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.
Acknowledgments
This work was completed while the first and second authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). We thank the Institute for its support and hospitality. The research of the first and second authors is supported by NAFOSTED, Vietnam [101.01-2017.300]; the research of the third author is supported by the Chiang Mai University, Thailand. The authors are grateful to the Editor-in-Chief and anonymous reviewers for their valuable comments to improve the paper. We also thank Dr. H.T. Tuan for helpful discussions and suggestions related to Lemma 2.6.
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; 10.1515/fca-2017-0048; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.Search in Google Scholar
[2] B. Chen and J. Chen, Razumikhin-type stability theorems for functional fractional-order differential systems and applications. Appl. Math. Computation254 (2015), 63–69.10.1016/j.amc.2014.12.010Search in Google Scholar
[3] L. Dai, Singular Control Systems. Springer, Berlin (1981).Search in 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), 650–659.10.1016/j.cnsns.2014.10.008Search in 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. Sinica3 (2016), 477–482.10.1109/JAS.2016.7510106Search in Google Scholar
[6] L. Kexue, P. Jigen, Laplace transform and fractional differential equations. Appl. Math. Letters24 (2011), 2019–2023.10.1016/j.aml.2011.05.035Search in 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), 4027–4041.10.1016/j.cam.2012.03.010Search in Google Scholar
[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier Science (2006).Search in Google Scholar
[9] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Sci. Techn., Harlow & John Wiley and Sons, New York (1994).Search in Google Scholar
[10] M.P. Lazarevic, A.M. Spasic, Finite-time stability analysis of fractional order time-delay systems: Gronwall approach. Math. Computer Modelling49 (2009), 475–481.10.1016/j.mcm.2008.09.011Search in Google Scholar
[11] M. Li and J. Zhang, Finite-time stability of fractional delay differential equations. Appl. Math. Letters64 (2017), 170–176.10.1016/j.aml.2016.09.004Search in 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), 32–39.10.1016/j.aml.2016.10.002Search in 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; 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.Search in Google Scholar
[14] K. Mathiyalagan, K. Balachandran, Finite-time stability of fractional-order stochastic singular systems with time delay and white noise. Complexity21 (2016), 370–379.10.1002/cplx.21815Search in Google Scholar
[15] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in 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), 1367–1373.10.1049/iet-cta.2018.5556Search in 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 Brief66 (2019), 1356–1360.10.1109/TCSII.2018.2880777Search in Google Scholar
[18] H. Trinh, H.T. Tuan, Stability of fractional-order nonlinear systems by Lyapunov direct method. IET Control Theory Appl. 12 (2018), 2417–2422.10.1049/iet-cta.2018.5233Search in 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; 10.1515/fca-2020-0010; https://www.degruyter.com/view/j/fca.2020.23.issue-1/issue-files/fca.2020.23.issue-1.xml.Search in 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. Letters53 (2016), 45–51.10.1016/j.aml.2015.10.001Search in 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), 1015–1025.10.1007/s11071-015-2214-ySearch in 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; 10.1515/fca-2018-0021; https://www.degruyter.com/view/j/fca.2018.21.issue-2/issue-files/fca.2018.21.issue-2.xml.Search in 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. Computation269 (2015), 351–362.10.1016/j.amc.2015.07.059Search in Google Scholar
[24] Z. Zhang, W. Jiang, Some results of the degenerate fractional differential system with delay. Computers Math. Appl. 62 (2011), 1284–1291.10.1016/j.camwa.2011.03.061Search in 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. Science49 (2018), 537–545.10.1080/00207721.2017.1412534Search in 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. Control20 (2018), 1–14.10.1002/asjc.1675Search in 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, 1–8.10.1155/2014/850279Search in Google Scholar
© 2020 Diogenes Co., Sofia
