A GUARANTEED PURSUIT TIME IN A DIFFERENTIAL GAME IN HILBERT SPACE
Article Sidebar
Published:
Feb 28, 2019
Main Article Content
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.
Downloads
Download data is not yet available.
Article Details
How to Cite
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
Section
ICMSS2018 (Published)
Transfer of Copyrights
- In the event of publication of the manuscript entitled [INSERT MANUSCRIPT TITLE AND REF NO.] in the Malaysian Journal of Science, I hereby transfer copyrights of the manuscript title, abstract and contents to the Malaysian Journal of Science and the Faculty of Science, University of Malaya (as the publisher) for the full legal term of copyright and any renewals thereof throughout the world in any format, and any media for communication.
Conditions of Publication
- I hereby state that this manuscript to be published is an original work, unpublished in any form prior and I have obtained the necessary permission for the reproduction (or am the owner) of any images, illustrations, tables, charts, figures, maps, photographs and other visual materials of whom the copyrights is owned by a third party.
- This manuscript contains no statements that are contradictory to the relevant local and international laws or that infringes on the rights of others.
- I agree to indemnify the Malaysian Journal of Science and the Faculty of Science, University of Malaya (as the publisher) in the event of any claims that arise in regards to the above conditions and assume full liability on the published manuscript.
Reviewer’s Responsibilities
- Reviewers must treat the manuscripts received for reviewing process as confidential. It must not be shown or discussed with others without the authorization from the editor of MJS.
- Reviewers assigned must not have conflicts of interest with respect to the original work, the authors of the article or the research funding.
- Reviewers should judge or evaluate the manuscripts objective as possible. The feedback from the reviewers should be express clearly with supporting arguments.
- If the assigned reviewer considers themselves not able to complete the review of the manuscript, they must communicate with the editor, so that the manuscript could be sent to another suitable reviewer.
Copyright: Rights of the Author(s)
- Effective 2007, it will become the policy of the Malaysian Journal of Science (published by the Faculty of Science, University of Malaya) to obtain copyrights of all manuscripts published. This is to facilitate:
(a) Protection against copyright infringement of the manuscript through copyright breaches or piracy.
(b) Timely handling of reproduction requests from authorized third parties that are addressed directly to the Faculty of Science, University of Malaya. - As the author, you may publish the fore-mentioned manuscript, whole or any part thereof, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given.
You may produce copies of your manuscript, whole or any part thereof, for teaching purposes or to be provided, on individual basis, to fellow researchers. - You may include the fore-mentioned manuscript, whole or any part thereof, electronically on a secure network at your affiliated institution, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given.
- You may include the fore-mentioned manuscript, whole or any part thereof, on the World Wide Web, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given.
- In the event that your manuscript, whole or any part thereof, has been requested to be reproduced, for any purpose or in any form approved by the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers), you will be informed. It is requested that any changes to your contact details (especially e-mail addresses) are made known.
Copyright: Role and responsibility of the Author(s)
- In the event of the manuscript to be published in the Malaysian Journal of Science contains materials copyrighted to others prior, it is the responsibility of current author(s) to obtain written permission from the copyright owner or owners.
- This written permission should be submitted with the proof-copy of the manuscript to be published in the Malaysian Journal of Science
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.