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Method of Resolving Functions for Game Problems of Approach of Controlled Objects with Different Inertia

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Cybernetics and Systems Analysis Aims and scope

Abstract

The authors consider the problem of approach of controlled objects with different inertia in dynamic game problems on the basis of the modern version of the method of resolving functions. For such objects, it is characteristic that the Pontryagin condition is not satisfied on a certain time interval, which significantly complicates the application of the method of resolving functions to this class of dynamic game problems. A method for solving such problems is proposed, which is associated with the construction of some scalar (resolving) functions, which qualitatively characterize the course of approach of controlled objects with different inertia and the efficiency of the decisions made. The method of resolving functions allows efficient use of the modern technique of multi-valued mappings in substantiating game constructions and obtaining meaningful results on their basis. The guaranteed times of game termintion are compared for different schemes of approaching of controlled objects. An illustrative example is given.

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References

  1. A. A. Chikrii, Conflict Controlled Processes, Springer Science and Business Media, Boston–London–Dordrecht (2013).

    Google Scholar 

  2. A. A. Chikrii, “An analytical method in dynamic pursuit games,” Proc. Steklov Inst. of Mathematics, Vol. 271, 69–85 (2010).

    Article  MathSciNet  Google Scholar 

  3. A. A. Chikrii and V. K. Chikrii, “Image structure of multi valued mappings in game problems of motion control,” J. Autom. Inform. Sci., Vol. 48, No. 3, 20–35 (2016).

    Article  Google Scholar 

  4. A. A. Chikrii, “The upper and lower resolving functions in dynamic game problems,” Tr. IMM UrO RAN, Vol. 23, No. 1, 293–305 (2017). https://doi.org/10.21538/0134-4889-2017-23-1-293-305.

  5. N. N. Krasovskii and A. I. Subbotin, Positional Differential Games [in Russian], Nauka, Moscow (1974).

  6. L. S. Pontryagin, Selected Scientific Works, Vol. 2 [in Russian], Nauka, Moscow (1988).

  7. A. I. Subbotin and A. G. Chentsov, Guarantee Optimization in Control Problems [in Russian], Nauka, Moscow (1981).

  8. O. Hajek, Pursuit Games, Vol. 12, Academic Press, New York (1975).

  9. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston–Basel–Berlin (1990).

    MATH  Google Scholar 

  10. R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics, Princeton Univ. Press (1996).

    Google Scholar 

  11. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems [in Russian], Nauka, Moscow (1974).

  12. A. A. Chikrii and I. S. Rappoport, “Method of resolving functions in the theory of conflict-controlled processes,” Cybern. Syst. Analysis, Vol. 48, No. 4, 512–531 (2012).

    Article  MathSciNet  Google Scholar 

  13. A. A. Chikrii and I. I. Matychyn, “On linear conflict-controlled processes with fractional derivatives,” Trudy Instituta Matematiki i Mekhaniki URo RAN, Vol. 17, No. 2, 256–270 (2011).

    MATH  Google Scholar 

  14. M. V. Pittsyk and A. A. Chikrii, “On group pursuit problem,” J. Applied Math. and Mech., Vol. 46, No. 5, 584–589 (1982).

    Article  MathSciNet  Google Scholar 

  15. A. A. Chikrii and K. G. Dzyubenko, “Bilinear Markov processes of findig the moving objects,” Problemy Upravl. i Inform., No. 1, 92–107 (1997).

  16. S. D. Eidelman and A. A. Chikrii, “Dynamic game problems of approach for fractional-order equations,” Ukrainian Math. J., Vol. 52, No. 11, 1787–1806 (2000).

    Article  MathSciNet  Google Scholar 

  17. A. A. Chikrii and S. F. Kalashnikova, “Pursuit of a group of evaders by a single controlled object,” Cybernetics, Vol. 23, No. 4, 437–445 (1987).

    Article  Google Scholar 

  18. Yu. V. Pilipenko and A. A. Chikrii, “Oscillatory conflict-control processes,” J. of Applied Math. and Mech., Vol. 57, No. 3, 407–417(1993).

  19. A. A. Chikrii and I. I. Matychyn, “Game problems for fractional-order linear systems,” Proc. Steklov Inst. of Mathemaics, Suppl. 1, s1–s17 (2010).

  20. A. A. Chikrii and S. D. Eidelman, “Control game problems for quasilinear systems with Riemann–Liouville fractional derivatives,” Cybern. Syst. Analysis, Vol. 37, No. 6, 836–864 (2001).

    Article  MathSciNet  Google Scholar 

  21. A. A. Chikrii and S. D. Eidelman, “Game problems for fractional quasilinear systems,” J. Comp. and Math. with Applications, Pergamon, New York, Vol. 44, 835–851 (2002).

  22. A. A. Chikrii and S. D. Eidelman, “Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order,” Cybern. Syst. Analysis, Vol. 36, No. 3, 315–338 (2000).

    Article  Google Scholar 

  23. A. F. Philippov, “On some issues of the optimal control theory,” in: Vestn. MGU, Ser. Matematika, Mekhanika, Astronomiya, Fizika, Khimiya, No. 2, 25–32 (1959).

  24. E. S. Polovinkin, Elements of the Theory of Multi-Valued Mappings [in Russian], Izd. MFTI, Moscow (1982).

  25. I. S. Rappoport, “On guaranteed result in game problems of controlled objects approach,” J. Autom. Inform. Sci., Vol. 52, Iss. 3, 48–64 (2020). https://doi.org/10.1615/JAutomatInfScien.v52.i3.40.

  26. A. A. Belousov, V. V. Kuleshyn, and V. I. Vyshenskiy, “Real-time algorithm for calculation of the distance of the interrupted take-off,” J. Autom. Inform. Sci., Vol. 52, Iss. 4, 38–46 (2020). https://doi.org/10.1615/JAutomatInfScien.v52.i4.40.

  27. A. A. Chikriy, G.Ts. Chikrii, and K.Yu. Volyanskiy, “Quasilinear positional integral games of approach,”. J. Autom. Inform. Sci., Vol. 33, Iss. 10, 31–43 (2001). https://doi.org/10.1615/JAutomatInfScien.v33.i10.40.

  28. G. Ts. Chikrii, “Principle of time stretching in evolutionary games of approach,” J. Autom. Inform. Sci., Vol. 48, Iss. 5, 12–26 (2016). https://doi.org/10.1615/JAutomatInfScien.v48.i5.20.

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Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, Academy of Sciences of Ukraine, Kyiv, Ukraine

    I. S. Rappoport

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  1. I. S. Rappoport
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Correspondence to I. S. Rappoport.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 2, March–April, 2021, pp. 147–166.

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Rappoport, I.S. Method of Resolving Functions for Game Problems of Approach of Controlled Objects with Different Inertia. Cybern Syst Anal 57, 296–312 (2021). https://doi.org/10.1007/s10559-021-00355-9

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  • DOI: https://doi.org/10.1007/s10559-021-00355-9

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