Pursuit Differential Game of Many Pursuers with Integral Constraints on Compact Convex Set

Abstract

We study a pursuit differential game of many pursuers and one evader in the plane. We are given a compact convex subset of \({\mathbb {R}}^2\), and the pursuers and evader move in this set. They cannot leave this set during the game. Control functions of players are subject to coordinate-wise integral constraints. If the state of a pursuer coincides with that of evader at some time, then we say that pursuit is completed. Pursuers try to complete the pursuit, and the aim of the evader is opposite. We obtain some conditions under which pursuit can be completed from any positions of the players in the given set. Moreover, we construct strategies for the pursuers. Also, we prove some properties of convex sets.

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References

  1. 1.

    Isaacs, R.: Differential Games. Wiley, New York (1965)

    Google Scholar 

  2. 2.

    Blaquiere, A., Gerard, F., Leitmann, G.: Quantitative and Qualitative Games. Academic Press, New York (1969)

    Google Scholar 

  3. 3.

    Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988)

    Google Scholar 

  4. 4.

    Pontryagin, L.S.: Selected Works. Nauka, Moscow (1988)

    Google Scholar 

  5. 5.

    Petrosyan, L.A.: Differential Games of Pursuit. World Scientific, Singapore (1993)

    Google Scholar 

  6. 6.

    Friedman, A.: Differential Games. Wiley, New York (1971)

    Google Scholar 

  7. 7.

    Hajek, O.: Pursuit Games. Mathematics, Science, Engineering. Academic Press, New York (1975)

    Google Scholar 

  8. 8.

    Nikol’skii, M.S.: The First Direct Method of L.S. Pontryagin in Differential Games. MSU Press, Moscow (1984)

    Google Scholar 

  9. 9.

    Pshenichnyi, B.N., Ostapenko, V.V.: Differential Games. Naukova Dumka, Kiev (1992)

    Google Scholar 

  10. 10.

    Chikrii, A.A.: Conflict-Controlled Processes. Kluwer, Dordrecht (1997)

    Google Scholar 

  11. 11.

    Satimov, N.Y., Rikhsiev, B.B.: Methods of Solving the Problem of Avoiding Encounter in Mathematical Control Theory. Fan, Tashkent (2000)

    Google Scholar 

  12. 12.

    Pshenichnyi, B.N.: Simple pursuit by several objects. Cybern. Syst. Anal. 12(3), 145–146 (1976)

    MathSciNet  Google Scholar 

  13. 13.

    Chernous’ko, F.L.: A problem of evasion from many pursuers. Prikladnaya Matematika i Mekhanika 40(1), 14–24 (1976)

    MATH  Google Scholar 

  14. 14.

    Zak, V.L.: On a problem of evading many pursuers. J. Appl. Math. 43(3), 456–465 (1978)

    MathSciNet  Google Scholar 

  15. 15.

    Blagodatskikh, A.I., Petrov, N.N.: Conflict Interaction Between Groups of Controlled Objects. Udmurt State University, Izhevsk (2009)

    Google Scholar 

  16. 16.

    Borowko, P., Rzymowski, W., Stachura, A.: Evasion from many pursuers in the simple motion case. J. Math. Anal. Appl. 135(1), 75–80 (1988)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Alexander, S., Bishop, R., Christ, R.: Capture pursuit games on unbounded domain. L’Enseignement Mathématique 55(1/2), 103–125 (2009)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ibragimov, G., Ferrara, M., Kuchkarov, A., Pansera, B.A.: Simple motion evasion differential game of many pursuers and evaders with integral constraints. Dyn. Games Appl. 8, 352–378 (2018). https://doi.org/10.1007/s13235-017-0226-6

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Alias, I.A., Ibragimov, G.I., Rakhmanov, A.: Evasion differential game of infinitely many evaders from infinitely many pursuers in Hilbert space. Dyn. Games Appl. 6(2), 1–13 (2016). https://doi.org/10.1007/s13235-016-0196-0

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Grigorenko, N.L.: Mathematical Methods of Control of Several Dynamic Processes. MSU Press, Moscow (1990). (in Russian)

    Google Scholar 

  21. 21.

    Huang, H., Zhangy, W., Ding, J., Stipanović, D.M., Tomlin, C.J.: Guaranteed decentralized pursuit-evasion in the plane with multiple pursuers. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference(CDC–ECC) Orlando, FL, USA (2011)

  22. 22.

    Zhou, Z., Zhang, W., Ding, J., Huang, H., Stipanović, D.M., Tomlin, C.J.: Cooperative pursuit with Voronoi partitions. Automatica 72(2016), 64–72 (2016)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Bakolas, E., Tsiotras, P.: Relay pursuit of a maneuvering target using dynamic voronoi diagrams. Automatica 48, 2213–2220 (2012)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sun, W., Tsiotras, P.: Sequential pursuit of multiple targets under external disturbances via Zermelo–Voronoi diagrams. Automatica 81, 253–260 (2017)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kurzhanski, A.B.: Problem of collision avoidance for a team motion with obstacles. Proc. Steklov Inst. Math. 293, 120–136 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kumkov, S.S., Le Ménec, S., Patsko, V.S.: Zero-sum pursuit-evasion differential games with many objects: survey of publications. Dyn. Games Appl. 7, 609–633 (2017). https://doi.org/10.1007/s13235-016-0209-z

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Liu, S.-Y., Zhou, Z., Tomlin, C., Hedrick, K.: Evasion as a team against a faster pursuer. Proc. Am. Control Conf. (2013). https://doi.org/10.1109/ACC.2013.6580676

    Article  Google Scholar 

  28. 28.

    Liu, S.-Y., Zhou, Z., Claire T.J., Hedrick, K.: Evasion of a team of Dubins vehicles from a hidden pursuer. In: Proceedings of IEEE International Conference on Robotics and Automation (2014). https://doi.org/10.1109/ICRA.2014.6907859

  29. 29.

    Chen, M., Zhou, Z., Tomlin, C.J.: Multiplayer reach-avoid games via pairwise outcomes. IEEE Trans. Autom. Control 62(3), 1–1 (2016). https://doi.org/10.1109/TAC.2016.2577619

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Chen, M., Zhou, Z., Tomlin, C.J.: Multiplayer reach-avoid games via low dimensional solutions and maximum matching. In: Proceedings of the American Control Conference, https://doi.org/10.1109/ACC.2014.6859219 (2014)

  31. 31.

    Chen, M., Zhou, Z., Tomlin, C.J.: A path defense approach to the multiplayer reach-avoid game. In: IEEE Conference on Decision and Control (2014)

  32. 32.

    Zhou, Z., Takei, R., Huang, H., Tomlin, C.J.: A general, open-loop formulation for reach-avoid games.https://doi.org/10.1109/CDC.2012.6426643 (2012)

  33. 33.

    Zhou, Z., Ding, J., Huan, H., Takei, R., Tomlin, C.: Efficient path planning algorithms in reach-avoid problems. Automatica 89(2018), 28–36 (2018). https://doi.org/10.1016/j.automatica.2017.11.035

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Awheda M.D., Schwartz, H.M.: Decentralized learning in pursuit-evasion differential games with multi-pursuer and single-superior evader. In: Annual IEEE Systems Conference (SysCon), vol. 4, IEEE, https://doi.org/10.1109/SYSCON.2016.7490516 (2016)

  35. 35.

    Ramana, M.V., Kothari, M.: Pursuit strategy to capture high-speed evaders using multiple pursuers. J. Guidance, Control, Dyn. 49(1), 139–149 (2017). https://doi.org/10.2514/1.G000584

  36. 36.

    Ramana, M.V., Kothari, M.: Pursuit-evasion games of high speed evader. J. Intell. Robot. Syst. 85(2), 293–306 (2017)

    Article  Google Scholar 

  37. 37.

    Huseyin, A., Huseyin, N., Guseinov, K.: Approximation of sections of the set of trajectories of a control system with bounded control resources. Tr. Inst. Mat. Mekh. 23(1), 116–127 (2017). https://doi.org/10.21538/0134-4889-2017-23-1-116-127

    MathSciNet  Article  Google Scholar 

  38. 38.

    Belousov, A.A.: Method of resolving functions for differential games with integral constraints. Theory Opt. Solut. 9, 10–16 (2010)

    Google Scholar 

  39. 39.

    Samatov, B.T.: Problems of group pursuit with integral constraints on controls of the players. I. Cybern. Syst. Anal. 49(5), 756–767 (2013)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Satimov, N.Y., Rikhsiev, B.B., Khamdamov, A.A.: On a pursuit problem for n-person linear differential and discrete games with integral constraints. Math. USSR-Sbornik 46(4), 459–471 (1983)

    Article  Google Scholar 

  41. 41.

    Ibragimov, G.I., Satimov, N.Y.: A multi player pursuit differential game on closed convex set with integral constraints. Abstr. Appl. Anal. 2012(Article ID 460171):12, https://doi.org/10.1155/2012/460171 (2012)

  42. 42.

    Ibragimov, G., Salimi, M., Amini, M.: Evasion from many pursuers in simple motion differential game with integral constraints. Eur. J. Oper. Res. 218(2), 505–511 (2012)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Ibragimov, G., Salleh, Y.: Simple motion evasion differential game of many pursuers and one evader with integral constraints on control functions of players. J. Appl. Math. (2012). https://doi.org/10.1155/2012/748096

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Alias, I.A., Ibragimov, G.I., Kuchkarov, A.S., Sotvoldiev, A.: Differential game with Many Pursuers when controls are subjected to coordinate-wise integral constraints. Malays. J. Math. Sci. 10(2), 195–207 (2016)

    MathSciNet  Google Scholar 

  45. 45.

    Ferrara, M., Ibragimov, G., Salimi, M.: Pursuit-evasion game of many players with coordinate-wise integral constraints on a convex set in the plane. AAPP \(\mid \) Atti della Accademia Peloritana dei Pericolanti Classe di Scienze Fisiche. Matematiche e Naturali 952, A6-1–A6-6 (2017)

    Google Scholar 

  46. 46.

    Berkovitz, L.D.: Convexity and Optimization in \(\mathbb{R}^n\). Wiley, New York (2002)

    Google Scholar 

  47. 47.

    Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill Book Company, Singapore (1986)

    Google Scholar 

  48. 48.

    Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow (1976)

    Google Scholar 

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Acknowledgements

The present research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia, No. 01-01-17-1921FR, and by MEDAlics, the Research Centre at the University Dante Alighieri, Reggio Calabria, Italy.

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Correspondence to Idham Arif Alias.

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Ferrara, M., Ibragimov, G., Alias, I.A. et al. Pursuit Differential Game of Many Pursuers with Integral Constraints on Compact Convex Set. Bull. Malays. Math. Sci. Soc. 43, 2929–2950 (2020). https://doi.org/10.1007/s40840-019-00844-3

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Keywords

  • Differential game
  • Control
  • Strategy
  • Integral constraint
  • State constraint

Mathematics Subject Classification

  • Primary: 91A23
  • Secondary: 49N75