Finite-time Stability Analysis of Fractional-order with Multi-state Time Delay

In this paper, the finite-time stabilization of a class of multi-state time delay of fractional-order system is proposed. First, we define finite-time stability with the fractional-order system. Second, by using Generalized Gronwall's approach and the methods of the inequality, we get some conditions of finite-time stability for the fractional system with multi-state delay. Finally, a numerical example is given to illustrate the result.

• Liqiong Liu
• Shouming Zhong

• Finite-time stabilization
• fractional-order system
• Gronwall inequality.

[1] Domenico Delbosco, Luigi Rodino, Existence and uniqueness for
a nonlinear fractional di?erential equation, J. Math. Anal. Appl.
204(1996)609-625.
[2] Xiuyun Zhang, Some results of linear fractional order time-delay system,
Appl. Math. Comput.197 (2008)407-411.
[3] Richard J-P. Time delay systems: An overview of some recent advances
and open problems. Automatica 2003; 39: 1667-94.
[4] M. Zavarei, M. Jamshidi, Time-Delay Systems: Analysis, Optimization
and Applications, North-Holland, Amsterdam, 1987.
[5] M. P. Lazarevi'c,Finite time stability analysis of fractional control of
robotic time-delay systems, Mech.Res. Comm. 33(2006)269-279.
[6] Mihailo P. Lazarevi'c, Aleksandar M. Spasi? b, Finite-time stability
analysis of fractional order time-delay systems: Gronwall-s approach,
Mathematical and Computer Modelling 49(2009)475-481.
[7] Kilbas A-A, Srivastava H-M, Trujillo J-J. Theory and applications
of fractional differential equations.Mathematics studies204. Elsevier,
North-Holland2006.
[8] Oldham KB, Spanier J.The Fractional Calculus.New York: Academic
Press; 1974.
[9] H.Ye, J. Gao, Y.Ding, Ageneralized Gronwall inequality and its application
to afractional differential equation, J.Math. Anal.Appl.328
(2007)1075-1081.