Finite time stability of linear fractional order dynam-ical system with variable delays


  • Jackreece P. C Department of mathematics/StatisticsUniversity of Port Harcourt, Nigeria
  • Aniaku S. E





Dynamical System, Fractional Calculus, Finite Time Stability, Time Varying Delays.


In this paper, some sufficient condition ensuring finite time stability are derived for a class of linear fractional order dynamical system with variable delay using generalized Gronwall inequality as well as classical Bellman-Gronwall inequality.


[1] F. Amato, M. Ariola, C. Cosentino, C. T. Abdallah, and P. Dorato, Necessary and sufficient conditions for finite-time stability of linear systems. Proc. American Control Conference, Denver, Colorado, USA (2003), 4452–4456.

[2] E. K. Boukas and N. F. Al-Muthairi, Delay-dependent stabilization of singular linear systems with delays. International Journal of Innovative Computing, Information and Control, 2(2), (2006), 283–291.

[3] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. R. Astron. Soc. 13, (1967), 529-539.

[4] Y. Q. Chen and K. L. Moore, Analytical Stability bound for a class of delayed fractional order dynamic system, Nonlin. Dyn. Vol. 29, (202), 191-200.

[5] S. Das, Functional fractional calculus, Academic press, San Diego (1999).

[6] D. L. Debeljkovi, M. P. Lazarevic, D. Koruga, S. A. Milinkovic, M. B. Jovanovic, and L. A. Jacic, Further Results on Non-Lyapunov stability of the linear nonautonomous systems with delayed state, Facta Universitatis, Mechanics, Automatic Control and Robotic series, 3(11), (2001), 231–241.

[7] D. L. Debeljkovi, S. B. Stojannovic, G. V. Simeunovic and N. J. Dimitrijevic, Further results on stability of singular time delay systems in the sense of non-Lyapunov: A new delay dependent conditions, Automatic Control and Information Sciences, 2(1), (2014) 13–19.

[8] D. L. Debeljkovic, S. B. Stojanovic, and A. M. Jovanovic, Further results on finite time and practical stability of linear continuous time delay systems. FME Transactions, 41(3), (2013), 241–249.

[9] W. Deng, Smoothness and stability of the solution for nonlinear fractional differential equation, Nonlinear Anal. 72(3-4), (2010), 1768- 1777.

[10] W. Deng, and C. Li, Numerical schemes for fractional ordinary differential equations. Numerical Modelling, Dr. Peep Miidla (Ed), ISBN: 978-953-51-0219-9, InTech, Retrieved from, (2012).

[11] W. Deng, C. Li, and J. Lü, Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 48(4), (2007), 409–416.

[12] P. Denghao, and J. Wei, Finite time stability analysis of fractional singular time delay systems, Advances in difference equation, 2014/1/259.

[13] C. A. Desoer, and M. Vidyasagar, (1975). Feedback Systems: Input- Output properties. Academic Press, New York, (1975).

[14] L. Dugard, and E. I. Verriest, Stability and control of time delay systems Springer-Verlag, (1997).

[15] N. M. F. Ferreira, F. B. Duarte, F. Miguel, M. G. Marcos, and J. A. T. Machado, Application of fractional calculus in the dynamical analysis and control of mechanical manipulators. Fractional calculus and applied analysis, 11(1), (2008), 91-113.

[16] J. K. Hale, Functional differential equations, Springer, New York, (1971).

[17] K. Gu, V. I. Kharitonov, and J. Chen, Stability of time delay systems, Boston, Birkhauser, (2003).

[18] R. Hilfer, Application of fractional calculus in Physics, World Scientific Singapore, (2000)

[19] Y. Hong, Y. Xu, and J. Huang, Finite time control for robot manipulation, Systems and Control Letters, 46, (2002), 243–253.

[20] Y. Hong, Finite time Stabilization and Stabilizability of a class of controllable systems. Systems and Control Letters, 46, (2002), 231– 236.

[21] G. D. Hu, and M. Z. Liu, The weighted logarithm matrix norm and bounds of the matrix exponential. Linear Algebra Appl., 390, (2004), 145–154.

[22] G. D. Hu and T. Mitsui, Bounds of the matrix eigenvalues and its exponential by lyapunov equation. Kybernetika, 48(5), (2012), 865– 878.

[23] P. C. Jackreece, Finite time stability of linear control system with multiple delays, Control Theory and Informatics, Vol. 6(3), (2016), 69-72.

[24] P. C. Jackreece, Finite time stability of linear control system with constant delay in the state, International Journal of Mathematics and Statistics Invention(IJMI), Vol.5(2), (2017), 49-52.

[25] H. Jia, X. Cao, X. Yu, and P. Zhang, A simple approach to determine power system delay margin, in Proceedings of the IEEE PES general meeting, Montreal, Quebec, (2007), 1-7

[26] N. A. Kablar, and D. L. Debeljkovic, Non-Lyapunov stability of linear singular systems: Matrix measure approach. Preprints 5th IFAC Symposium on Low Cost Automation, Shenyang, China, September 8- 10, TS13, (1998), 16–20.

[27] E. Kaslik, and S. Sivasundaram, An analytical and numerical methods for the stability analysis of linear fractional delay differential equations Journal of computational and Applied Mathematics, 236(16), (2012), 4027-4041.

[28] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, New York, (2006).

[29] V. V. Kulish, and J. L. Lage, Application of fractional calculus to Mechanics, Journal of Fluids Engineering, Vol. 124, No. 3, (2002), 803-806.

[30] M. P. Lazarevic, Finite time stability analysis of fractional control of robotic time delay systems, Mech. Res. Commun.33 (2), (2006), 269- 279.

[31] M. P. Lazarevic, and D. L. Debeljkovi, Finite time stability analysis of linear autonomous fractional order systems with delayed state, Asian Journal of Control, 7(4), (2005), 440–447.

[32] M. P. Lazarevic, A. Obradovic and V. Vasic, Robust finite time stability analysis of fractional order time delay systems: New results, Advances in dynamical systems and control (2010), 101-106, retrieved from

[33] T. N. Lee, and S. Diant, Stability of time delay system, IEEE Trans. Automat. Control AC 31(3), (1981), 951-953.

[34] C. Li, K. Chen, J. Lu, and R. Tang, Stability and Stabilization analysis of fractional order linear systems subject to actuator saturation and distribution, IFAC paper online 50-1, (2017), 9718-9723.

[35] Y. Li, Y. Chen, and I. Podlubny, Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag- Leffler stability, Comput. Math. Appl. 59(5), (2010), 1810-1821.

[36] P. L. Liu, Exponential stability for linear time delay systems with delay dependence. Journal of the Franklin Institute, 340, (2003), 481– 488.

[37] P. L. Liu, Robust exponential stabilization for uncertain systems with state and control Delay. International Journal of Systems Science, 34(12–13), (2003), 675–682.

[38] D. Matignon, Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications, (1996), 963–968.

[39] F. A. Mohd, S. Manoj, and J. Renu, An application of fractional calculus in electrical engineering, Advanced Engineering Technology and Application, 5(2), (2016), 41-45.

[40] S. Momani, and S. Hadid, Lyapunov stability solution of fractional integrodifferential equations, Int. J. math. Math. Sc. 47, (2004), 2503- 2507.

[41] T. Mori, Criteria for Asymptotic stability of linear time delay systems, IEEE trans. Automat. Control, AC 30, (1985), 158-161.

[42] E. Moulay, M. Dambrine, N. Yeganefar, and W. Perruquetti, Finite Time Stability and Stabilization of time delay systems. Systems and control letters, 57, (2008), 561–566.

[43] M. Naber, Time fractional Schrodinger equation, J. Maths. Phys. 45, (2004), 3339-3352.

[44] K. B. Oldham, and J. Spanier, The fractional calculus, Academic Press, New York, (1974).

[45] I. Podlubny, Fractional differential equation, Academic Press, San Diego, (1999).

[46] Y. Ryabov, and A. Puzenko, Damped oscillation in view of the fractional oscillator equation, Phys. Rev., 66, (2002), 184-201.

[47] J. Sabatier, M. Moze, and C. Farges, Stability conditions for fractional order systems, Comput. Math. Appl. 59(5), (2010), 1594-1609.

[48] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives theory and applications, Gordon and Breach, New York, (1993).

[49] Y. Shen, L. Zhu, and Q. Guo, Finite time boundedness analysis of uncertain neural networks with time delay: An LMI approach. Proc. 4th Int Symp. Neural Networks, Nanjing, China. (2007), 904–909.

[50] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, Vol. 23, (2002), 397-404.

[51] C. Tunç, and E. Biçer, Stability to a Kind of Functional differential equations of second order with multiple delays by fixed points, Abstract and Applied Analysis, 2014(3).

[52] H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328(2007), 1075-1081.

[53] F. Yu, Integrable coupling system of fractional differential equations, Eur. Phys. J. Spec. Top., (2011), 193, 27-47.

[54] M. Zavarei, and M. Jamshidi, Time Delay systems: Analysis, Optimization and Applications, North-Holland, Amsterdam, (1987).

[55] X. Zhang, Some results of linear fractional order time delay system, Appl. Math. Comp., 197, (2008), 407-411.

[56] S. Zhou, and J. Lam, Robust stabilization of delay singular systems with linear fractional parametric uncertainties. Circuits Systems Signal Processing, 22(6), (2003), 578–588.

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