Since Lyapunov method has not been well developed for fractional-order systems, stability of fractional-order nonlinear delayed systems remains a formidable problem. In this letter, finite-time stability of a class of fractional-order nonlinear delayed systems with order between 0 and 1 is addressed. By using the technique of inequalities, a new and simple delay-independent sufficient condition guaranteeing stability of fractional-order nonlinear delayed system over the finite time interval is obtained. Numerical examples are presented to demonstrate the validity and feasibility of the obtained results.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic
,
New York
.
2.
Butzer
,
P. L.
, and
Westphal
,
U.
,
2000
,
An Introduction to Fractional Calculus
,
World Scientific
,
Singapore
.
3.
Hilfer
,
R.
,
2001
,
Applications of Fractional Calculus in Physics
,
World Scientific
,
Singapore
.
4.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Application of Fractional Differential Equations
,
Elsevier
,
Amsterdam
.
5.
Naber
,
M.
,
2004
, “
Time Fractional Schrödinger Equation
,”
J. Math. Phys.
,
45
(
8
), pp.
3339
3352
.10.1063/1.1769611
6.
Ryabov
,
Y.
, and
Puzenko
,
A.
,
2002
, “
Damped Oscillation in View of the Fractional Oscillator Equation
,”
Phys. Rev. B
,
66
(
18
), pp.
184
201
.10.1103/PhysRevB.66.184201
7.
Das
,
S.
, and
Gupta
,
P.
,
2011
, “
A Mathematical Model on Fractional Lotka–Volterra Equations
,”
J. Theor. Biol.
,
277
(
1
), pp.
1
6
.10.1016/j.jtbi.2011.01.034
8.
Burov
,
S.
, and
Barkai
,
E.
,
2008
, “
Fractional Langevin Equation: Overdamped, Underdamped, and Critical Behaviors
,”
Phys. Rev. E
,
78
(
3
), p.
031112
.10.1103/PhysRevE.78.031112
9.
Matignon
,
D.
,
1996
, “
Stability Results for Fractional Differential Equations With Applications
,”
Proceedings of IMACS-IEEE Computational Engineering in Systems. Applications (CESA) Conference
, pp.
963
968
.
10.
Lu
,
J. G.
, and
Chen
,
Y. Q.
,
2010
, “
Robust Stability and Stabilization of Fractional-Order Interval Systems With the Fractional Order α: The 0 < α < 1 Case
,”
IEEE Trans. Autom. Control
,
55
(
1
), pp.
152
158
.10.1109/TAC.2009.2033738
11.
Tavazoe
,
M. S.
, and
Haeri
,
M.
,
2009
, “
A Note on the Stability of Fractional Order Systems
,”
Math. Comput. Simul.
,
79
(
5
), pp.
1566
1576
.10.1016/j.matcom.2008.07.003
12.
Ahn
,
H. S.
, and
Chen
,
Y. Q.
,
2008
, “
Necessary and Sufficient Stability Condition of Fractional-Order Interval Linear Systems
,”
Automatica
,
44
(
11
), pp.
2985
2988
.10.1016/j.automatica.2008.07.003
13.
Deng
,
W. H.
,
Li
,
C. P.
, and
Lu
,
J. H.
,
2007
, “
Stability Analysis of Linear Fractional Differential System With Multiple Time Delays
,”
Nonlinear Dyn.
,
48
(
4
), pp.
409
416
.10.1007/s11071-006-9094-0
14.
Hua
,
C. C.
,
Liu
,
D.
, and
Guan
,
X. P.
,
2014
, “
Necessary and Sufficient Stability Criteria for a Class of Fractional-Order Delayed Systems
,”
IEEE Trans. Circuits Syst. Express Briefs
,
61
(
1
), pp.
59
63
.10.1109/TCSII.2013.2291137
15.
Mesbahi
,
A.
, and
Haeri
,
M.
,
2013
, “
Stability of Linear Time Invariant Fractional Delay Systems of Retarded Type in the Space of Delay Parameters
,”
Automatica
,
49
(
5
), pp.
1287
1294
.10.1016/j.automatica.2013.01.041
16.
Delavari
,
H.
,
Baleanu
,
D.
, and
Sadati
,
J.
,
2012
, “
Stability Analysis of Caputo Fractional-Order Nonlinear Systems Revisited
,”
Nonlinear Dyn.
,
67
(
4
), pp.
2433
2439
.10.1007/s11071-011-0157-5
17.
Wen
,
X. J.
,
Wu
,
Z. M.
, and
Lu
,
J. G.
,
2008
, “
Stability Analysis of a Class of Nonlinear Fractional-Order Systems
,”
IEEE Trans. Circuits Syst. Express Briefs
,
55
(
11
), pp.
1178
1182
.10.1109/TCSII.2008.2002571
18.
Deng
,
W. H.
,
2010
, “
Smoothness and Stability of the Solutions for Nonlinear Fractional Differential Equations
,”
Nonlinear Anal. Theory Methods Appl.
72
(
3
), pp.
1768
1777
.10.1016/j.na.2009.09.018
19.
Chen
,
L. P.
,
Chai
,
Y.
,
Wu
,
R. C.
, and
Yang
,
J.
,
2012
, “
Stability and Stabilization of a Class of Nonlinear Fractional-Order Systems With Caputo Derivative
,”
IEEE Trans. Circuits Syst. Express Briefs
,
59
(9), pp.
602
606
.10.1109/TCSII.2012.2206936
20.
Chen
,
L. P.
,
He
,
Y. G.
,
Chai
,
Y.
, and
Wu
,
R. C.
,
2014
, “
New Results on Stability and Stabilization of a Class of Nonlinear Fractional-Order Systems
,”
Nonlinear Dyn.
75
(
4
), pp.
633
641
.10.1007/s11071-013-1091-5
21.
Stamova
,
I.
,
2014
, “
Global Stability of Impulsive Fractional Differential Equations
,”
Appl. Math. Comput.
,
237
(6), pp.
605
612
.10.1016/j.amc.2014.03.067
22.
Wang
,
J. R.
,
Zhou
,
Y.
, and
Feckan
,
M.
,
2012
, “
Nonlinear Impulsive Problems for Fractional Differential Equations and Ulam Stability
,”
Comput. Math. Appl.
,
64
(
10
), pp.
3389
3405
.10.1016/j.camwa.2012.02.021
23.
Sadati
,
S. J.
,
Baleanu
,
D.
,
Ranjbar
,
D. A.
,
Ghaderi
,
R.
, and
Abdeljawad
,
T.
,
2010
, “
Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems With Delay
,”
Abstr. Appl. Anal.
,
2010
, pp. 1–7.10.1155/2010/10865110.1155/2010/108651
24.
Liu
,
S.
,
Li
,
X. Y.
,
Jiang
,
W.
, and
Zhou
,
X. F.
,
2012
, “
Mittag-Leffler Stability of Nonlinear Fractional Neutral Singular Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
17
(10), pp.
3961
3966
.10.1016/j.cnsns.2012.02.012
25.
Mohammad
,
A. P.
,
Sara
,
P.
, and
Mohammad
,
A. N.
,
2014
, “
On the Stability of Fractional Order Systems of Neutral Type
,”
J. Comput. Nonlinear Dyn.
(to be published).10.1115/1.4027593
26.
Grujić
,
L. T.
,
1975
, “
Non-Lyapunov Stability Analysis of Large-Scale Systems on Time-Varying Sets
,”
Int. J. Control
,
21
(
3
), pp.
401
405
.10.1080/00207177508921999
27.
Grujić
,
L. T.
,
1975
, “
Practical Stability With Settling Time on Composite Systems
,”
Automatika
9
, pp.
1
11
.
28.
Zhang
,
X.
,
2008
, “
Some Results of Linear Fractional Order Time-Delay System
,”
Appl. Math. Comput
.
197
(
1
), pp.
407
411
.10.1016/j.amc.2007.07.069
29.
Lazarevic
,
M.
,
2009
, “
A Finite-Time Stability Analysis of Fractional Order Time-Delay Systems: Gronwall’s Approach
,”
Math. Comput. Modell.
,
49
(
3–4
), pp.
475
481
.10.1016/j.mcm.2008.09.011
30.
Lazarevic
,
M.
, and
Debeljkovic
,
D.
,
2005
, “
Finite Time Stability Analysis of Linear Autonomous Fractional Order Systems With Delayed State
,”
Asian J. Control
,
7
(
4
), pp.
440
447
.10.1111/j.1934-6093.2005.tb00407.x
31.
Lazarevic
,
M.
,
2006
, “
Finite Time Stability Analysis of PDα Fractional Control of Robotic Time-Delay Systems
,”
Mech. Res. Commun.
,
33
(
2
), pp.
269
279
.10.1016/j.mechrescom.2005.08.010
32.
Chen
,
L. P.
,
He
,
Y. G.
,
Wu
,
R. C.
,
Chai
,
Y.
, and
Yin
,
L. S.
,
2014
, “
Robust Finite Time Stability of Fractional-Order Linear Delayed Systems With Nonlinear Perturbations
,”
Int. J. Control Autom. Syst.
,
12
(
3
), pp.
697
702
.10.1007/s12555-013-0436-7
33.
Rakkiyappan
,
R.
,
Velmurugan
,
G.
, and
Cao
,
J. D.
,
2014
, “
Finite-Time Stability Analysis of Fractional-Order Complex-Valued Memristor-Based Neural Networks With Time Delays
,”
Nonlinear Dyn.
78
(
4
), pp.
2823
2836
.10.1007/s11071-014-1628-2
34.
Wu
,
R. C.
,
Hei
,
X. D.
, and
Chen
,
L. P.
,
2013
, “
Finite-Time Stability of Fractional-Order Neural Networks With Delay
,”
Commun. Theor. Phys.
60
(
2
), pp.
189
195
.10.1088/0253-6102/60/2/08
35.
Cao
,
Y. P.
, and
Bai
,
C. Z.
,
2014
, “
Finite-Time Stability of Fractional-Order BAM Neural Networks With Distributed Delay
,”
Abstr. Appl. Anal.
,
2014
, p.
634803
.10.1155/2014/634803
36.
Pang
,
D. H.
, and
Jiang
,
W.
,
2014
, “
Finite-Time Stability of Neutral Fractional Time-Delay Systems Via Generalized Gronwalls Inequality
,”
Abstr. Appl. Anal.
,
2014
, p.
610547
10.1155/2014/610547.
37.
Wu
,
R. C.
,
Lu
,
Y. F.
, and
Chen
,
L. P.
,
2015
, “
Finite-Time Stability of Fractional Delayed Neural Networks
,”
Neurocomputing
,
149
, pp.
700
707
.10.1016/j.neucom.2014.07.060
38.
Mitrinovi
,
S. D.
,
1970
,
Analytic Inequalities
,
Springer-Verlag
,
New York
.
39.
Kuczma
,
M.
,
2009
,
An Introduction to the Theory of Functional Equations and Inequalities: Cauchys Equation and Jensens Inequality, Birkhauser, Boston.
40.
Lakshmikantham
,
V.
,
2008
, “
Theory of Fractional Functional Differential Equations
,”
Nonlinear Anal. Theory Methods Appl.
,
69
(
10
), pp.
3337
3343
.10.1016/j.na.2007.09.025
41.
Yang
,
X.
,
Song
,
Q.
,
Liu
,
Y.
, and
Zhao
,
Z.
,
2015
, “
Finite-Time Stability Analysis of Fractional-Order Neural Networks With Delay
,”
Neurocomputing
,
152
, pp. 19–26.10.1016/j.neucom.2014.11.023
42.
Ke
,
Y.
, and
Miao
,
C.
,
2014
, “
Stability Analysis of Fractional-Order Cohen-Grossberg Neural Networks With Time Delay
,”
Inter. J. Computer Math
(in press).10.1080/00207160.2014.935734
43.
Wang
,
H.
,
Yu
,
Y. G.
,
Wen
,
G. G.
, and
Zhang
,
S.
,
2014
, “
Stability Analysis of Fractional-Order Neural Networks With Time Delay
,”
Neural Process Lett.
(to be published).
10.1007/s11063-014-9368-3
44.
Wang
,
H.
,
Yu
,
Y. G.
, and
Wen
,
G. G.
,
2014
, “
Stability Analysis of Fractional-Order Hopfield Neural Networks With Time Delays
,”
Neural Networks
,
55
, pp.
98
109
.10.1016/j.neunet.2014.03.012
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