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

New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 415--425 | DOI:10.5890/DNC.2016.12.007

Tianzeng Li$^{1}$,$^{2}$, Yu Wang$^{1}$,$^{3}$, Yong Yang

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China

2Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing, Zigong 643000, China

3Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, 643000, China

Abstract

In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.

Acknowledgments

The work is supported by Found of Science &Technology Department of Sichuan Province (Grant No.2016JQ0046), Artificial Intelligence Key Laboratory of Sichuan Province (Grant No.2016RYJ06), Found of Sichuan University of Science and Engineering (Grant 2014PY06, 2015RC10), the Opening Project of Sichuan Province University Key Laborstory of Bridge Non-destruction Detecting and Engineering Computing (Grant No.2015QYJ02, 2014QZJ03), Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (Grant No.2016WYJ04).

References

1.  [1] Das, S. and Gupta, P. (2011), A mathematical model on fractional Lptka-Volterra equations, Joural of Theoretical Biology, 277, 1-6.
2.  [2] Naber, M. (2004), Time fractional Schrödinger equation, Journal of Mathematical Physics, 45, 3339-3352.
3.  [3] Burov, S. and Barkai, E. (2008), Fractional Langevin equation: overdammped, underdamped, and cirtical behaviors, Physical Review E, 78, 031112.
4.  [4] Ryabov, Y. and Puzenko, A. (2002), Damped oscillation in view of the fractional oscillator equation, Physical Review B, 66, 184-201.
5.  [5] Bonnet, C. and Partington, J.R. (2000), Coprime factorizations and stability fo fractional defferential systems, System & Control Letters 41, 167-174.
6.  [6] Deng, W.H., Li, C.P. and Lü, J.H. (2007), Stability analysis of linear fractional dirrerential system with multiple timedelays, Nolinear Dynamics, 48, 409-416.
7.  [7] Kheirizad, I., Tavazoei,M.s. and Jalali, A.A. (2010), Stability criteria for a class of fractional order systems, Nonlinear Dynamics, 61, 153-161.
8.  [8] Zhang, F. and Li, C.P. (2011), Stability analysis of fractional differential systems with order lying in (1,2), Advances in Difference Equations, 2011, 213485.
9.  [9] Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R. and Abdeljawad, T. (2010), Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstract and Applied Analysis 2010, 108651.
10.  [10] Liu, L. and Zhong, S. (2011), Finite-time stability analysis of fractional-order with multi-state time delay, Word Academy of Science, Eniineering and Technology, 76, 874-877.
11.  [11] Li, Y., Chen, Y.Q. and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965-1969.
12.  [12] 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 & Mathematics with Application, 59, 1810-1821.
13.  [13] Wang, Y. and Li, T.Z. (2014), Stability analysis of fractional-order nonlinera systems with delay. Mathematical Problems in Engineering 2014, 301235.
14.  [14] Chen, Y.Q. and Moore, K.L. (2002), Analytical stability bound for a class of delayed fractional order dynamic systems, Nonlenear Dynamics. 29, 191-200.
15.  [15] Li, T.Z.,Wang, Y., Yang, Y. (2014), Designing synchronization schemes for fractional-order chaotic system via a single state fractional-order controller, Optik 125, 6700-6705.
16.  [16] Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.
17.  [17] Li, T.Z., Wang, Y. and Luo, M.K. (2014), Control of fractional chaotic and hyperchaotic systems based on a fractional order controller, Chinese Physics B, 23, 080501.
18.  [18] Li, T.Z., Wang, Y. and Yang, Y. (2014), Synchronization of fractional-order hyperchaotic systems via fractional-order controllers, Discrete Dynamics in Nature and Society, 2014, 408972.
19.  [19] Sabatier, J., Agrawal, Q.P. and Machado, T.J.A. (2007), Advances in fractional calculus-theoretical developments and applications in physics and engineering, Springer.
20.  [20] De la Sen, M. (2011), About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications, 2011, 867932.
21.  [21] Chen, L.P., He, Y.G., Chai, Y. and Wu, R.C. (2014), New results on stability and stabilization of a calss of nonlinear fractional-order systems, Nonlinear Dynamics., 75, 633-641.
22.  [22] Ye, H., Gao, J. and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.