Skip to main content
SpringerLink
Log in

Pursuit and Evasion Games for an Infinite System of Differential Equations

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper we study a linear pursuit differential game described by an infinite system of first-order differential equations in Hilbert space. The control functions of players are subject to geometric constraints. The pursuer attempts to bring the state of system from a given initial state to the origin for a finite time and the evader’s purpose is opposite. We obtain a formula for the guaranteed pursuit time and construct a strategy for pursuer. Also, we obtain a formula for the guaranteed evasion time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avdonin, S.A., Ivanov, S.A.: Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge UniversityPress, Cambridge (1995)

    MATH  Google Scholar 

  2. Azamov, A.A., Ruziboev, M.B.: The time-optimal problem for evolutionary partial differential equations. J. Appl. Math. Mech. 77(2), 220–224 (2013)

    Article  MathSciNet  Google Scholar 

  3. Berkovitz, L.D.: A Survey of Differential Games, Mathematical Theory of Control, pp. 373–385. Academic Press, New York (1967)

    Google Scholar 

  4. Butkovskiy, A.G.: Theory of Optimal Control of Distributed Parameter Systems. Elsevier, New York (1969)

    MATH  Google Scholar 

  5. Chernous’ko, F.L.: Bounded controls in distributed-parameter systems. J. Appl. Math. Mech. 56(5), 707–723 (1992)

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

    Book  Google Scholar 

  7. Daletskii, Yu.L., Krein, M.G.: Stability of solutions of differential equations in Banach space. Transl. Math. Monographs. Vol. 43, Am. Math. Soc., Providence, RI, (1974)

  8. Elliot, R.J., Kalton, N.J.: The existence of value for differential games. Am. Math. Soc., (1972)

  9. Fattorini, H.O.: Time-optimal control of solutions of operational differential equations. SIAM J. Control 2(1), 54–59 (1964)

    MATH  Google Scholar 

  10. Fleming, W.H.: A note on differential games of prescribed duration. Contribut. Theory of Games 3, 407–416 (1957)

    MathSciNet  MATH  Google Scholar 

  11. Fleming, W.H.: The convergence problem for differential games. J. Math. Anal. Appl. 3, 102–116 (1961)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  13. Fursikov, A.V.: Optimal Control of Distributed Systems, Theory and Applications, Translations of Math. Monographs, 187 , Am. Math. Soc., Providence, Rhode Island, (2000)

  14. Hajek, O.: Pursuit Games. Academic Press, New York (1975)

    MATH  Google Scholar 

  15. Ho, Y., Bryson, A., Baron, S.: Differential games and optimal pursuit-evasion strategies. IEEE Trans. Autom. Control 10, 385–389 (1965)

    Article  MathSciNet  Google Scholar 

  16. Ibragimov, G.I., Allahabi, F., Kuchkarov, A.S.: A pursuit problem in an infinite system of second-order differential equations. Ukrain. Math. J. 65(8), 1203–1216 (2014)

    Article  MathSciNet  Google Scholar 

  17. Ibragimov, G.I.: The optimal pursuit problem reduced to an infinite system of differential equation. J. Appl. Math. Mech. 77(5), 470–476 (2013). https://doi.org/10.1016/j.jappmathmech.2013.12.002

    Article  MathSciNet  MATH  Google Scholar 

  18. Ibragimov, G.I.: Optimal pursuit with countably many pursuers and one evader. Differ. Equ. 41(5), 627–635 (2005)

    Article  MathSciNet  Google Scholar 

  19. Ibragimov, G.I., Risman, M.H.: Pursuit and evasion differential game in Hilbert space. Int. Game Theory Rev. 12(3), 239–251 (2010)

    Article  MathSciNet  Google Scholar 

  20. Ibragimov, G.I.: Optimal pursuit time for a differential game in the Hilbert space \(l_2\). ScienceAsia 39S, 25–30 (2013)

    Article  Google Scholar 

  21. Ibragimov, G., Norshakila, A.R., Kuchkarov, A., Fudziah, I.: Multi pursuer differential game of optimal approach with integral constraints on controls of players. Taiwan. J. Math. 19(3), 963–976 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Ibragimov, G.I.: A problem of optimal pursuit in systems with distributed parameters. J. Appl. Math. Mech. 66(5): 719–724. (Prikladnaya Matematika i Mekhanika. 66(5): 753–759, 2002) (2002)

  23. Alias, I.A., Ibragimov, G., Rakhmanov, A.: Evasion differential game of infinitely many evaders from infinitely many pursuers in Hilbert space. Dynamic Games Appl. (2017). https://doi.org/10.1007/s13235-016-0196-0,7:347-359

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  Google Scholar 

  26. Ladyzhenskaya, O.A.: Kraevye zadachi matematicheskoy fiziki. (Boundary value problems of mathematical physics) Moscow, (1973)

  27. Lions, J.L.: Contrôle Optimal de Systémes Gouvernées par des Equations aux Dérivées Partielles. Dunod, Paris (1968)

    MATH  Google Scholar 

  28. Osipov, Y.S.: The theory of differential games in systems with distributed parameters. Dokl Akad Nauk SSSR 223(6): 1314–1317 [SovietMath. Dokl. 16, 1093–1097 (1975)], (1975)

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  31. Pshenichnii, B.N.: Simple pursuit by several objects. Cybern. Syst. Anal. 12(3), 484–485 (1976). https://doi.org/10.1007/BF01070036

    Article  MathSciNet  Google Scholar 

  32. Salimi, M., Ferrara, M.: Differential game of optimal pursuit of one evader by many pursuers. Int. J. Game Theory 48(2), 481–490 (2019)

    Article  MathSciNet  Google Scholar 

  33. Salimi, M., Ibragimov, G.I., Siegmund, S., Sharifi, S.: On a fixed duration pursuit differential game with geometric and integral constraints. Dyn. Games Appl. 6(2), 409–425 (2016). https://doi.org/10.1007/s13235-015-0161-3

    Article  MathSciNet  MATH  Google Scholar 

  34. Satimov, N.Y., Tukhtasinov, M.: Game problems on a fixed interval in controlled first-order evolution equations. Math. Notes 80(3–4), 578–589 (2006)

    Article  MathSciNet  Google Scholar 

  35. Satimov, N.Y., Tukhtasinov, M.: On some game problems for first-order controlled evolution equations. Differ. Equ. 41(8), 1169–1177 (2005)

    Article  MathSciNet  Google Scholar 

  36. Satimov, N.Y., Tukhtasinov, M.: On game problems for second-order evolution equations. Russian Math. 51(1), 49–57 (2007)

    Article  MathSciNet  Google Scholar 

  37. Subbotin, A.I., Chentsov, A.G.: Optimization of Guaranteed Result in Control Problems. Nauka, Moscow (1981)

    Google Scholar 

  38. Tukhtasinov, M.: Some problems in the theory of differential pursuit games in systems with distributed parameters. J. Appl. Math. Mech. 59(6), 979–984 (1995)

    Article  MathSciNet  Google Scholar 

  39. Tukhtasinov, M., Mamatov, M.S.: On pursuit problems in controlled distributed parameters systems. Math. Notes. 84(2), 256–262 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work completed during the stay of the author Ibragimov G.I. at University Mediterranea of Reggio Calabria—Dept Di.Gi.ES—as Visiting Researcher, and it was partially supported by Geran Putra Berimpak UPM/700-2/1/GPB/2017/9590200 of Universiti Putra Malaysia and the financial support by Decisions_LAB—Dept. of Law, Economics and Human Sciences—University Mediterranea of Reggio Calabria, Italy.

Author information

Authors and Affiliations

  1. Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Malaysia

    Gafurjan Ibragimov, Idham Arif Alias & Nurzeehan Ismail

  2. Department of Law, Economics and Human Sciences, University Mediterranea of Reggio Calabria, Calabria, Italy

    Gafurjan Ibragimov

  3. Department of Law, Economics and Human Sciences and Decisions Lab, University Mediterranea of Reggio Calabria, Calabria, Italy

    Massimiliano Ferrara

  4. Department of Management and Technology, ICRIOS - The Invernizzi Centre for Research in Innovation, Organization, Strategy and Entrepreneurship Bocconi University, Milan, Italy

    Massimiliano Ferrara

  5. Department of Mathematics & Statistics, St.Francis Xavier University, Antigonish, NS, Canada

    Mehdi Salimi

Authors
  1. Gafurjan Ibragimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Massimiliano Ferrara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Idham Arif Alias
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mehdi Salimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nurzeehan Ismail
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Idham Arif Alias.

Additional information

Communicated by See Keong Lee.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibragimov, G., Ferrara, M., Alias, I.A. et al. Pursuit and Evasion Games for an Infinite System of Differential Equations. Bull. Malays. Math. Sci. Soc. 45, 69–81 (2022). https://doi.org/10.1007/s40840-021-01176-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01176-x

Keywords

Mathematics Subject Classification

Search

Navigation