Finite-time stability analysis of fractional singular time-delay systems. (English) Zbl 1348.34135

Summary: This paper studies the finite-time stability of fractional singular time-delay systems. First, by the method of the steps, we discuss the existence and uniqueness of the solutions for the equivalent systems to the fractional singular time-delay systems. Furthermore, we give the Mittag-Leffler estimation of the solutions for the equivalent systems and obtain the sufficient conditions of the finite-time stability for the original systems.


34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
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[1] Miller KS, Boss B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993. · Zbl 0789.26002
[2] Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999. · Zbl 0924.34008
[3] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006. · Zbl 1092.45003
[4] Kaczorek T: Selected Problems of Fractional Systems Theory. Springer, Berlin; 2011. · Zbl 1221.93002
[5] Weitzner H, Zaslavsky GM: Some applications of fractional equations.Commun. Nonlinear Sci. Numer. Simul. 2003,8(3-4):273-281. · Zbl 1041.35073
[6] Machado JT, Kiryakova V, Mainardi F: Recent history of fractional calculus.Commun. Nonlinear Sci. Numer. Simul. 2011,16(3):1140-1153. · Zbl 1221.26002
[7] Machado JAT, Costa AC, Quelhas MD: Fractional dynamics in DNA.Commun. Nonlinear Sci. Numer. Simul. 2011,16(8):2963-2969. · Zbl 1218.92038
[8] Zhou Y, Jiao F, Li J: Existence and uniqueness for fractional neutral differential equations with infinite delay.Nonlinear Anal. 2009,71(7-8):3249-3256. · Zbl 1177.34084
[9] Zhou X, Jiang W, Hu L: Controllability of a fractional linear time-invariant neutral dynamical system.Appl. Math. Lett. 2013,26(4):418-424. · Zbl 1258.93030
[10] Deng W: Smoothness and stability of the solutions for nonlinear fractional differential equations.Nonlinear Anal. 2010,72(3-4):1768-1777. · Zbl 1182.26009
[11] Li Y, Chen Y, Podlubny I: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability.Comput. Math. Appl. 2010,59(5):1810-1821. · Zbl 1189.34015
[12] Sabatier J, Moze M, Farges C: LMI stability conditions for fractional order systems.Comput. Math. Appl. 2010,59(5):1594-1609. · Zbl 1189.34020
[13] Lazarević MP:Finite time stability analysis of[InlineEquation not available: see fulltext.]fractional control of robotic time-delay systems.Mech. Res. Commun. 2006,33(2):269-279. · Zbl 1192.70008
[14] Zhang X: Some results of linear fractional order time-delay system.Appl. Math. Comput. 2008,197(1):407-411. · Zbl 1138.34328
[15] Lazarević MP, Spasić AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach.Math. Comput. Model. 2009,49(3-4):475-481. · Zbl 1165.34408
[16] Pang, D.; Jiang, W., Finite-time stability of neutral fractional time-delay systems via generalized Gronwall’s inequality, No. 2014 (2014)
[17] Wang J, Lv L, Zhou Y: New concepts and results in stability of fractional differential equations.Commun. Nonlinear Sci. Numer. Simul. 2012,17(6):2530-2538. · Zbl 1252.35276
[18] Dai L: Singular Control Systems. Springer, Berlin; 1989. · Zbl 0669.93034
[19] Fridman E: Stability of linear descriptor systems with delay: a Lyapunov-based approach.J. Math. Anal. Appl. 2002,273(1):24-44. · Zbl 1032.34069
[20] Jiang W: Eigenvalue and stability of singular differential delay systems.J. Math. Anal. Appl. 2004,297(1):305-316. · Zbl 1063.34052
[21] Campbell SL, Linh VH: Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions.Appl. Math. Comput. 2009,208(2):397-415. · Zbl 1169.65079
[22] Liu X, Zhong S, Ding X: A Razumikhin approach to exponential admissibility of switched descriptor delayed systems.Appl. Math. Model. 2014,38(5-6):1647-1659. · Zbl 1427.93205
[23] MacCluer BD: Elementary Functional Analysis. Springer, New York; 2009. · Zbl 1170.46002
[24] Ye H, Gao J, Ding Y: A generalized Gronwall inequality and its application to a fractional differential equation.J. Math. Anal. Appl. 2007,328(2):1075-1081. · Zbl 1120.26003
[25] Jiang W: The constant variation formulae for singular fractional differential systems with delay.Comput. Math. Appl. 2010,59(3):1184-1190. · Zbl 1189.34153
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