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On Continuous Random Processes with Fuzzy States

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Abstract

Continuous random processes with fuzzy states are studied. The properties of their numerical characteristics (expectations and correlation functions) corresponding to those of numerical random processes are established. The results obtained are based on the properties of fuzzy random variables. Applications to the problem of transforming a random signal with fuzzy states by a linear dynamic system are considered.

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REFERENCES

  1. Averkin, A.N., Nechetkie mnozhestva v modelyakh upravleniya i iskusstvennogo intellekta (Fuzzy Sets in Models of Control and Artificial Intelligence), Moscow: Nauka, 1986.

  2. Buckley, J.J., Eslami, E., and Feuring, T., Fuzzy Mathematics in Economics and Engineering, Heidelberg–New York: Physica-Verl., 2002.

    Book  MATH  Google Scholar 

  3. Pegat, A., Nechetkoe modelirovanie i upravlenie (Fuzzy Modeling and Control), Moscow: BINOM. Laboratoriya Znanii, 2015.

  4. Mochalov, I.A., Khrisat, M.S., and Shihab Eddin, M.Ya., Fuzzy Differential Equations in Control. Part II, Information Technologies, 2015, vol. 21, no. 4, pp. 243–250.

    Google Scholar 

  5. Demenkov, N.P., Mikrin, E.A., and Mochalov, I.A., Fuzzy Optimal Control of Linear Systems. Part 1. Positional Control, Information Technologies, 2019, vol. 25, no. 5, pp. 259–270.

    Google Scholar 

  6. Venttsel’, E.S. and Ovcharov, L.A., Teoriya sluchainykh protsessov i ikh inzhenernye prilozheniya (Theory of Random Processes and Their Engineering Applications), Moscow: Knorus, 2016.

  7. Puri, M.L. and Ralesku, D.A., Fuzzy Random Variables, Journal of Mathematical Analysis and Applications, 1986, vol. 114, pp. 409–422.

    Article  MathSciNet  Google Scholar 

  8. Nguyen, H.T. and Wu, B., Fundamentals of Statistics with Fuzzy Data, Berlin–Heidelberg: Springer-Verlag, 2006.

    Book  MATH  Google Scholar 

  9. Shvedov, A.A., Estimating the Means and the Covariances of Fuzzy Random Variables, Applied Econometrics, 2016, vol. 42, pp. 121–138.

    Google Scholar 

  10. Puri, M.L. and Ralescu, D.A., Differential of Fuzzy Functions, J. Math. Anal. Appl., 1983, vol. 91, pp. 552–558.

    Article  MathSciNet  MATH  Google Scholar 

  11. Seikkala, S., On the Fuzzy Initial Value Problem, Fuzzy Sets Syst., 1987, vol. 24, no. 3, pp. 319–330.

    Article  MathSciNet  MATH  Google Scholar 

  12. Aumann, R.J., Integrals of Set-Valued Functions, J. Math. Anal. Appl., 1987, no. 3, pp. 1–12.

  13. Hukuhara, M., Integration des applications mesurables dont la valeur est un compact convexe, Func. Ekvacioj., 1967, no. 11, pp. 205–223.

  14. Demenkov, N.P., Mikrin, E.A., and Mochalov, I.A., Markov and Semi-Markov Processes with Fuzzy States. Part 1. Markov Processes, Information Technologies, 2020, vol. 26, no. 6, pp. 323–334.

    Google Scholar 

  15. Demenkov, N.P., Mikrin, E.A., and Mochalov, I.A., Markov and Semi-Markov Processes with Fuzzy States. Part 2. Semi-Markov Processes, Information Technologies, 2020, vol. 26, no. 7, pp. 387–393.

    Google Scholar 

  16. Diamond, P. and Kloeden, P., Metric Spaces of Fuzzy Sets, Fuzzy Sets Syst., 1990, vol. 35, no. 2, pp. 241–249.

    Article  MathSciNet  MATH  Google Scholar 

  17. Khatskevich, V.L., On Some Properties of Fuzzy Expectations and Nonlinear Fuzzy Expectations of Fuzzy-Random Variables, Russian Math. (Iz. VUZ), 2022, vol. 66, no. 11, pp. 86–96.

  18. Khatskevich, V.L., Extremal Properties of Means of Fuzzy Random Variables, in Itogi nauki i tekhniki. Sovre-mennaya matematika i prilozheniya. Tematicheskie obzory (Results of Science and Engineering. Contemporary Mathematics and Applications. Thematic Surveys), Moscow: VINITI, Moscow, 2022, vol. 204, pp. 160–169.

  19. Dubois, D. and Prade, H., The Mean Value of Fuzzy Number, Fuzzy Sets and Syst., 1987, vol. 24, no. 3, pp. 279–300.

    Article  MathSciNet  MATH  Google Scholar 

  20. Feng, Y., Hu, L., and Shu, H., The Variance and Covariance of Fuzzy Random Variables, Fuzzy Syst., 2001, vol. 120, no. 3, pp. 487–497.

    Article  MathSciNet  MATH  Google Scholar 

  21. Rozanov, Yu.A., Teoriya veroyatnostei, sluchainye protsessy i matematicheskaya statistika (Probability Theory, Random Processes, and Mathematical Statistics), Moscow: Nauka, 1985.

  22. Kaleva, O., Fuzzy Differential Equations, Fuzzy Sets and Syst., 1987, vol. 24, no. 3, pp. 301–317.

    Article  MathSciNet  MATH  Google Scholar 

  23. Park, J.Y. and Han, H., Existence and Uniqueness Theorem for a Solution of Fuzzy Differential Equations, Int. J. Math. Mathem. Sci., 1999, no. 22(2), pp. 271–280.

  24. Ahmad, L., Farooq, M., and Abdullah, S., Solving nth Order Fuzzy Differential Equation by Fuzzy Laplace Transform, Ind. J. Pure Appl. Math., 2014, no. 2, pp. 1–20.

  25. Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions of Differential Equations in a Banach Space), Moscow: Nauka, 1970.

  26. Krasnosel’skii, M.A., Burd, V.Sh., and Kolesov, Yu.S., Nelineinye pochti periodicheskie kolebaniya (Nonlinear Almost Periodic Oscillations), Moscow: Nauka, 1970.

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  1. Military Training and Research Center of the Air Force, Air Force Academy named after N.E. Zhukovsky and Yu.A. Gagarin, Voronezh, Russia

    V. L. Khatskevich

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  1. V. L. Khatskevich
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Correspondence to V. L. Khatskevich.

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This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board

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Khatskevich, V.L. On Continuous Random Processes with Fuzzy States. Autom Remote Control 84, 687–698 (2023). https://doi.org/10.1134/S0005117923070081

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  • DOI: https://doi.org/10.1134/S0005117923070081

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