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New finite-time stability analysis of singular fractional differential equations with time-varying delay

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Abstract

The Lyapunov function method is a powerful tool to stability analysis of functional differential equations. However, this method is not effectively applied for fractional differential equations with delay, since the constructing Lyapunov-Krasovskii function and calculating its fractional derivative are still difficult. In this paper, to overcome this difficulty we propose an analytical approach, which is based on the Laplace transform and “inf-sup” method, to study finite-time stability of singular fractional differential equations with interval time-varying delay. Based on the proposed approach, new delay-dependent sufficient conditions such that the system is regular, impulse-free and finite-time stable are developed in terms of a tractable linear matrix inequality and the Mittag-Leffler function. A numerical example is given to illustrate the application of the proposed stability conditions.

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References

  1. D. Boyadzhiev, H. Kiskinov, M. Veselinova, A. Zahariev, Stability analysis of linear distributed order fractional systems with distributed delays. Fract. Calc. Appl. Anal. 20, No 4 (2017), 914–935; DOI: 10.1515/fca-2017-0048; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.

    Article  MathSciNet  Google Scholar 

  2. B. Chen, J. Chen, Razumikhin-type stability theorems for functional fractional-order differential systems and applications. Appl. Math. Computation 254 (2015), 63–69.

    Article  MathSciNet  Google Scholar 

  3. L. Dai, Singular Control Systems. Springer, Berlin (1981).

    Google Scholar 

  4. M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat. 22 (2015), 650–659.

    Article  MathSciNet  Google Scholar 

  5. C. Hua, T. Zhang, Y. Li, X. Guan, Robust output feedback control for fractional-order nonlinear dystems with time-varying delays. IEEE/CAA J. Auto. Sinica 3 (2016), 477–482.

    Article  Google Scholar 

  6. L. Kexue, P. Jigen, Laplace transform and fractional differential equations. Appl. Math. Letters 24 (2011), 2019–2023.

    Article  MathSciNet  Google Scholar 

  7. E. Kaslik, S. Sivasundaram, Analytical and numerical methods for the stability analysis of linear fractional delay differential equations. J. Comput. Appl. Math. 236 (2012), 4027–4041.

    Article  MathSciNet  Google Scholar 

  8. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier Science (2006).

    MATH  Google Scholar 

  9. V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Sci. Techn., Harlow & John Wiley and Sons, New York (1994).

    MATH  Google Scholar 

  10. M.P. Lazarevic, A.M. Spasic, Finite-time stability analysis of fractional order time-delay systems: Gronwall approach. Math. Computer Modelling 49 (2009), 475–481.

    Article  MathSciNet  Google Scholar 

  11. M. Li, J. Zhang, Finite-time stability of fractional delay differential equations. Appl. Math. Letters 64 (2017), 170–176.

    Article  MathSciNet  Google Scholar 

  12. S. Liu, X.F. Zhou, X. Li, W. Jiang, Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays. Appl. Math. Lett. 65 (2017), 32–39.

    Article  MathSciNet  Google Scholar 

  13. J.G. Lu, Y.Q. Chen, Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fract. Calc. Appl. Anal. 16, No 1 (2013), 142–157; DOI: 10.2478/s13540-013-0010-2; https://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.

    Article  MathSciNet  Google Scholar 

  14. K. Mathiyalagan, K. Balachandran, Finite-time stability of fractional-order stochastic singular systems with time delay and white noise. Complexity 21 (2016), 370–379.

    Article  MathSciNet  Google Scholar 

  15. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).

    MATH  Google Scholar 

  16. N.T. Thanh, V.N. Phat, Switching law design for finite-time stability of singular fractional-order systems with delay. IET Control Theory Appl. 13 (2019), 1367–1373.

    Article  MathSciNet  Google Scholar 

  17. N.T. Thanh, V.N. Phat, Improved approach for finite-time stability of nonlinear fractional-order systems with interval time-varying delay. IEEE TCAS II: Express Brief 66 (2019), 1356–1360.

    Google Scholar 

  18. H. Trinh, H.T. Tuan, Stability of fractional-order nonlinear systems by Lyapunov direct method. IET Control Theory Appl. 12 (2018), 2417–2422.

    Article  MathSciNet  Google Scholar 

  19. H.T. Tuan, S. Siegmund, Stability of scalar nonlinear fractional differential equations with linearly dominated delay. Fract. Calc. Appl. Anal. 23, No 1 (2020), 250–267; DOI: 10.1515/fca-2020-0010; https://www.degruyter.com/view/j/fca.2020.23.issue-1/issue-files/fca.2020.23.issue-1.xml.

    Article  MathSciNet  Google Scholar 

  20. F.F. Wang, D.Y. Chen, X.G. Zhang, Y. Wu, The existence and uniqueness theorem of the solution to a class of nonlinear fractional order systems with time delay. Appl. Math. Letters 53 (2016), 45–51.

    Article  MathSciNet  Google Scholar 

  21. Y. Wen, X.F. Zhou, Z. Zhang, S. Liu, Lyapunov method for nonlinear fractional differential systems with delay. Nonlinear Dyn. 82 (2015), 1015–1025.

    Article  MathSciNet  Google Scholar 

  22. G.C. Wu, D. Baleanu, Stability analysis of impulsive fractional difference equations. Fract. Calc. Appl. Anal. 21, No 2 (2018), 354–375; DOI: 10.1515/fca-2018-0021; https://www.degruyter.com/view/j/fca.2018.21.issue-2/issue-files/fca.2018.21.issue-2.xml.

    Article  MathSciNet  Google Scholar 

  23. C. Yin, S. Zhong, X. Huang, Y. Cheng, Robust stability analysis of fractional-order uncertain singular nonlinear system with external disturbance. Appl. Math. Computation 269 (2015), 351–362.

    Article  MathSciNet  Google Scholar 

  24. Z. Zhang, W. Jiang, Some results of the degenerate fractional differential system with delay. Computers Math. Appl. 62 (2011), 1284–1291.

    Article  MathSciNet  Google Scholar 

  25. H. Zhang, R. Ye, S. Liu, J. Cao, A. Alsaedi, X. Li, LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays. Int. J. Syst. Science 49 (2018), 537–545.

    Article  MathSciNet  Google Scholar 

  26. H. Zhang, R. Ye, J. Cao, A. Ahmed, X. Li, Y. Wan, Lyapunov functional approach to stability analysis of Riemann-Liouville fractional neural networks with time-varying delays. Asian J. Control 20 (2018), 1–14.

    Article  MathSciNet  Google Scholar 

  27. H. Zhang, D. Wu, J. Cao, H. Zhang, Stability analysis for fractional-order linear singular delay differential systems. Discrete Dyn. Nature Soc. 2014 (2014), Art. ID 850279, 1–8.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Mining and Geology, Hanoi, Vietnam

    Nguyen T. Thanh

  2. ICRTM, Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Hanoi, 10307, Vietnam

    Vu N. Phat

  3. RCMAM, Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai, 50200, Thailand

    Piyapong Niamsup

Authors
  1. Nguyen T. Thanh
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  2. Vu N. Phat
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  3. Piyapong Niamsup
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Correspondence to Nguyen T. Thanh.

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Thanh, N.T., Phat, V.N. & Niamsup, P. New finite-time stability analysis of singular fractional differential equations with time-varying delay. Fract Calc Appl Anal 23, 504–519 (2020). https://doi.org/10.1515/fca-2020-0024

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  • DOI: https://doi.org/10.1515/fca-2020-0024

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