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Homogeneity-Based Exponential Stability Analysis for Conformable Fractional-Order Systems

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Ukrainian Mathematical Journal Aims and scope

We study the exponential stability of homogeneous fractional time-varying systems and the existence of Lyapunov homogeneous function for the conformable fractional homogeneous systems. We also prove that the local and global behaviors are similar. A numerical example is given to illustrate the efficiency of the obtained results.

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References

  1. T. Abdeljawad, J. Alzabut, and F. Jarad, “A generalized Lyapunov type inequality in the frame of conformable derivatives,” Adv. Difference Equat., 2017 (2017); DOI https://doi.org/10.1186/s13662-017-1383-z.

  2. T. Abdeljawad, Q. M. Al-Mdallal, and F. Jarad, “Fractional logistic models in the frame of fractional operators generated by conformable derivatives,” Chaos Solitons Fractals, 119, 94–101 (2019).

    Article  MathSciNet  Google Scholar 

  3. M. Al-Refai and T. Abdeljawad, “Fundamental results of conformable Sturm–Liouville eigenvalue problems,” Complexity, 2017, 1–7 (2017).

    Article  MathSciNet  Google Scholar 

  4. T. Abdeljawad, “On conformable fractional calculus,” J. Comput. Appl. Math., 279, 57–66 (2015).

    Article  MathSciNet  Google Scholar 

  5. M. Arfan, I. Mahariq, K. Shah, T. Abdeljawad, G. Laouini, and P. O. Mohammed, “Numerical computations and theoretical investigations of a dynamical system with fractional order derivative,” Alex. Eng. J., 61, 1982–1994 (2022).

    Article  Google Scholar 

  6. R. Almeida, M. Guzowska, and T. Odzijewicz, “A remark on local fractional calculus and ordinary derivatives,” Open Math., 14, 1122–1124 (2016).

    Article  MathSciNet  Google Scholar 

  7. N. Benkhettou, S. Hassani, and D. F. M. Torres, “A conformable fractional calculus on arbitrary time scales,” J. King Saud Univ. Sci., 28, 93–98 (2016).

    Article  Google Scholar 

  8. S. P. Bhat and D. S. Bernstein, “Geometric homogeneity with applications to finite-time stability,” Math. Control Signals Systems, 17, 101–127 (2005).

    Article  MathSciNet  Google Scholar 

  9. Y. Cenesiz, D. Baleanu, A. Kurt, and O. Tasbozan, “New exact solutions of Burgers type equations with conformable derivative,” Waves Random Complex Media, 27, No. 1, 103–116; https://doi.org/10.1080/17455030.2016.1205237.

  10. W. S. Chung, “Fractional Newton mechanics with conformable fractional derivative,” J. Comput. Appl. Math., 290, 150–158 (2015).

    Article  MathSciNet  Google Scholar 

  11. M. Farman, A. Akgl, T. Abdeljawad, P. Ahmad Naik, N. Bukhari, and A. Ahmad, “Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel,” Alex. Eng. J., 61, 2062–2073 (2022).

  12. P. M. Guzmán, G. Langton, L. M. Lugo, J. Medina, and J. E. Nápoles Valds, “A new definition of a fractional derivative of local type,” J. Math. Anal., 9, No. 2, 88–98 (2018).

  13. H. Hermes, “Nilpotent and high-order approximations of vector field systems,” SIAM Rev., 33, No. 2, 238–264 (1991).

    Article  MathSciNet  Google Scholar 

  14. H. Jerbi, T. Kharrat, and F. Mabrouk, “Stabilization of polynomial systems in R3 via homogeneous feedback,” J. Appl. Anal., 28, No. 2, 189–197 (2022); https://doi.org/10.1515/jaa-2021-2080.

    Article  MathSciNet  Google Scholar 

  15. M. Kawski, “Homogeneous stabilizing feedback laws,” Control Theory Adv. Tech., 6, No. 4, 497–516 (1990).

    MathSciNet  Google Scholar 

  16. M. Kawski, “Homogeneous feedback stabilization,” Progress in Systems and Control Theory, vol. 7 (1991).

  17. R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comput. Appl. Math., 264, 65–70 (2014).

  18. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Math. Stud., Elsevier Sci. B.V., Amsterdam (2006).

  19. E. Moulay and W. Perruquetti, “Finite time stability and stabilization of a class of continuous systems,” J. Math. Anal. Appl., 323, No. 2, 1430–1443 (2006).

    Article  MathSciNet  Google Scholar 

  20. J. E. Nápoles Valds, P. M. Guzmán, and L. M. Lugo, “Some new results on non conformable fractional calculus,” Adv. Dyn. Syst. Appl., 13, No. 2, 167–175 (2018).

  21. J. E. Nápoles Valds, P. M. Guzmán, and L. M. Lugo, “On the stability of solutions of fractional non conformable differential equations,” Stud. Univ. Babeş-Bolyai Math., 65, No. 4, 495–502 (2020).

  22. I. Podlubny, “Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,” Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, CA (1999).

  23. L. Rosier, “Homogeneous Lyapunov function for homogeneous continuous vector field,” Syst. Control Lett., 19, 467–473 (1992).

    Article  MathSciNet  Google Scholar 

  24. L. P. Rothschild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups,” Acta Math., 137, 247–320 (1976).

    Article  MathSciNet  Google Scholar 

  25. A. Souahi, A. Ben Makhlouf, and M. A. Hammami, “Stability analysis of conformable fractional-order nonlinear systems,” Indag. Math., 28, 1265–1274 (2017).

  26. L. L. Wang and J. L. Fu, “Non-Noether symmetries of Hamiltonian systems with conformable fractional derivatives,” Chinese Phys. B, Article 014501 (2016).

  27. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ., River Edge, NJ (2000).

    Book  Google Scholar 

  28. H. Rezazadeh, H. Aminikhah, and A. H. Refahi Sheikhani, “Stability analysis of conformable fractional systems,” Iran. J. Numer. Anal. Optim., 7, No. 1, 13–32 (2017).

  29. K. Shah, M. Sher, A. Ali, and T. Abdeljawad, “Extremal solutions of generalized Caputo-type fractional-order boundary value problems using monotone iterative method,” Fractal Fract., 2022, No. 6 (2022).

  30. R. Singh, T. Abdeljawad, E. Okyere, and L. Guran, “Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse,” Adv. Difference Equat., Article 390 (2021).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax, Tunisia

    Fehmi Mabrouk

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  1. Fehmi Mabrouk
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Correspondence to Fehmi Mabrouk.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 10, pp. 1402–1410, October, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i10.7280.

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Mabrouk, F. Homogeneity-Based Exponential Stability Analysis for Conformable Fractional-Order Systems. Ukr Math J 75, 1590–1600 (2024). https://doi.org/10.1007/s11253-024-02280-4

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  • DOI: https://doi.org/10.1007/s11253-024-02280-4

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