Algebraic stability of impulsive fractional-order systems

Electronic Journal of Qualitative Theory of Differential Equations

Volumn 2014, Issue 32, 2014, Pages 1-13

  Wu, Ranchao ^a   Hei, Xindong ^a  

^a ANHUI UNIVERSITY   (China)

Author keywords

fractional order system; impulsive system; Lyapunov s direct method; Mittag Leffler stability

Indexed keywords

EID: 84987871759     PISSN: None     EISSN: 14173875     Source Type: Journal    
DOI: 10.14232/ejqtde.2014.1.32     Document Type: Article

References (24)

1. H. Schiesse, R. Metzlert, A. Blument, T. F. Nonnemacher, Generalized viscoelastic models: their fractional equations with solutions, J. Phys. Math. Gen. 28(1995), No. 23 6567–6584. url
2. E. Kaslik, S. Sivasundaram, Nonlinear dynamics and chaos in fractional-order neural networks, Neural Networks 32(2012), 245–256. url
3. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1998. MR1658022
4. D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational engineering in systems applications, Vol. 2, 963–968, 1996.
5. W. Deng, C. Li, J. Lv, Stability analysis of linear fractional differential system with mul-tiple time delays, Nonlinear Dynam. 48(2007), No. 4, 409–416. MR2312588; url
6. C. Li, F. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Spec. Top. 193(2011), No. 1, 27–47. url
7. T. Yang, L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circ. Syst. Fund. Theor. Appl. 44(1997), No. 10, 976–988. MR1488197
8. M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, Hindawi Publishing Corporation, New York, 2006. MR2322133
9. W. M. Haddad, V. Chellaboina, S. G. Nersesov, Impulsive and hybrid dynamical systems, Princeton University Press, Princeton, 2006. MR2245760
10. M. Fečkan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), No. 7, 3050–3060. MR2880474
11. T. Guo, W. Jiang, Impulsive fractional functional differential equations, Comput. Math. Appl. 64(2012), No. 10, 3414–3424. MR2989369
12. T. Guo, Controllability and observability of impulsive fractional linear time-invariant system, Comput. Math. Appl. 64(2012), No. 10, 3171–3182. MR2989345
13. J. Wang, M. Feckan, Y. Zhou, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64(2012), No. 10, 3389–3405. MR2989367
14. J. Lu, Y. Chen, Stability and stabilization of fractional order linear systems with convex polytopic uncertainties, Fract. Calc. Appl. Anal. 16(2013), No. 1, 142–157. MR3016647
15. Z. Jiao, Y. Chen, Stability analysis of fractional-order systems with double noncommen-surate orders for matrix case, Fract. Calc. Appl. Anal. 14(2011), No. 3, 436–453. MR2837640; url
16. C. Zeng, Y. Chen, Q. Yang, Almost sure and moment stability properties of fractional order Black-Scholes model, Fract. Calc. Appl. Anal. 16(2013), No. 2, 317–331. MR3033691
17. Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl. 59(2010), No. 5, 1810–1821. MR2595955; url
18. F. Zhang, C. Li, Y. Chen, Asymptotical stability of nonlinear fractional differential system with Caputo derivative, Inter. J. Diff. Eqns. 2011, Art. ID 635165, 12 pp. MR2821248; url
19. H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional order nonlin-ear systems revisited, Nonlinear Dynam. 67(2012), No. 4, 2433–2439. MR2881553
20. H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applica-tions, J. Appl. Math. 2011, Art. ID 298628, 51 pp. MR2800586; url
21. H. K. Khalil, Nonlinear systems, 3rd ed., Prentice Hall, 2002.
22. L. Huang, Stability Theory, Peking University Press, Beijing, China, 1992.
23. M. Sabin, Analysis and design of univariate subdivision schemes, Geometry and Computing, Vol. 6, Springer-Verlag, Berlin–Heidelberg, 2010. MR2723195; url
24. V. Lakshmikantham, A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. 11(2007), No. 3–4, 395–402. MR2368191; url