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**Mittag-Leffler stability of fractional order nonlinear dynamic systems.**
*(English)*
Zbl 1185.93062

Summary: We propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.

### MSC:

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

34A08 | Fractional ordinary differential equations |

93C10 | Nonlinear systems in control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

### Keywords:

fractional order dynamic system; nonautonomous system; fractional Lyapunov direct method; Mittag-Leffler stability; fractional comparison principle
Full Text:
DOI

### References:

[1] | Bagley, R. L.; Torvik, P. J., Fractional calculus — A different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21, 5, 741-748 (1983) · Zbl 0514.73048 |

[2] | Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Theology, 27, 3, 201-210 (1983) · Zbl 0515.76012 |

[3] | Chen, Y. Q. (2006). Ubiquitous fractional order controls? In Proceedings of the second IFAC workshop on fractional differentiation and its applications; Chen, Y. Q. (2006). Ubiquitous fractional order controls? In Proceedings of the second IFAC workshop on fractional differentiation and its applications |

[4] | Chen, Y. Q.; Moore, K. L., Analytical stability bound for a class of delayed fractional order dynamic systems, Nonlinear Dynamics, 29, 191-200 (2002) · Zbl 1020.34064 |

[5] | Chua, L. O., Memristor — The missing circuit element, IEEE Transactions on Circuit Theory, CT-18, 5, 507-519 (1971) |

[6] | Cohen, I.; Golding, I.; Ron, I. G.; Ben-Jacob, E., Bio-uiddynamics of lubricating bacteria, Mathematical Methods in the Applied Sciences, 24, 1429-1468 (2001) · Zbl 1097.76618 |

[7] | Khalil, H. K., Nonlinear systems (2002), Prentice Hall · Zbl 0626.34052 |

[8] | Li, Y., Chen, Y. Q., Podlubny, I., & Cao, Y. (2008). Mittag-leffler stability of fractional order nonlinear dynamic systems. In Proceedings of the 3rd IFAC workshop on fractional differentiation and its applications; Li, Y., Chen, Y. Q., Podlubny, I., & Cao, Y. (2008). Mittag-leffler stability of fractional order nonlinear dynamic systems. In Proceedings of the 3rd IFAC workshop on fractional differentiation and its applications · Zbl 1185.93062 |

[9] | Li, C.; Deng, W., Remarks on fractional derivatives, Applied Mathematics and Computation, 187, 2, 777-784 (2007) · Zbl 1125.26009 |

[10] | Miller, K. S.; Samko, S. G., Completely monotonic functions, Integral Transforms and and Special Functions, 12, 4, 389-402 (2001) · Zbl 1035.26012 |

[11] | Momani, S.; Hadid, S., Lyapunov stability solutions of fractional integrodifferential equations, International Journal of Mathematics and Mathematical Sciences, 47, 2503-2507 (2004) · Zbl 1074.45006 |

[12] | Podlubny, I., Fractional-order systems and \(PI^\lambda D^\mu \)-controllers, IEEE Transactions on Automatic Control, 44, 1, 208-214 (1999) · Zbl 1056.93542 |

[13] | Podlubny, I., Fractional differential equations (1999), Academic Press · Zbl 0918.34010 |

[14] | Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5, 4, 367-386 (2002) · Zbl 1042.26003 |

[15] | Podlubny, I.; Heymans, N., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45, 5, 765-772 (2006) |

[16] | Sabatier, J. (2008). On stability of fractional order systems. In Plenary lecture VIII on 3rd IFAC workshop on fractional differentiation and its applications; Sabatier, J. (2008). On stability of fractional order systems. In Plenary lecture VIII on 3rd IFAC workshop on fractional differentiation and its applications |

[17] | Sabatier, J.; Agrawal, O. P.; Tenreiro Machado, J. A., Advances in fractional calculus — Theoretical developments and applications in physics and engineering (2007), Springer · Zbl 1116.00014 |

[18] | Tarasov, V. E., Fractional derivative as fractional power of derivative, International Journal of Mathematics, 18, 3, 281-299 (2007) · Zbl 1119.26011 |

[19] | Xu, M.; Tan, W., Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics, Science in China: Series G Physics, Mechanics and Astronomy, 49, 3, 257-272 (2006) · Zbl 1109.26005 |

[20] | Zhang, L.-g.; Li, J.-m.; Chen, G.-p., Extension of Lyapunov second method by fractional calculus, Pure and Applied Mathematics, 21, 3 (2005), 1008-5513(2005)03-0291-04 · Zbl 1118.33001 |

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