ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Asymptotic Stability of Caputo Fractional Singular Differential Systems with Multiple Delays

Discontinuity, Nonlinearity, and Complexity 7(3) (2018) 243--251 | DOI:10.5890/DNC.2018.09.003

Sivaraj Priyadharsini$^{1}$, Venkatesan Govindaraj$^{2}$

$^{1}$ Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore-641 008, India

$^{2}$ Department of Science and Humanities, National Institute of Technology Puducherry, Karaikal-609 609, India

Abstract

In this work, Lyapunov functions combining Matrix inequalities or Matrix equations are developed to analyze the asymptotic stability of Caputo fractional singular systems with multiple time-varying delay. Integer-order derivatives of the Lyapunov functions are used to derive asymptotic stability criteria. The main advantage of applying stability criteria is that no need of solving roots of transcendental equations. Some examples are provided to explain the effectiveness of the proposed criteria.

Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions.

References

1.  [1] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equation, Elsevier, Amsterdam.
2.  [2] Podlubny, I. (1999), Fractional differential equation, Academic Press, New York.
3.  [3] Miller, K.S. and Ross, B. (1993), An introduction to the fractional calculus and fractional differential equation,Wiley, New York.
4.  [4] Diethelm, K. and Ford, N.J. (2004), Analysis of fractional differential equations, Springer: New York.
5.  [5] Yao, Y., Zhuang, J,. and Chang-Yin, S. (2013), Sufficient and necessary condition of admissibility for fractional-order singular system, Acta Automatica Sinica, 39, 2160-2164.
6.  [6] Gu, K. and Niculescu, S.L. (2003), Survey on recent results in the stability and control of time-delay systems, Journal of Dynamical Systems, 125, 158-165.
7.  [7] Li, Y., Chen, Y.Q., and Podlubny, I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers Mathematical Applications, 5, 1810-1821.
8.  [8] Tan,M.C. (2008), Asymptotic stability of nonlinear systems with unbounded delays, Journal of Mathematical Analysis and Applications, 337, 1010-1021.
9.  [9] Krol, K. ( 2011), Asymptotic properties of fractional delay differential equations, Applied Mathematical Computation, 218, 1515-1532.
10.  [10] Zhang, Z. and Jiang, W. ( 2011), Some results of the degenerate fractional differential system with delay, Computers and Mathematics with Applications, 62, 1284-1291.
11.  [11] Govindaraj, V. and Balachandran, K. (2014), Stability of basset equation, Journal of Fractional Calculus and Applications, 20, 1-15.
12.  [12] Deng, W.H., Li, C.P., and Lu, J.H. (2007), Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynamics, 48, 409-416.
13.  [13] Priyadharsini, S. (2016), Stability analysis of fractional differential systems with constant delay, Journal of Indian Mathematical Society, 83, 337-350.
14.  [14] Liu, S., Li, X., Zhou, X.F., and Jiang,W. (2016),Lyapunov stability analysis of fractional nonlinear systems, Applied Mathematical Letters, 51, 13-19.
15.  [15] Li, Y., Chen, Y., and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica , 45, 1965-1969,
16.  [16] Liu, S., Li, X., Jiang, W., and Zhou, X.F. (2012), Mittag-Leffler stability of nonlinear fractional neutral singular systems, Communications in Nonlinear Science Numerical Simulations, 17, 3961-3966.
17.  [17] Masubuchi, I., Kamitane, Y., Ohara, A., and Suda, N. (1997), Control for descriptor systems: a matrix inequalities approach, Automatica, 33, 669-673.
18.  [18] Doye, I.N., Zasadzinski, M., Darouach, M., and Radhy, N. (2013), Robust stabilization of uncertain descriptor fractional-order systems, Automatica, 49, 1907-1913.
19.  [19] Liu, S., Wu, X., Zhou, X.F., and Jiang, W. (2016), Asymptotical stability of Riemann-Liouville fractional nonlinear systems, Nonlinear Dynamics, 86, 65-71.
20.  [20] Liu, S., Li, X., Zhou, X.F., and Jiang, W. (2017), Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays, Applied Mathematics Letters, 65, 32-39 .
21.  [21] Qian, D., Li, C., Agarwal, R.P., and Wong, P.J.Y. (2010), Stability analysis of fractional differential system with Riemann-Liouville derivative, Mathematical Computation Modelling, 52, 862-874.
22.  [22] Liu, S., Zhou, X.F., Li X., and Jiang, W. (2016), Stability of fractional nonlinear singular systems and its applications in synchronization of complex dynamical networks, Nonlinear Dynamics, 84, 2377-2385.
23.  [23] Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., and Castro-Linares, R. (2015), Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications in Nonlinear Science Numerical Simulations, 22, 650-659, .
24.  [24] Liu, S., Li, X., Zhou, X.F., and Jiang, W. (2015), Synchronization analysis of singular dynamical networks with unbounded time-delays, Advances in Difference Equations, 193, 1-9 .