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Regularization scheme for uncertain fuzzy differential equations: Analysis of solutions

  • Received: 02 February 2023 Revised: 21 April 2023 Accepted: 26 April 2023 Published: 11 May 2023
  • In this paper a regularization scheme for a family of uncertain fuzzy systems of differential equations with respect to the uncertain parameters is introduced. Important fundamental properties of the solutions are discussed on the basis of the established technique and new results are proposed. More precisely, existence and uniqueness criteria of solutions of the regularized equations are established. In addition, an estimation is proposed for the distance between two solutions. Our results contribute to the progress in the area of the theory of fuzzy systems of differential equations and extend the existing results to the uncertain case. The proposed results also open the horizon for generalizations including equations with delays and some modifications of the Lyapunov theory. In addition, the opportunities for applications of such results to the design of efficient fuzzy controllers are numerous.

    Citation: Anatoliy Martynyuk, Gani Stamov, Ivanka Stamova, Yulya Martynyuk–Chernienko. Regularization scheme for uncertain fuzzy differential equations: Analysis of solutions[J]. Electronic Research Archive, 2023, 31(7): 3832-3847. doi: 10.3934/era.2023195

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  • In this paper a regularization scheme for a family of uncertain fuzzy systems of differential equations with respect to the uncertain parameters is introduced. Important fundamental properties of the solutions are discussed on the basis of the established technique and new results are proposed. More precisely, existence and uniqueness criteria of solutions of the regularized equations are established. In addition, an estimation is proposed for the distance between two solutions. Our results contribute to the progress in the area of the theory of fuzzy systems of differential equations and extend the existing results to the uncertain case. The proposed results also open the horizon for generalizations including equations with delays and some modifications of the Lyapunov theory. In addition, the opportunities for applications of such results to the design of efficient fuzzy controllers are numerous.



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    [1] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
    [2] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, New York, 1992. Available from: https://www.degruyter.com/document/doi/10.1515/9783110874228/html.
    [3] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994. https://doi.org/10.1142/2326
    [4] C. V. Negoita, D. A. Ralescu, Applications of Fuzzy Sets to System Analysis, Springer, Basel, 1975. Available from: https://link.springer.com/book/10.1007/978-3-0348-5921-9.
    [5] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552–558. https://doi.org/10.1016/0022-247X(83)90169-5 doi: 10.1016/0022-247X(83)90169-5
    [6] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1–12. https://doi.org/10.1016/0022-247X(65)90049-1 doi: 10.1016/0022-247X(65)90049-1
    [7] J. J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets Syst., 110 (2000), 43–54. https://doi.org/10.1016/S0165-0114(98)00141-9 doi: 10.1016/S0165-0114(98)00141-9
    [8] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets Syst., 35 (1990), 389–396. https://doi.org/10.1016/0165-0114(90)90010-4 doi: 10.1016/0165-0114(90)90010-4
    [9] O. Kaleva, The Peano theorem for fuzzy differential equations revisited, Fuzzy Sets Syst., 98 (1998), 147–148. https://doi.org/10.1016/S0165-0114(97)00415-6 doi: 10.1016/S0165-0114(97)00415-6
    [10] A. Kandel, W. J. Byatt, Fuzzy processes, Fuzzy Sets Syst., 4 (1980), 117–152. https://doi.org/10.1016/0165-0114(80)90032-9 doi: 10.1016/0165-0114(80)90032-9
    [11] V. Lakshmikantham, R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, CRC Press, London, 2003. https://doi.org/10.1201/9780203011386
    [12] A. A. Martynyuk, Y. A. Martynyuk-Chernienko, Analysis of the set of trajectories of fuzzy equations of perturbed motion, Ukr. Math. J., 66 (2015), 1512–1527. Available from: https://umj.imath.kiev.ua/index.php/umj/article/view/2242.
    [13] D. Vorobiev, S. Seikkala, Towards the theory of fuzzy differential equations, Fuzzy Sets Syst., 125 (2002), 231–237. https://doi.org/10.1016/S0165-0114(00)00131-7 doi: 10.1016/S0165-0114(00)00131-7
    [14] H. J. Zimmermann, Fuzzy Set Theory–and Its Applications, Springer, New York, 2001. Available from: https://link.springer.com/book/10.1007/978-94-010-0646-0.
    [15] Z. Cai, L. Huang, Z. Wang, X. Pan, L. Zhang, Fixed-time stabilization of IT2 T-S fuzzy control systems with discontinuous interconnections: indefinite derivative Lyapunov method, J. Franklin Inst., 359 (2022), 2564–2592. https://doi.org/10.1016/j.jfranklin.2022.02.002 doi: 10.1016/j.jfranklin.2022.02.002
    [16] X. Jian, Z. Wang, A. Xin, Y. Chen, S. Xie, An improved finite-time stabilization of discontinuous non-autonomous IT2 T-S fuzzy interconnected complex-valued systems: a fuzzy switching state-feedback control method, Electron. Res. Arch., 31 (2023), 273–298. https://doi.org/10.3934/era.2023014 doi: 10.3934/era.2023014
    [17] Z. Jin, J. Wu, On the Ulam stability of fuzzy differential equations, AIMS Math., 5 (2020), 6006–6019. https://doi.org/10.3934/math.2020384 doi: 10.3934/math.2020384
    [18] M. Mazandarani, L. Xiu, A review on fuzzy differential equations, IEEE Access, 9 (2021), 62195–62211. https://doi.org/10.1109/ACCESS.2021.3074245 doi: 10.1109/ACCESS.2021.3074245
    [19] Y. Wu, H. Lan, C. Liu, On implicit coupled systems of fuzzy fractional delay differential equations with triangular fuzzy functions, AIMS Math., 6 (2021), 3741–3760. https://doi.org/10.3934/math.2021222 doi: 10.3934/math.2021222
    [20] R. Baranitha, R. Rakkiyappan, X. Li, T-S fuzzy model based single-master multi-slave teleoperation systems with decentralized communication structure and varying time delays, IEEE Trans. Fuzzy Syst., 28 (2019), 3406–3417. https://doi.org/10.1109/TFUZZ.2019.2952789 doi: 10.1109/TFUZZ.2019.2952789
    [21] M. Cui, M. Pan, J. Wang, P. Li, A parameterized level set method for structural topology optimization based on reaction diffusion equation and fuzzy PID control algorithm, Electron. Res. Arch., 30 (2022), 2568–2599. https://doi.org/10.3934/era.2022132 doi: 10.3934/era.2022132
    [22] X. Li, T. Huang, J. A. Fang, Event-triggered stabilization for Takagi-Sugeno fuzzy complex-valued Memristive neural networks with mixed time-varying delay, IEEE Trans. Fuzzy Syst., 29 (2020), 1853–1863. https://doi.org/10.1109/TFUZZ.2020.2986713 doi: 10.1109/TFUZZ.2020.2986713
    [23] P. Liu, H. Li, Fuzzy Neural Network Theory and Application, World Scientific, Singapore, 2004. Available from: https://www.worldscientific.com/worldscibooks/10.1142/5493#t = aboutBook.
    [24] J. Tavoosi, A. Mohammadzadeh, K. Jermsittiparsert, A review on type-2 fuzzy neural networks for system identification, Soft Comput., 25 (2021), 7197–7212. https://doi.org/10.1007/s00500-021-05686-5 doi: 10.1007/s00500-021-05686-5
    [25] G. Wang, J. Qiao, An efficient self-organizing deep fuzzy neural network for nonlinear system modeling, IEEE Trans. Fuzzy Syst., 30 (2022), 2170–2182. https://doi.org/10.1109/TFUZZ.2021.3077396 doi: 10.1109/TFUZZ.2021.3077396
    [26] A. A. Martynyuk, Y. A. Martynyuk-Chernienko, Uncertain Dynamical Systems–-Stability and Motion Control, CRC Press, Boca Raton, 2012. https://doi.org/10.1201/b11314
    [27] B. Liu, X. Liu, X. Liao, Robust stability of uncertain impulsive dynamical systems, J. Math. Anal. Appl., 290 (2004), 519–533. https://doi.org/10.1016/j.jmaa.2003.10.035 doi: 10.1016/j.jmaa.2003.10.035
    [28] M. Defoort, K. C. Veluvolu, J. J. Rath, M. Djemai, Adaptive sensor and actuator fault estimation for a class of uncertain Lipschitz nonlinear systems, Int. J. Adapt. Control Signal Process., 30 (2016), 271–283. https://doi.org/10.1002/acs.2556 doi: 10.1002/acs.2556
    [29] D. Li, X. Li, Robust exponential stability of uncertain impulsive delays differential systems, Neurocomputing, 191 (2016), 12–18. https://doi.org/10.1016/j.neucom.2016.01.011 doi: 10.1016/j.neucom.2016.01.011
    [30] B. Mansouri, N. Manamanni, K. Guelton, M. Djemai, Robust pole placement controller design in LMI region for uncertain and disturbed switched systems, Nonlinear Anal. Hybrid Syst., 2 (2008), 1136–1143. https://doi.org/10.1016/j.nahs.2008.09.010 doi: 10.1016/j.nahs.2008.09.010
    [31] G. Stamov, I. M. Stamova, Uncertain impulsive differential systems of fractional order: almost periodic solutions, Int. J. Syst. Sci., 49 (2018), 631–638. https://doi.org/10.1080/00207721.2017.1416428 doi: 10.1080/00207721.2017.1416428
    [32] G. T. Stamov, I. M. Stamova, J. Cao, Uncertain impulsive functional differential systems of fractional order and almost periodicity, J. Franklin Inst., 355 (2018), 5310–5323. https://doi.org/10.1016/j.jfranklin.2018.05.021 doi: 10.1016/j.jfranklin.2018.05.021
    [33] F. Z. Taousser, M. Defoort, M. Djemai, Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions, Nonlinear Anal. Hybrid Syst., 16 (2015), 13–23. https://doi.org/10.1016/j.nahs.2014.12.001 doi: 10.1016/j.nahs.2014.12.001
    [34] X. Xu, C. Huang, C. Li, G. Zhao, X. Li, C. Ma, Uncertain design optimization of automobile structures: a survey, Electron. Res. Arch., 31 (2023), 1212–1239. https://doi.org/10.3934/era.2023062 doi: 10.3934/era.2023062
    [35] R. Jafari, W. Yu, Fuzzy modeling for uncertainty nonlinear systems with fuzzy equations, Math. Probl. Eng., 2017 (2017), 8594738. https://doi.org/10.1155/2017/8594738 doi: 10.1155/2017/8594738
    [36] F. Hausdorff, Dimension und äußeres Maß, Math. Ann., 79 (1918), 157–179. https://doi.org/10.1007/BF01457179 doi: 10.1007/BF01457179
    [37] L. T. P. Ngoc, N. T. Long, On a first-order differential system with initial and nonlocal boundary conditions, Demonstr. Math., 55 (2022), 277–296. https://doi.org/10.1515/dema-2022-0012 doi: 10.1515/dema-2022-0012
    [38] V. Lakshmikantham, S. Leela, A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-27200-9
    [39] A. A. Martynyuk, Novel bounds for solutions of nonlinear differential equations, Appl. Math., 6 (2015), 182–194. https://doi.org/10.4236/am.2015.61018 doi: 10.4236/am.2015.61018
    [40] Y. Louartassi, H. Mazoudi, N. Elalami, A new generalization of lemma Gronwall-Bellman, Appl. Math. Sci., 6 (2012), 621–628. Available from: http://www.m-hikari.com/ams/ams-2012/ams-13-16-2012/louartassiAMS13-16-2012.pdf.
    [41] X. Li, J. Shen, R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14–22. https://doi.org/10.1016/j.amc.2018.01.036 doi: 10.1016/j.amc.2018.01.036
    [42] X. Li, X. Yang, T. Huang, Persistence of delayed cooperative models: impulsive control method, Appl. Math. Comput., 342 (2019), 130–146. https://doi.org/10.1016/j.amc.2018.09.003 doi: 10.1016/j.amc.2018.09.003
    [43] D. Peng, X. Li, R. Rakkiyappan, Y. Ding, Stabilization of stochastic delayed systems: event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 126054. https://doi.org/10.1016/j.amc.2021.126054 doi: 10.1016/j.amc.2021.126054
    [44] T. Wei, X. Xie, X. Li, Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses, AIMS Math., 6 (2021), 5786–5800. https://doi.org/10.3934/math.2021342 doi: 10.3934/math.2021342
    [45] S. Singh, J. Dabas, Tikhonov solutions of approximately controllable second-order semilinear control systems, Rend. Circ. Mat. Palermo, 2022. https://doi.org/10.1007/s12215-022-00802-2 doi: 10.1007/s12215-022-00802-2
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