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UDC 519.837
L.V. Baranovska1


1 Institute of Applied Systems Analysis of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

 lesia@baranovsky.org

PURSUIT PROBLEM FOR FRACTIONAL DIFFERENTIAL SYSTEMS WITH PURE DELAY

Abstract. In this paper, we consider the pursuit problem for linear fractional differential systems with pure delay. A scheme of the method of resolving functions for such conflict-controlled processes has been developed using the latest representation of the Cauchy formula. Sufficient conditions for the completion of the game and the method of practical finding of resolving functions are formulated.

Keywords: conflict-controlled process, differential games, fractional differential games, pursuit games, game theory.


FULL TEXT

REFERENCES

  1. Isaacs R. Differential games. New York: John Wiley, 1965. 480 p.

  2. Krasovsky N.N. Game taskproblems on the meeting of movements [in Russian]. Moscow: Nauka, 1970. 420 p.

  3. Krasovsky N.N., Subbotin A.I. Positional differential games. [in Russian]. Moscow: Nauka, 1974. 455 p.

  4. Pontryagin L.S. Selected scientific works [in Russian]. Vol. 2. Moscow: Nauka, 1988. 575 p.

  5. Nikolsky M.S. The first direct method of L.S. Pontryagin in differential games [in Russian]. Moscow: MSU Publishing House, 1984. 65 p.

  6. Chikrii A.A. Conflict controlled processes. Boston; London; Dordrecht: Springer Science and Business Media, 2013. 424 p.

  7. Baranovskaya L.V., Chikrij A.A., Chikrij Al.A. Inverse Minkowski functional in a nonstationary problem of group pursuit. Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya. 1997. N 1. P. 109–114.

  8. Baranovskaya L.V., Chikrii A.A., Chikrii Al.A. Inverse Minkowski functional in a nonstationary problem of group pursuit. Journal of Computer and Systems Sciences International. 1997. Vol. 36, N 1. P. 101–106.

  9. Samko S.G., Kilbas A.A. and Marychev O.I. Fractional order integrals and derivatives, and some its applications [in Russian]. Minsk: Nauka i tekhnika, 1987. 688 p.

  10. Chikrij A.A. and Eidelman S.D. Generalized Mittag–Leffler matrix functions in game problems for evolutionary equations of fractional order. Cybernetics and Systems Analysis. 2000. Vol. 36, N 3. P. 315–338. https://doi.org/10.1007/BF02732983.

  11. Chikrii A.A., Eidelman S.D. Control game problems for quasilinear systems with Riemann–Liouville fractional derivatives. Cybernetics and Systems Analysis. 2001. Vol. 37, N 6. P. 836–864.

  12. Dzhrbashyan M.M. Integral transformations and function representations in complex plane. Moscow: Nauka, 1966. 671 p.

  13. Eidelman S.D., Chikrii A.A. Game problems for systems with Volterra evolution. Fractal games. J. Game Theory and Appl. 2000. N 6. P. 21–58.

  14. Chikrii A., Matychyn I. Riemann–Liouville, Сaputo, and sequential fractional derivatives in differential games. Annals of the International Society of Dynamic Games. 2011. Vol. 11. P. 61–81. https://doi.org/10.1007/978-0-8176-8089-3_4.

  15. Khusainov D.Ya., Shuklin G.V. Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina Math. Ser. 2003. Vol. 17. P. 101–108.

  16. PospЗil M. Representation of solutions of systems of linear differential equations with multiple delays and nonpermutable variable coefficients. Mathematical Modeling and Analysis. 2020. Vol. 25, N 2. P. 303–322.

  17. PospЗil M., Jaro F. On the representation of solutions of delayed differential equations via Laplace transform. Electronic Journal of Qualitative Theory of Differential Equations. 2016. N 117. P. 1–13. https://doi.org/10.14232/ejqtde.2016.1.117.

  18. Baranovska L.V. Group pursuit differential games with pure time-lag. Understanding Complex Systems. 2021. P. 475–488. https://doi.org/10.1007/978-3-030-50302-4_23.

  19. Baranovska L.V. Quasi-linear differential-difference game of approach. Understanding Complex Systems. 2019. P. 505–524. https://doi.org/10.1007/978-3-319-96755-4_26.

  20. Baranovska L.V. Pursuit differential-difference games with pure time-lag. Discrete and Continuous Dynamical Systems — B. 2019. Vol. 24, N 3. P. 1021–1031. https://doi.org/10.3934/dcdsb.2019004.

  21. Baranovska L.V. On quasilinear differential-difference games of approach. J. of Automation and Information Sciences. 2017. Vol. 49, N 8. P. 53–67. https://doi.org/10.1615/JAutomatInfScien.v49.i8.40.

  22. Baranovska L., Hyriavets D., Dovzhanytsia K., Mukhin V. Visualization of pursuit differential game on a plane. CEUR Workshop Proceedings, 2021. P. 99–113. http://ceur-ws.org/ Vol-2859/paper9.pdf.

  23. Baranovska G.G., Baranovska L.V. Group pursuit in quasilinear differential-difference games. J. of Automation and Information Sciences. 1997. Vol. 29, N 1. P. 55–62. https://doi.org/10.1615/JAutomatInfScien.v29.i1.70.

  24. Baranovskaya L.V. A method of resolving functions for one class of pursuit problems. Eastern-European Journal of Enterprise Technologies. 2015. Vol. 2, N 4. P. 4–8. https://doi.org/10.15587/1729-4061.2015.39355.

  25. Baranovska L.V. Method of resolving functions for the differential-difference pursuit game for different-inertia objects. Studies in Systems, Decision and Control. 2016. Vol. 69. P. 159–176. https://doi.org/10.1007/978-3-319-40673-2_7.

  26. Chikrii A.A., Chikrii G.T. Matrix resolving functions in game problems of dynamics. Proceedings of the Steklov Institute of Mathematics. 2015. Vol. 291. P. 56–65. https://doi.10.1134/ S0081543815090047.

  27. Chikrii G.T. Using the effect of information delay in differential pursuit games. Cybernetics and Systems Analysis. 2007. Vol. 43, N 2. P. 233–245. https://doi.10.1007/s10559-007-0042-x

  28. Chikrii G.T. Principle of time stretching in evolutionary games of approach. Journal of Automation and Information Sciences. 2016. Vol. 48, N 5. P. 12–26. https://doi.10.1615/JAutomatInfScien.v48.i5.20.

  29. Chikrii G.T. On one problem of approach for damped oscillations. Journal of Automation and Information Sciences. 2009. Vol. 41, N 10. P. 1–9. https://doi.10.1615/JAutomatInfScien.v41.i10.10.

  30. Chikriy A.A., Baranovskaya L.V., Chikriy Al.A. An approach game problem under the failure of controlling devices. J. of Automation and Information Sciences. 2000. Vol. 32, N 5. P. 1–8. https://doi.org/10.1615/JAutomatInfScien.v32.i5.10.

  31. Chikrij A.A., Baranovskaya L.V., Chikrij Al.A. The game problem of approach under the condition of failure of controlling devices. Problemy Upravleniya i Informatiki (Avtomatika). 1997. N 4. P. 5–13. https://doi.org/10.1615/JAutomatInfScien.v32.i5.10.

  32. Baranovskaya L.V., Chikrii Al.A. Game problems for a class of hereditary systems. J. of Automation and Information Sciences. 1997. Vol. 29, N 2. P. 87–97. https://doi.org/10.1615/JAutomatInfScien.v29.i2-3.120.

  33. Li M., Wang J.R. Finite time stability of fractional delay differential equations. Appl. Math. Lett. 2017. Vol. 64. P. 170–176.

  34. Li M., Wang J.R. Exploring delayed Mittag–Leffler type matrix functions to study finite time stability of fractional delay differential equations. Applied Mathematics and Computation. 2018. Vol. 324. P. 254–265.

  35. Liang C., Wang J.R., O’Regan D. Representation of solution of a fractional linear system with pure delay. Appl. Math. Lett. 2018. Vol. 77. P. 72–78.

  36. Mahmudov N.I. Delayed perturbation of Mittag–Leffler functions and their applications to fractional linear delay differential equations. Math. Meth. Appl. Sci. 2019. Vol. 42. P. 5489–5497.

  37. Huseynov I.T., Mahmudov N.I. Delayed analogue of three-parameter Mittag–Leffler functions and their applications to Caputo-type fractional time delay differential equations. Math. Meth. Appl. Sci., 2020. P. 1–25. https://doi.org/10.1002/mma.6761.

  38. Liu L., Dong Q., Li G. Exact solutions and Hyers–Ulam stability for fractional oscillation equations with pure delay. Appl. Math. Lett. 2021. Vol. 112. P. 1–7. https://doi.org/10.1016/j.aml.2020.106666.

  39. Ahmed M. Elshenhab, Xing Tao Wang. Representation of solution for linear fractional systems with pure delay and multiple delays. Mathematical Methods in the Applied Sciences. 2021. Vol. 44. P. 12835–12850. https://doi.org/10.1002/mma.7585.

  40. Hajek O. Pursuit games. New York: Academic Press. 1975. Vol. 12. 266 p.

  41. Aubin J.-P., Frankowska H. Set-valued analysis. Boston; Basel; Berlin: Birkhauser, 1990. 461 p.




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