A GUARANTEED PURSUIT TIME IN A DIFFERENTIAL GAME IN HILBERT SPACE
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Published:
Feb 28, 2019
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Abstract
We study a pursuit differential game problem of one pursuer and one evader in the Hilbert space . The differential game is described by an infinite number of first-order 2-systems of linear differential equations. The control functions of players are subjected to integral constraints. A game is started from the given initial position
. It is given another point
in the space
. If the state of the infinite system coincides with the point
at some time, then pursuit is considered completed. Our purpose is to obtain an equation to find a guaranteed pursuit time and construct a strategy for the pursuer.
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Ibragimov, G., Waziri, U., Alias, I. A., & Ibrahim, Z. B. (2019). A GUARANTEED PURSUIT TIME IN A DIFFERENTIAL GAME IN HILBERT SPACE. Malaysian Journal of Science, 38(Sp 1), 43–54. https://doi.org/10.22452/mjs.sp2019no1.4
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References
Alias, I. A., Ibragimov, G., & Rakhmanov, A. (2017). Evasion differential game of infinitely many evaders from infinitely many pursuers in Hilbert space. Dynamic Games and Applications, 7(3), 347-359.
Avdonin, I., Avdonin, S. A., & Ivanov, S. A. (1995). Families of exponentials: the method of moments in controllability problems for distributed parameter systems. Cambridge University Press.
Azamov, A. A., & Ruziboyev, M. B. (2013). The time-optimal problem for evolutionary partial differential equations. Journal of Applied Mathematics and Mechanics, 77(2), 220-224.
Azamov, A. A., & Samatov, B. (2000). -strategy. An elementary introduction to the theory of differential games. Taskent: National Univ. of Uzb.
Butkovsky, A. (1969). Theory of optimal control of distributed parameter systems, New York: Elsevier.
Chernous' ko, F. L. (1992). Bounded controls in distributed-parameter systems. Journal of Applied Mathematics and Mechanics, 56(5), 707-723.
Chikrii, A. A., & Belousov, A. A. (2010). On linear differential games with integral constraints. Proceedings of the Steklov Institute of Mathematics, 269(1), 69.
Friedman, A. (1971). Differential games, New York: John Wiley and Sons.
Huseyin, A., & Huseyin, N. (2012). Precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra integral equation. Mathematical Modelling and Analysis, 17(5), 686-695.
Ibragimov, G. (2002). An optimal pursuit problem in systems with distributed parameters. Prikladnaya Matematika i Mekhanika, 66(5):753–759.
Ibragimov, G. (2013). Optimal pursuit time for a differential game in the Hilbert space . Differential Equations, 6(8), 9.
Ibragimov, G. I. & Ja'afaru, A. B. (2011). Australian Journal of Basic and Applied Science, 5:736-742.
Ibragimov, G. I. (2013). The optimal pursuit problem reduced to an infinite system of differential equations. Journal of Applied Mathematics and Mechanics, 77(5):470–476.
Ibragimov, G. I., Azamov, A. A., & Khakestari, M. (2011). Solution of a linear pursuit-evasion game with integral constraints. ANZIAM Journal, 52, 59-75.
Ibragimov, G. I., Azamov, A., & Hasim, R. M. (2008). Existence and uniqueness of the solution for an infinite system of differential equations. Journal KALAM, International Journal of Mathematics and Statistics, 1(2), 9-14.
Ibragimov, G., & Salleh, Y. (2012). Simple motion evasion differential game of many pursuers and one evader with integral constraints on control functions of players. Journal of Applied Mathematics, 2012.
Ibragimov, G., Alias, I. A., Waziri, U., & Ja’afaru, A. B. (2017). Differential Game of Optimal Pursuit for an Infinite System of Differential Equations. Bulletin of the Malaysian Mathematical Sciences Society, 1-13.
Ibragimov, G., Rasid, N. A., Kuchkarov, A., & Ismail, F. (2015). Multi pursuer differential game of optimal approach with integral constraints on controls of players. Taiwanese Journal of Mathematics, 19(3), 963-976.
Isaacs, R. (1965). Differential games, a mathematical theory with applications to optimization, control and warfare, New York: Wiley.
Konovalov, A. P. (1987). Linear differential evasion games with lag and integral constraints. Cybernetics, 41-45.
Krasovskiǐ, N. N. & Subbotin, A. I. (1988). Game-theoretical control problems, New York: Springer.
Kuchkarov, A. S. (2013). On a differential game with integral and phase constraints. Automation and Remote Control, 74(1), 12-25.
Lewin, J. (1994). Differential Games, London: Springer.
Lokshin, M. D. (1990). Differential games with integral constraints on disturbances. Journal of Applied Mathematics and Mechanics, 54(3), 331-337.
Mamatov, M. S. (2008). On the theory of differential pursuit games in distributed parameter systems. Automatic Control and Computer Sciences, 43(1), 1-8.
Nikolskii, M. S. (1969). The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, 2, 49-59.
Okhezin, S. P. (1977). Differential encounter-evasion game for parabolic system under integral constraints on the player’s controls. Prikl Mat I Mekh,41(2):202–209.
Petrosyan, L. A. (1993). Differential Games of Pursuit, Singapore, London: World Scientific.
Pontryagin, L. S. (1988). Selected Scientific Works. Vol. II.
Samatov, B. T. (2013). Problems of group pursuit with integral constraints on controls of the players. I. Cybernetics and Systems Analysis, 49(5), 756-767.
Satimov, N. Y., & Tukhtasinov, M. (2006). Game problems on a fixed interval in controlled first-order evolution equations. Mathematical notes, 80(3-4), 578-589.
Satimov, N., Rikhsiev, B. B., & Khamdamov, A. A. (1983). On a pursuit problem for n-person linear differential and discrete games with integral constraints. Sbornik: Mathematics, 46(4), 459-471.
Solomatin, A. M. (1984). A game theoretic approach-evasion problem for a linear system with integral constraints imposed on the player control. Journal of Applied Mathematics and Mechanics, 48(4), 401-405.
Subbotin, A. and Ushakov, V. N. (1975). Alternative for an encounter-evasion differential game with integral constraints on the players’ controls. J. Appl. Maths Mekhs, 39(3):367–375.
Tukhtasinov, M., & Mamatov, M. S. (2008). On pursuit problems in controlled distributed systems. Mathematical notes, 84(1-2), 256-262.
Tukhtasinov, M., & Mamatov, M. S. (2009). On transfer problems in control systems. Differential Equations, 45(3),439-444.
Avdonin, I., Avdonin, S. A., & Ivanov, S. A. (1995). Families of exponentials: the method of moments in controllability problems for distributed parameter systems. Cambridge University Press.
Azamov, A. A., & Ruziboyev, M. B. (2013). The time-optimal problem for evolutionary partial differential equations. Journal of Applied Mathematics and Mechanics, 77(2), 220-224.
Azamov, A. A., & Samatov, B. (2000). -strategy. An elementary introduction to the theory of differential games. Taskent: National Univ. of Uzb.
Butkovsky, A. (1969). Theory of optimal control of distributed parameter systems, New York: Elsevier.
Chernous' ko, F. L. (1992). Bounded controls in distributed-parameter systems. Journal of Applied Mathematics and Mechanics, 56(5), 707-723.
Chikrii, A. A., & Belousov, A. A. (2010). On linear differential games with integral constraints. Proceedings of the Steklov Institute of Mathematics, 269(1), 69.
Friedman, A. (1971). Differential games, New York: John Wiley and Sons.
Huseyin, A., & Huseyin, N. (2012). Precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra integral equation. Mathematical Modelling and Analysis, 17(5), 686-695.
Ibragimov, G. (2002). An optimal pursuit problem in systems with distributed parameters. Prikladnaya Matematika i Mekhanika, 66(5):753–759.
Ibragimov, G. (2013). Optimal pursuit time for a differential game in the Hilbert space . Differential Equations, 6(8), 9.
Ibragimov, G. I. & Ja'afaru, A. B. (2011). Australian Journal of Basic and Applied Science, 5:736-742.
Ibragimov, G. I. (2013). The optimal pursuit problem reduced to an infinite system of differential equations. Journal of Applied Mathematics and Mechanics, 77(5):470–476.
Ibragimov, G. I., Azamov, A. A., & Khakestari, M. (2011). Solution of a linear pursuit-evasion game with integral constraints. ANZIAM Journal, 52, 59-75.
Ibragimov, G. I., Azamov, A., & Hasim, R. M. (2008). Existence and uniqueness of the solution for an infinite system of differential equations. Journal KALAM, International Journal of Mathematics and Statistics, 1(2), 9-14.
Ibragimov, G., & Salleh, Y. (2012). Simple motion evasion differential game of many pursuers and one evader with integral constraints on control functions of players. Journal of Applied Mathematics, 2012.
Ibragimov, G., Alias, I. A., Waziri, U., & Ja’afaru, A. B. (2017). Differential Game of Optimal Pursuit for an Infinite System of Differential Equations. Bulletin of the Malaysian Mathematical Sciences Society, 1-13.
Ibragimov, G., Rasid, N. A., Kuchkarov, A., & Ismail, F. (2015). Multi pursuer differential game of optimal approach with integral constraints on controls of players. Taiwanese Journal of Mathematics, 19(3), 963-976.
Isaacs, R. (1965). Differential games, a mathematical theory with applications to optimization, control and warfare, New York: Wiley.
Konovalov, A. P. (1987). Linear differential evasion games with lag and integral constraints. Cybernetics, 41-45.
Krasovskiǐ, N. N. & Subbotin, A. I. (1988). Game-theoretical control problems, New York: Springer.
Kuchkarov, A. S. (2013). On a differential game with integral and phase constraints. Automation and Remote Control, 74(1), 12-25.
Lewin, J. (1994). Differential Games, London: Springer.
Lokshin, M. D. (1990). Differential games with integral constraints on disturbances. Journal of Applied Mathematics and Mechanics, 54(3), 331-337.
Mamatov, M. S. (2008). On the theory of differential pursuit games in distributed parameter systems. Automatic Control and Computer Sciences, 43(1), 1-8.
Nikolskii, M. S. (1969). The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, 2, 49-59.
Okhezin, S. P. (1977). Differential encounter-evasion game for parabolic system under integral constraints on the player’s controls. Prikl Mat I Mekh,41(2):202–209.
Petrosyan, L. A. (1993). Differential Games of Pursuit, Singapore, London: World Scientific.
Pontryagin, L. S. (1988). Selected Scientific Works. Vol. II.
Samatov, B. T. (2013). Problems of group pursuit with integral constraints on controls of the players. I. Cybernetics and Systems Analysis, 49(5), 756-767.
Satimov, N. Y., & Tukhtasinov, M. (2006). Game problems on a fixed interval in controlled first-order evolution equations. Mathematical notes, 80(3-4), 578-589.
Satimov, N., Rikhsiev, B. B., & Khamdamov, A. A. (1983). On a pursuit problem for n-person linear differential and discrete games with integral constraints. Sbornik: Mathematics, 46(4), 459-471.
Solomatin, A. M. (1984). A game theoretic approach-evasion problem for a linear system with integral constraints imposed on the player control. Journal of Applied Mathematics and Mechanics, 48(4), 401-405.
Subbotin, A. and Ushakov, V. N. (1975). Alternative for an encounter-evasion differential game with integral constraints on the players’ controls. J. Appl. Maths Mekhs, 39(3):367–375.
Tukhtasinov, M., & Mamatov, M. S. (2008). On pursuit problems in controlled distributed systems. Mathematical notes, 84(1-2), 256-262.
Tukhtasinov, M., & Mamatov, M. S. (2009). On transfer problems in control systems. Differential Equations, 45(3),439-444.