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STABILITY PROPERTIES IN IMPULSIVE DIFFERENTIAL SYSTEMS OF NON-INTEGER ORDER

Kang, Bowon;Koo, Namjip

  • Received : 2018.02.13
  • Accepted : 2018.04.09
  • Published : 2019.01.01

Abstract

In this paper we establish some new explicit solutions for impulsive linear fractional differential equations with impulses at fixed times, which provides a handy tool in deriving singular integral-sum inequalities and an impulsive fractional comparison principle. Thus we study the Mittag-Leffler stability of impulsive differential equations with the Caputo fractional derivative by using the impulsive fractional comparison principle and piecewise continuous functions of Lyapunov's method. Also, we give some examples to illustrate our results.

Keywords

aimpulsive fractional differential equation;Mittag-Leffler system;impulsive fractional comparison principle;piecewise continuous auxiliary function

References

  1. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1989.
  2. D. D. Bainov and P. S. Simeonov, Impulsive differential equations, translated from the Bulgarian manuscript by V. Covachev [V. Khr. Kovachev], Series on Advances in Mathematics for Applied Sciences, 28, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  3. M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 2009 (2009), no. 10, 11 pp.
  4. M. Caputo, Linear models of dissipation whose Q is almost frequency independent. II, Geophysical J. Royal Astronomical Soc. 13 (1967), no. 5, 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  5. S. K. Choi, B. Kang, and N. Koo, Stability for fractional differential equations, Proc. Jangjeon Math. Soc. 16 (2013), no. 2, 165-174.
  6. S. K. Choi, B. Kang, and N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal. 2014 (2014), Art. ID 631419, 6 pp.
  7. S. K. Choi and N. Koo, The monotonic property and stability of solutions of fractional differential equations, Nonlinear Anal. 74 (2011), no. 17, 6530-6536. https://doi.org/10.1016/j.na.2011.06.037
  8. S. K. Choi and N. Koo, Variationally stable impulsive differential systems, Dyn. Syst. 30 (2015), no. 4, 435-449. https://doi.org/10.1080/14689367.2015.1068742
  9. S. K. Choi and N. Koo, A note on linear impulsive fractional differential equations, J. Chungcheong Math. Soc. 28 (2015), 583-590. https://doi.org/10.14403/jcms.2015.28.4.583
  10. S. K. Choi and N. Koo, A converse theorem on h-stability via impulsive variational systems, J. Korean Math. Soc. 53 (2016), no. 5, 1115-1131. https://doi.org/10.4134/JKMS.j150428
  11. S. K. Choi, N. Koo, and C. Ryu, h-stability of linear impulsive differential equations via similarity, J. Chungcheong Math. Soc. 24 (2011), 393-400.
  12. S. K. Choi, N. Koo, and C. Ryu, Stability of linear impulsive differential equations via $t_{\infty}$-similarity, J. Chung-cheong Math. Soc. 26 (2013), 811-819. https://doi.org/10.14403/jcms.2013.26.4.811
  13. M. Feckan, Y. Zhou, and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 7, 3050-3060. https://doi.org/10.1016/j.cnsns.2011.11.017
  14. B. Kang and N. Koo, A note on generalized singular Gronwall inequalities, J. Chungcheong Math. Soc. 31 (2018), 161-166.
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  16. G. K. Kulev and D. D. Bainov, Lipschitz stability of impulsive systems of differential equations, Internat. J. Theoret. Phys. 30 (1991), no. 5, 737-756. https://doi.org/10.1007/BF00671986
  17. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. 69 (2008), no. 10, 3337-3343. https://doi.org/10.1016/j.na.2007.09.025
  18. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
  19. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. II, Academic Press, New York, 1969.
  20. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers Ltd., 2009.
  21. V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), no. 8, 2677-2682. https://doi.org/10.1016/j.na.2007.08.042
  22. Y. Li, Y. Chen, and I. Podlubny, Mittag-Leer stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), no. 8, 1965-1969. https://doi.org/10.1016/j.automatica.2009.04.003
  23. Y. Li, Y. Chen, and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leer stability, Comput. Math. Appl. 59 (2010), no. 5, 1810-1821. https://doi.org/10.1016/j.camwa.2009.08.019
  24. G. M. Mittag-Leer, Sur l'integrale de Laplace-Abel, C. R. Acad. Sci. Paris (Ser. II) 136 (1902), 937-939.
  25. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  26. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.
  27. P. S. Simeonov and D. D. Bainov, Exponential stability of the solutions of singularly perturbed systems with impulse effect, J. Math. Anal. Appl. 151 (1990), no. 2, 462-487. https://doi.org/10.1016/0022-247X(90)90161-8
  28. I. Stamova, Global stability of impulsive fractional differential equations, Appl. Math. Comput. 237 (2014), 605-612.
  29. I. Stamova, Global Mittag-Leer stability and synchronization of impulsive fractional-order neural networks with time-varying delays, Nonlinear Dynam. 77 (2014), no. 4, 1251-1260. https://doi.org/10.1007/s11071-014-1375-4
  30. I. Stamova, Mittag-Leer stability of impulsive differential equations of fractional order, Quart. Appl. Math. 73 (2015), no. 3, 525-535. https://doi.org/10.1090/qam/1394
  31. I. Stamova and G. Stamov, Stability analysis of impulsive functional systems of fractional order, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 3, 702-709. https://doi.org/10.1016/j.cnsns.2013.07.005
  32. J. Wang, Y. Zhou, and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (2012), no. 10, 3389-3405. https://doi.org/10.1016/j.camwa.2012.02.021

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)