Nonlinear Dynamics

, Volume 84, Issue 3, pp 1353–1361 | Cite as

Extension of Lyapunov direct method about the fractional nonautonomous systems with order lying in \(\mathbf{(1,2)}\)

Original Paper


In this paper, Lyapunov direct method is employed to study the stability problem of Caputo-type fractional nonautonomous systems with order between 1 and 2. By utilizing Riemann–Liouville fractional integral, some sufficient conditions on stability are derived. In the proof of the obtained results, Bellman–Gronwall’s inequality, the generalized Bihari inequality and estimates of Mittag-Leffler functions are employed. Besides, two examples and corresponding numerical simulations are provided to show the validity and feasibility of the proposed stability criterion.


Caputo derivative Fractional nonautonomous systems  Bihari-type inequality Bellman–Gronwall’s inequality Globally uniformly asymptotically stable 



The authors would like to thank the anonymous reviewers for their helpful suggestions and comments. This work was supported by the National Science and Technology Major Project (No. 2012CB821202), the National Nature Science Foundation ( No. 61327807) and Beijing Natural Science Foundation (No. 4122043).


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.The Seventh Research DivisionBeijing University of Aeronautics and AstronauticsBeijingPeople’s Republic of China

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