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Group Pursuit Differential Games with Pure Time-Lag

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Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

The paper is devoted to the group pursuit differential-difference game with pure time-lag. An approach to the solution of this problem based on the method of resolving functions is proposed. For the group problem, the integral presentation of game solution based on the time-delay exponential is proposed at the first time. The guaranteed time of the game termination is found, and corresponding control law is constructed. The results are illustrated by a model example. In such game of two persons, it is possible to avoid meeting with the terminal set with any control of the pursuer. It is shown that if the pursuers are several then the pursuit game can be completed.

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References

  1. Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley Inc., New York (1965)

    Google Scholar 

  2. Pontryagin, L.S.: Selected Scientific Works, vol. 2. Nauka, Moscow (1988)

    Google Scholar 

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

    Google Scholar 

  4. Chikrii, A.A.: Conflict-Controlled Processes. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  5. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Springer Netherlands, Dordrecht (1988). https://doi.org/10.1007/978-94-015-7793-9

  6. Chikrii, A.A.: An analytical method in dynamic pursuit games. Proc. Steklov Inst. Math. 271(1), 69–85 (2010). https://doi.org/10.1134/S0081543810040073

    Article  MathSciNet  MATH  Google Scholar 

  7. Chikrii, A.A.: Multivalued mappings and their selections in game control problems. J. Autom. Inf. Sci. 27(1), 27–38 (1995)

    MathSciNet  Google Scholar 

  8. Chikrii, A.A., Rappoport, I.S., Chikrii, K.A.: Multivalued mappings and their selectors in the theory of conflict-controlled processes. Cybern. Syst. Anal. 43(5), 719–730 (2007). https://doi.org/10.1007/s10559-007-0097-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Chikrii, A.A., Rappoport, I.S.: Method of resolving functions in the theory of conflict-controlled processes. Cybern. Syst. Anal. 48(4), 512–531 (2012). https://doi.org/10.1007/s10559-012-9430-y

    Article  MathSciNet  MATH  Google Scholar 

  10. Chikrii, A.A., Eidelman, S.D.: Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order. Cybern. Syst. Anal. 36(3), 315–338 (2000). https://doi.org/10.1007/BF02732983

    Article  MATH  Google Scholar 

  11. Chikriy, A.A., Matichin, I.I.: Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann-Liouville, Caputo and Miller-Ross. J. Autom. Inf. Sci. 40(6), 1–11 (2008). https://doi.org/10.1615/JAutomatInfScien.v40.i6.10

    Article  Google Scholar 

  12. Chikrii, A.A.: Optimization of game interaction of fractional-order controlled systems. Optim. Methods Softw. 23(1), 39–72 (2008). https://doi.org/10.1080/10556780701281309

    Article  MathSciNet  MATH  Google Scholar 

  13. Chikrii, A.A.: Game dynamic problems for systems with fractional derivatives. Springer Optimization and Its Applications, vol. 17, pp. 349–386 (2008). https://doi.org/10.1007/978-0-387-77247-9_1

  14. Chikrii, A.A., Eidelman, S.D: Game problems of control for quasilinear systems with fractional Riemann-Liouville derivatives. Kibern. Sist. Anal. 6, 66–99 (2001)

    Google Scholar 

  15. Chikrii, A.A., Eidelman, S.D.: Control game problems for quasilinear systems with Riemann-Liouville fractional derivatives. Cybern. Syst. Anal. 37(6), 836–864 (2001). https://doi.org/10.1023/A:1014529914874

  16. Chikrii, A.A., Eidelman, S.D.: Game problems for fractional quasilinear systems. Comput. Math. Appl. 44(7), 835–851 (2002). https://doi.org/10.1016/S0898-1221(02)00197-9

  17. Chikrii A.A.: On nonstationary game problem of motion control. J. Autom. Inf. Sci. 47(11), 74–83 (2015). https://doi.org/10.1615/JAutomatInfScien.v47.i11.60

  18. Chikrii A.A., Pepelyaev, V.A.: On the game dynamics problems for nonstationary controlled processes. J. Autom. Inf. Sci. 49(3), 13–23 (2017). https://doi.org/10.1615/JAutomatInfScien.v49.i3.30

  19. Baranovskaya, L.V., Chikrij, A.A., Chikrij, A.A.: Inverse Minkowski functionals in a nonstationary problem of group pursuit. Izv. Akad. Nauk Teor. Sist. Upr. (1), 109–114 (1997)

    Google Scholar 

  20. Kryvonos, I.I., Chikrii, A.A., Chikrii, K.A.: On an approach scheme in nonstationary game problems. J. Autom. Inf. Sci. 45(11), 15–21 (2013). https://doi.org/10.1615/JAutomatInfScien.v45.i11.30

  21. Baranovskaya, L.V., Chikrii, A.A., Chikrii, A.A.: Inverse Minkowski functional in a nonstationary problem of group pursuit. J. Comput. Syst. Sci. Int. 36(1), 101–106 (1997)

    Google Scholar 

  22. Baranovskaya, G.G., Baranovskaya, L.V.: Group pursuit in quasilinear differential-difference games. J. Autom. Inf. Sci. 29(1), 55–62 (1997). https://doi.org/10.1615/JAutomatInfScien.v29.i1.70

  23. Baranovskaya, L.V.: A method of resolving functions for one class of pursuit problems. East.-Eur. J. Enterp. Technol. 2(4), 4–8 (2015). https://doi.org/10.15587/1729-4061.2015.39355

    Article  Google Scholar 

  24. Baranovskaya, L.V.: Method of resolving functions for the differential-difference pursuit game for different-inertia objects. Adv. Dyn. Syst. Control 69, 159–176 (2016). https://doi.org/10.1007/978-3-319-40673-2

    Article  MathSciNet  Google Scholar 

  25. Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic, Cambridge (1963)

    Google Scholar 

  26. Khusainov, D.Y., Benditkis, D.D., Diblik, J.: Weak Delay in Systems with an Aftereffect. Funct. Differ. Equ. 9(3–4), 385–404 (2002)

    Google Scholar 

  27. Baranovska, L.V.: On quasilinear differential-difference games of approach. J. Autom. Inf. Sci. 49(8), 53–67 (2017). https://doi.org/10.1615/JAutomatInfScien.v49.i8.40

  28. Baranovska, L.V.: Quasi-Linear Differential-Deference Game of Approach. Understanding Complex Systems, pp. 505–524 (2019). https://doi.org/10.1007/978-3-319-96755-4_26

  29. Baranovska, L.V.: Pursuit differential-difference games with pure time-lag. Discret. Contin. Dyn. Syst. - B 24(3), 1021–1031 (2019). https://doi.org/10.3934/dcdsb.2019004

  30. Khusainov, D.Y., Diblik, J., Ruzhichkova, M.: Linear dynamical systems with aftereffect. Representation of Decisions, Stability, Control, Stabilization. GP Inform-Analytics Agency, Kiev (2015)

    Google Scholar 

  31. Aubin, J.-P., Frankovska, H.: Set-Valued Analysis. Birkhäuser, Bostom (1990)

    Google Scholar 

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Acknowledgements

The author is grateful to Academician Zgurovsky M.Z. for the possibility of the publication and to professor Kasyanov P.O. for assistance in publication this article.

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Authors and Affiliations

  1. Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, 37 Peremogy Ave., building 35, Kyiv, 03056, Ukraine

    Lesia V. Baranovska

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  1. Lesia V. Baranovska
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Correspondence to Lesia V. Baranovska .

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  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

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Baranovska, L.V. (2021). Group Pursuit Differential Games with Pure Time-Lag. In: Sadovnichiy, V.A., Zgurovsky, M.Z. (eds) Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-50302-4_23

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