Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 3, pp 477–494 | Cite as

Averaging of Fuzzy Systems

  • N. A. Perestyuk
  • N. V. Skripnik
Article
  • 15 Downloads

We develop the ideas of the method of averaging for some classes of fuzzy systems (fuzzy differential equations with delay, fuzzy differerntial equations with pulsed action, fuzzy integral equations, fuzzy differential inclusions, and differential inclusions with fuzzy right-hand sides without and with pulsed action).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. A. Baidosov, “Differential inclusions with fuzzy right-hand sides,” Dokl. Akad. Nauk SSSR, 309, No. 4, 781–783 (1989).MathSciNetGoogle Scholar
  2. 2.
    V. A. Baidosov, “Fuzzy differential inclusions,” Prikl. Mat. Mekh., 54, Issue 1, 12–17 (1990).MathSciNetGoogle Scholar
  3. 3.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).zbMATHGoogle Scholar
  4. 4.
    E. A. Grebennikov, Averaging Method in Applied Problems [in Russian], Nauka, Moscow (1986).Google Scholar
  5. 5.
    O. D. Kichmarenko and N. V. Skrypnyk, “Fuzzy differential equations with delay,” Visn. Kyiv. Nats. Univ., Mat. Mekh., Issue 19, 18–23 (2008).Google Scholar
  6. 6.
    O. D. Kichmarenko and N. V. Skripnik, “Averaging of fuzzy differential equations with delay,” Nelin. Kolyv., 11, No. 3, 316–328 (2008); English translation: Nonlin. Oscillat., 11, No. 3, 331–344 (2008).Google Scholar
  7. 7.
    T. A. Komleva, A.V. Plotnikov, and L. I. Plotnikova, “Averaging of fuzzy differential equations,” Tr. Odes. Politekh. Univ., Issue 1 (27), 185–190 (2007).Google Scholar
  8. 8.
    T. A. Komleva, A.V. Plotnikov, and L. I. Plotnikova, “Averaging of differential inclusions with fuzzy right-hand sides on a finite interval,” Tr. Odes. Politekh. Univ., Issue 1(33)–2(34), 192–196 (2009).Google Scholar
  9. 9.
    T. A. Komleva, L. I. Plotnikova, A.V. Plotnikov, and N. V. Skripnik, “Averaging in fuzzy controlled systems,” Nelin. Kolyv., 14, No. 3, 325–332 (2011); English translation: Nonlin. Oscillat., 14, No. 3, 342–349 (2012).Google Scholar
  10. 10.
    M. A. Krasnosel’skii and S. G. Krein, “On the principle of averaging in nonlinear mechanics,” Usp. Mat. Nauk, 10, No. 3(65), 147–152 (1955).Google Scholar
  11. 11.
    N. M. Krylov and N. N. Bogolyubov, Introduction to Nonlinear Mechanics [in Russian], Akad. Nauk Ukr. SSR, Kiev (1937).Google Scholar
  12. 12.
    Yu. A. Mitropol’skii, Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).Google Scholar
  13. 13.
    Yu. A. Mitropol’skii and G. L. Khoma, Mathematical Justification of the Asymptotic Methods of Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1983).Google Scholar
  14. 14.
    N. N. Moiseev, Asymptotic Methods in Nonlinear Mechanics [in Russian], Nauka, Moscow (1981).Google Scholar
  15. 15.
    N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Impulsive Differential Equations with Multivalued and Discontinuous Right-Hand Sides [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).zbMATHGoogle Scholar
  16. 16.
    A.V. Plotnikov and N. V. Skripnik, Differential Equations with “Clear” and Fuzzy Multivalued Right-Hand Side. Asymptotic Methods [in Russian], AstroPrint, Odessa (2009).Google Scholar
  17. 17.
    A.V. Plotnikov and T. A. Komleva, “Averaging of fuzzy differential equations on a finite interval,” Nelin. Kolyv., 14, No. 4, 516–527 (2011); English translation: Nonlin. Oscillat., 14, No. 4, 547–559 (2012).Google Scholar
  18. 18.
    A.V. Plotnikov, “Averaging of fuzzy controlled differential inclusions with terminal quality criterion,” Nelin. Kolyv., 16, No. 1, 105–110 (2013); English translation: J. Math. Sci., 197, No. 1, 108–113 (2014).Google Scholar
  19. 19.
    A.V. Plotnikov, “A procedure of complete averaging for fuzzy differential inclusions on a finite segment,” Ukr. Mat. Zh., 67, No. 3, 366–374 (2015); English translation: Ukr. Math. J., 67, No. 3, 421–430 (2015).Google Scholar
  20. 20.
    V. A. Plotnikov, Averaging Method in Control Problems [in Russian], Lybid’, Kiev (1992).Google Scholar
  21. 21.
    V. A. Plotnikov, A.V. Plotnikov, and A. N. Vityuk, Differential Equations with Multivalued Right-Hand Side. Asymptotic Methods [in Russian], AstroPrint, Odessa (1999).Google Scholar
  22. 22.
    L. I. Plotnikova and N. V. Skripnik, “Averaging of linear fuzzy differential equations with delay,” Tr. Odes. Politekh. Univ., Issue 1 (29), 224–230 (2008).Google Scholar
  23. 23.
    O. V. Poletaeva and N. V. Skripnik, “Averaging of linear fuzzy differential equations with delay,” Visn. Odes. Nats. Univ., Mat. Mekh., 17, Issue 3 (5), 27–37 (2012).Google Scholar
  24. 24.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).Google Scholar
  25. 25.
    M. S. Sasonkina and N. V. Skripnik, “Averaging of fuzzy differential equations,” Visn. Odes. Nats. Univ., Mat. Mekh., 15, Issue 19, 109–118 (2010).Google Scholar
  26. 26.
    N. V. Skripnik, “Existence of classical solutions of fuzzy differential inclusions,” Ukr. Mat. Vestn., 5, 244–257 (2008).MathSciNetGoogle Scholar
  27. 27.
    N. V. Skripnik, “Averaging of impulsive fuzzy differential equations,” Nelin. Kolyv., 11, No. 4, 530–540 (2008); English translation: Nonlin. Oscillat., 11, No. 4, 559–570 (2008).Google Scholar
  28. 28.
    N. V. Skripnik, “The Krasnosel’skii–Krein theorem for fuzzy differential equations,” Visn. Odes. Nats. Univ., Mat. Mekh., 14, Issue 20, 115–122 (2009).Google Scholar
  29. 29.
    N. V. Skripnik, “Quasisolutions of fuzzy differential inclusions,” Nelin. Kolyv., 14, No. 4, 528–535 (2011); English translation: Nonlin. Oscillat., 14, No. 4, 560–567 (2012).Google Scholar
  30. 30.
    N. V. Skripnik, “Averaging of impulsive differential inclusions with fuzzy right-hand sides,” Ukr. Mat. Zh., 66, No. 11, 1563–1577 (2014); English translation: Ukr. Math. J., 66, No. 11, 1756–1772 (2015).Google Scholar
  31. 31.
    N. V. Skripnik, “Scheme of partial averaging for impulsive differential inclusions with fuzzy right-hand sides,” Mat. Stud., 43, No. 2, 129–139 (2015).MathSciNetzbMATHGoogle Scholar
  32. 32.
    N. V. Skripnik, “Averaging of fuzzy integral equations with constant delay,” Doslid. Mat. Mekh., 21, Issue 2(28), 55–63 (2016).Google Scholar
  33. 33.
    A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations [in Russian], Fan, Tashkent (1974).Google Scholar
  34. 34.
    F. L. Chernous’ko, L. D. Akulenko, and B. N. Sokolov, Control over Oscillations [in Russian], Nauka, Moscow (1980).zbMATHGoogle Scholar
  35. 35.
    A. Allaoui, S. Melliani, Y. Allaoui, and L. S. Chadli, “Averaging of intuitionistic fuzzy differential equations,” Notes Intuit. Fuzzy Sets, 23, No. 2, 44–54 (2017).zbMATHGoogle Scholar
  36. 36.
    J.-P. Aubin, “Fuzzy differential inclusions,” Probl. Control Inform. Theory, 19, No. 1, 55–67 (1990).MathSciNetzbMATHGoogle Scholar
  37. 37.
    S. Chakraverty, S. Tapaswini, and D. Behera, Fuzzy Differential Equations and Applications for Engineers and Scientists, CRC Press (2017).Google Scholar
  38. 38.
    R. Gama and G. Smirnov, “Stability and optimality of solutions to differential inclusions via averaging method,” Set-Valued Var. Anal., 22, No. 2, 349–374 (2014).MathSciNetCrossRefGoogle Scholar
  39. 39.
    E. Hullermeier, “An approach to modelling and simulation of uncertain dynamical systems,” Internat. J. Uncertain., Fuzziness Knowledge-Based Systems, No. 7, 117–137 (1997).MathSciNetCrossRefGoogle Scholar
  40. 40.
    O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets Systems, 24, No. 3, 301–317 (1987).MathSciNetCrossRefGoogle Scholar
  41. 41.
    O. D. Kichmarenko and N. V. Skripnik, “One scheme of averaging of fuzzy differential equations with maxima,” J. Adv. Res. Appl. Math., 3, No. 1, 94–103 (2011).MathSciNetCrossRefGoogle Scholar
  42. 42.
    O. D. Kichmarenko and N. V. Skripnik, “The partial averaging of fuzzy differential equations with maxima,” Adv. Dyn. Syst. Appl., 6, No. 2, 199–207 (2011).MathSciNetGoogle Scholar
  43. 43.
    S. Klimchuk, A. Plotnikov, and N. Skripnik, “Overview of V. A. Plotnikov’s research on averaging of differential inclusions,” Physica D, 241, No. 22, 1932–1947 (2012).MathSciNetCrossRefGoogle Scholar
  44. 44.
    T. A. Komleva, “The full averaging of linear fuzzy differential equations with 2π-periodic right-hand side,” J. Adv. Res. Dyn. Control Systems, 3, No. 1, 12–25 (2011).MathSciNetGoogle Scholar
  45. 45.
    T. A. Komleva, L. I. Plotnikova, and A.V. Plotnikov, “The averaging of fuzzy integrodifferential equations on a finite interval,” J. Adv. Res. Appl. Math., 6, No. 3, 47–59 (2014).MathSciNetCrossRefGoogle Scholar
  46. 46.
    M. Lakrib, R. Guen, and A. Bourada, “Averaging for fuzzy differential equations,” Surv. Math. Appl., No. 9, 93–104 (2014).Google Scholar
  47. 47.
    M. Lakrib, R. Guen, and A. Bourada, “An averaging result for fuzzy differential equations with a small parameter,” in: Annual Review of Chaos Theory and Bifurcations and Dynamical Systems, 5 (2015), pp. 1–9.Google Scholar
  48. 48.
    V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London (2003).CrossRefGoogle Scholar
  49. 49.
    N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities, Walter De Gruyter, Berlin (2011).CrossRefGoogle Scholar
  50. 50.
    A.V. Plotnikov and N. V. Skripnik, “The generalized solutions of the fuzzy differential inclusions,” Int. J. Pure Appl. Math., 56, No. 2, 165–172 (2009).MathSciNetzbMATHGoogle Scholar
  51. 51.
    A.V. Plotnikov and T. A. Komleva, “The full averaging of linear fuzzy differential equations,” J. Adv. Res. Different. Equat., 2, No. 3, 21–34 (2010).Google Scholar
  52. 52.
    A.V. Plotnikov, T. A. Komleva, and L. I. Plotnikova, “The partial averaging of differential inclusions with fuzzy right-hand side,” J. Adv. Res. Dynam. Control Syst., 2, No. 2, 26–34 (2010).MathSciNetGoogle Scholar
  53. 53.
    A.V. Plotnikov, T. A. Komleva, and L. I. Plotnikova, “On the averaging of differential inclusions with fuzzy right-hand side when the average of the right-hand side is absent,” Iran. J. Optimiz., 2, No. 3, 506–517 (2010).Google Scholar
  54. 54.
    A.V. Plotnikov, “Averaging of fuzzy integrodifferential inclusions,” Int. J. Control Sci. Eng., 1, No. 1, 8–14 (2011).CrossRefGoogle Scholar
  55. 55.
    A.V. Plotnikov, “The averaging of control linear fuzzy differential equations,” J. Adv. Res. Appl. Math., 3, No. 3, 1–20 (2011).MathSciNetCrossRefGoogle Scholar
  56. 56.
    A.V. Plotnikov and T. A. Komleva, “The averaging of control linear fuzzy 2π-periodic differential equations,” DCDIS, Ser. B Appl. Algorithms, 18, No. 6, 833–847 (2011).MathSciNetzbMATHGoogle Scholar
  57. 57.
    A.V. Plotnikov and T. A. Komleva, “Averaging of the fuzzy differential equations,” J. Uncertain Systems, 6, No. 1, 30–37 (2012).MathSciNetzbMATHGoogle Scholar
  58. 58.
    A.V. Plotnikov and T. A. Komleva, “The full averaging of fuzzy integrodifferential equations,” J. Adv. Res. Dyn. Control Systems, 4, No. 1, 48–59 (2012).MathSciNetGoogle Scholar
  59. 59.
    A.V. Plotnikov and T. A. Komleva, “Full averaging of control fuzzy integrodifferential inclusions with terminal criterion of quality,” Int. J. Control Sci. Eng., 3, No. 2, 68–72 (2013).Google Scholar
  60. 60.
    A.V. Plotnikov and T. A. Komleva, “The partial averaging of fuzzy differential inclusions on finite interval,” Int. J. Different. Equat., 2014 (2014), Article ID 307941.MathSciNetCrossRefGoogle Scholar
  61. 61.
    A.V. Plotnikov, “The averaging of fuzzy linear differential inclusions on finite interval,” DCDIS, Ser. B Appl. Algorithms, 23, No. 1, 1–9 (2016).MathSciNetzbMATHGoogle Scholar
  62. 62.
    M. L. Puri and D. A. Ralescu, “Differential of fuzzy functions,” J. Math. Anal. Appl., 91, 552–558 (1983).MathSciNetCrossRefGoogle Scholar
  63. 63.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).CrossRefGoogle Scholar
  64. 64.
    N. V. Skripnik, “The full averaging of fuzzy differential inclusions,” Iran. J. Optim., 1, No. 3, 302–317 (2009).Google Scholar
  65. 65.
    N. V. Skripnik, “The full averaging of fuzzy impulsive differential inclusions,” Surv. Math. Appl., 5, 247–263 (2010).MathSciNetzbMATHGoogle Scholar
  66. 66.
    N. V. Skripnik, “Linear fuzzy differential equations,” J. Uncertain Systems, 5, No. 4, 305–313 (2011).MathSciNetGoogle Scholar
  67. 67.
    N. V. Skripnik, “The partial averaging of fuzzy differential inclusions,” J. Adv. Res. Different. Equat., 3, No. 1, 52–66 (2011).MathSciNetzbMATHGoogle Scholar
  68. 68.
    N. V. Skripnik, “The partial averaging of fuzzy impulsive differential inclusions,” Different. Integral Equat., 24, No. 7-8, 743–758 (2011).MathSciNetzbMATHGoogle Scholar
  69. 69.
    N. V. Skripnik, “Averaging of fuzzy differential inclusions when the average of the right-hand side is absent,” DCDIS, Ser. A Math. Anal., 19, No. 2, 187–198 (2012).MathSciNetzbMATHGoogle Scholar
  70. 70.
    N. V. Skripnik, “Averaging of fuzzy integral equations,” Appl. Math. Phys., 1, No. 3, 39–44 (2013).Google Scholar
  71. 71.
    N. Skripnik, “Step scheme of averaging method for impulsive differential inclusions with fuzzy right-hand side,” Contemp. Methods Math. Phys. Gravitation, 1, No. 1, 9–26 (2015).Google Scholar
  72. 72.
    N. Skripnik, “Averaging of impulsive differential inclusions with fuzzy right-hand side when the average is absent,” Asian-Eur. J. Math., 8, No. 4, 1550086-1–1550086-12 (2015).MathSciNetCrossRefGoogle Scholar
  73. 73.
    N. Skripnik, “Averaging of fuzzy integral equations,” Discrete Contin. Dyn. Systems, Ser. B, 22, No. 5, 1999–2010 (2017).MathSciNetCrossRefGoogle Scholar
  74. 74.
    L. Zadeh, “Fuzzy sets,” Inform. Control, No. 8, 338–353 (1965).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • N. A. Perestyuk
    • 1
  • N. V. Skripnik
    • 2
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine
  2. 2.Mechnikov Odessa National UniversityOdessaUkraine

Personalised recommendations