is to force the state of the system towards the origin of the space l 2 against any action of the evader and the aim of the evader is opposite. We obtain an equation for a guaranteed pursuit time and construct a pursuit strategy explicitly. Note that a differential game problem was studied for the system (1) in Ibragimov (213b) when controls of players are subjected to integral constraints. In the present paper, we consider the case of geometric constraints. 2. Statement of Problem Let ρ, ρ and σ be positive numbers (ρ > σ). Definition 2.1. A function w( ) = (w 11 ( ), w 12 ( ), w 21 ( ), w 22 ( ),... ), w : [, T ] l 2, Malaysian Journal of Mathematical Sciences 183

4 Ibragimov, G. et al. with measurable components w k1 (t), w k2 (t), t T, k = 1, 2,..., subject to the condition (wk1(s) 2 + wk2(s)) 2 ρ 2 is referred to as the admissible control, where T > is sufficiently large fixed number. Denote the set of all admissible controls by S(ρ ). Definition 2.2. Functions u( ) S(ρ) and v( ) S(σ) are referred to as the admissible controls of pursuer and evader, respectively. Definition 2.3. A function u(t, v) = (u 1 (t, v), u 2 (t, v),... ), u : [, T ] l 2 l 2, with 2-dimensional components u k = (u k1, u k2 ) of the form u k (t, v) = v k (t) + w k (t), w k = (w k1, w k2 ), v k = (v k1, v k2 ), w( ) = (w 1 ( ), w 2 ( ),...) S(ρ σ), subject to the condition (admissibility) u k (t, v(t)) 2 ρ 2 for any v( ) S(σ), is called strategy of pursuer. Definition 2.4. If there exists a strategy of the pursuer such that, for any admissible control of the evader, the equality z(τ) = occurs at some τ, τ θ, then we say that differential game (1) can be completed for the time θ. The time θ is called a guaranteed pursuit time. Problem. Find a guaranteed pursuit time in differential game (1). Let z(t) = (z 1 (t), z 2 (t),... ) = (x 1 (t), y 1 (t), x 2 (t), y 2 (t),... ), ( 1/2 z k (t) = (x k (t), y k (t)), z k = x 2k + y2k, z = (x 2 k + yk)) 2, ( 1/2 z = (z 1, z 2,... ) = (x 1, y 1, x 2, y 2,... ), z = (x 2 k + yk)) 2, [ ] e α k A k (t) = t cosβ k t e αkt sinβ k t e αkt sinβ k t e αkt, k = 1, 2,.... (2) cosβ k t 184 Malaysian Journal of Mathematical Sciences

5 Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations It is not difficult to show that the matrices A k (t) have the following properties: A k (t + h) = A k (t)a k (h) = A k (h)a k (t), A k (t)z k = A k(t)z k = e αk(t) z k, (3) where A denotes the transpose of the matrix A. 3. Control Problem In this section, we construct a control and find a time required to steer the state of a system from a given initial point z to the origin. Let C(, T ; l 2 ) be the space of continuous functions z( ) such that z : [, T ] l 2. We need the following proposition Ibragimov et al. (28). Proposition 3.1. If w( ) S(ρ ) and α k, then for any given T > the following infinite system of differential equations ẋ k = α k x k β k y k + w 1k, x k () = x k, ẏ k = β k x k α k y k + w 2k, y k () = y k,, k = 1, 2,..., (4) has a unique solution z(t) = (z 1 (t), z 2 (t),... ), t T, in the space C(, T ; l 2 ). Of course, z k (t) = A k (t)z k + t A k (t s)w k (s)ds, k = 1, 2,.... It should be noted that this existence-uniqueness theorem for the system (4) was proved for any finite interval [, T ] (see Ibragimov et al. (28)). Before studying differential game for system (1), first we consider the following control problem for the system (4): Find an admissible control w k (t) and time θ such that z(θ) =. Let ϕ k (t) = { e 2α k t 1 2α k, α k >, t, α k =,, t >. Consider the equation e 2α kt z k 2 ϕ 2 k (t) = ρ 2. (5) Malaysian Journal of Mathematical Sciences 185

6 Ibragimov, G. et al. Since e 2αkt /ϕ 2 k (t) + as t + for each k, therefore left-hand side of the equation (5) approaches + as t +. Because of e 2α kt ϕ 2 k (t) = 1 [ ] 2 α k t t 2 < 1 sinh(α k t) t 2, α k >, and e2αkt ϕ 2 k (t) = 1 t 2, α k =, we obtain e 2α kt z k 2 ϕ 2 k (t) 1 t 2 z k 2 = 1 t 2 z 2, t. Moreover, the function f(t) = e2α k t decreases on (, ) and hence, equation ϕ 2 k (t) (5) has a unique root t = θ >. Lemma 3.1. The control w k (t) = 1 ϕ k (θ) A k( t)z k, t θ, (6) steers the system (4) to the origin at the time θ. Proof. A. Show that the control w k (t) is admissible. Using (3) and the fact that t = θ is a root of the equation (5), we obtain w k (t) 2 1 = ϕ 2 k( t)z k 2 k (θ) A = z k 2 ϕ 2 k (θ)e2α kt Therefore the control w k (t) is admissible. e 2α kθ z k 2 ϕ 2 k (θ) = ρ 2. B. Next, show that for the state of system (4) the equation z(θ) = is satisfied. Indeed, ξ k (θ). = z k + θ = z k 1 ϕ k (θ) A k ( s)w k (s)ds θ A k ( s)a k( s)z k ds = z k z k =, k = 1, 2,.... (7) 186 Malaysian Journal of Mathematical Sciences

7 Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations Therefore z k (θ) = A k (θ)ξ k (θ) =, k = 1, 2,.... This completes the proof of the lemma. 4. Differential Game Problem We now turn to the system (1) and formulate the main result of the paper. Let t = θ 1 be a root of the equation As mentioned above, this root is unique. e 2α kt z k 2 ϕ 2 k (t) = (ρ σ) 2. (8) Theorem 4.1. The number θ 1 is a guaranteed pursuit time in the game (1). Proof. First, show that pursuit is completed at the time θ 1. Let the pursuer use the following strategy: u k (t, v k (t)) = v k (t) + ω k (t), ω k (t) = 1 ϕ k (θ 1 ) A k( t)z k, k = 1, 2,..., (9) Pursuit is completed at the time θ 1 since using (9) and (8) we obtain θ1 z k (θ 1 ) = A k (θ 1 )z k + = A k (θ 1 )z k + [ = A k (θ 1 ) Thus, z k (θ 1 ) =, k = 1, 2,... θ1 z k 1 ϕ k (θ 1 ) A k (θ 1 s) (u k (s, v k (s)) v k (s)) ds A k (θ 1 s)ω k (s)ds θ1 A k ( s)a k( s)z k ds ] =. Next, show admissibility of strategy (9). Using the Minkowskii inequality and (9) yields Malaysian Journal of Mathematical Sciences 187

8 Ibragimov, G. et al. ( 1/2 ( ) 1/2 uk(t)) 2 = v k (t) + ω k (t) 2 ( ) 1/2 ( v k (t) + ω k (t) ) 2 ( ) 1/2 ( ) 1/2 v k (t) 2 e 2α kθ 1 + ϕ 2 k (θ 1) z k 2 σ + ρ σ = ρ. This completes the proof of the theorem. 5. Conclusion We have studied a pursuit differential game described by an infinite system of first-order differential equations when control functions are subjected to integral constraints. First, we solved an auxiliary control problem: we have constructed a control function and found a time for which state of the system can be steered to the origin. We have then solved pursuit problem: constructed a pursuit strategy, and found guaranteed pursuit time. Acknowledgements The present research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia, FR. References Alias, I., Ibragimov, G., and Rakhmanov, A. (216). Evasion differential game of infinitely many evaders from infinitely many pursuers in hilbert space. Dynamic Games and Applications, pages Avdonin, S. and Ivanov, S. (1995). Families of exponentials: the method of moments in controllability problems for distributed parameter systems. Cambridge University Press. 188 Malaysian Journal of Mathematical Sciences

9 Pursuit Differential Game Described by Infinite First Order 2-Systems of Differential Equations Butkovsky, A. (1969). Theory of optimal control of distributed parameter systems. New York: Elsevier. Chernous ko, F. (1992). Bounded controls in distributed-parameter systems. Journal of Applied Mathematics and Mechanics, 56(5): David, L. (1978). Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev., 2(4): Ibragimov, G. (22,(Prikladnaya Matematika i Mekhanika. 66(5): )). A problem of optimal pursuit in systems with distributed parameters. Journal of Applied Mathematics and Mechanics, 66(5): Ibragimov, G. (25). Optimal pursuit with countably many pursuers and one evader. Differential Equations, 41(5): Ibragimov, G. (213a). The optimal pursuit problem reduced to an infinite system of differential equations. Journal of Applied Mathematics and Mechanics, 77(5):47 476, Ibragimov, G. (213b). Optimal pursuit time for a differential game in the hilbert space l 2. ScienceAsia, 39S:25 3. Ibragimov, G., Abd Rasid, N., Kuchkarov, A., and Ismail, F. (215). Multi pursuer differential game of optimal approach with integral constraints on controls of players. Taiwanese Journal of Mathematics, 19(3): , Doi: /tjm Ibragimov, G., Allahabi, F., and Kuchkarov, A. (214). A pursuit problem in an infinite system of second-order differential equations. Ukrainian Mathematical Journal, 65(8): Ibragimov, G., Azamov, A., and Hasim, R. (28). Existence and uniqueness of the solution for an infinite system of differential equations. Journal KALAM, International Journal of Mathematics and Statistics, 1:9 14. Ibragimov, G. and Hasim, R. (21). Pursuit and evasion differential games in hilbert space. International Game Theory Review, 12(3): Ilin, V. (21). Boundary control of a string oscillating at one end, with the other end fixed and under the condition of the existence of finite energy. Dokl. RAN, 378(6): Isaacs, R. (1965). Differential games, a mathematical theory with applications to optimization, control and warfare. Wiley, New York. Malaysian Journal of Mathematical Sciences 189

10 Ibragimov, G. et al. Lions, J. (1968). Contrôle optimal de systemes gouvernés par des équations aux dérivées partielles. Paris:Dunod. Osipov, Y. (1975). Theory of differential games in systems with distributed parameters. Doklady Akademii Nauk SSSR, 223(6): Salimi, M., Ibragimov, G., Siegmund, S., and Sharifi, S. (216). On a fixed duration pursuit differential game with geometric and integral constraints. Dynamic Games and Applications, 6(2): Satimov, N. and Tukhtasinov, M. (25). On some game problems for firstorder controlled evolution equations. Differential Equations, 41(8): Satimov, N. and Tukhtasinov, M. (26). Game problems on a fixed interval in controlled first-order evolution equations. Mathematical notes, 8(3-4): Satimov, N. and Tukhtasinov, M. (27). On game problems for second-order evolution equations. Russian Mathematics (Iz VUZ), 51(1): Tukhtasinov, M. (1995). Some problems in the theory of differential pursuit games in systems with distributed parameters. Journal of Applied Mathematics and Mechanics, 59(6): Tukhtasinov, M. and Mamatov, M. (28). On pursuit problems in controlled distributed systems. Mathematical notes, 84(1-2): Zuazua, E. (26). Controllability and observability of partial differential equations: Some results and open problems. Handbook of differential equations: evolutionary differential equations, (eds C.M. Dafermos and E.Feireisl), volume 3. Amsterdam: Elsevier. 19 Malaysian Journal of Mathematical Sciences