Mittag-Leffler stability of impulsive differential equations of fractional order
Author:
Ivanka M. Stamova
Journal:
Quart. Appl. Math. 73 (2015), 525-535
MSC (2010):
Primary 26A33, 34A37, 34D20; Secondary 44A10
DOI:
https://doi.org/10.1090/qam/1394
Published electronically:
June 12, 2015
MathSciNet review:
3400757
Full-text PDF Free Access
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References |
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Additional Information
Abstract: In this paper we consider a nonlinear system of impulsive differential equations of fractional order. Applying the definition of Mittag-Leffler stability introduced by Podlubny and his co-authors and the fractional Lyapunov method, we give sufficient conditions for Mittag-Leffler stability and uniform asymptotic stability of the zero solution of the system under consideration.
References
- Bashir Ahmad and Juan J. Nieto, Existence of solutions for impulsive anti-periodic boundary value problems of fractional order, Taiwanese J. Math. 15 (2011), no. 3, 981–993. MR 2829892, DOI https://doi.org/10.11650/twjm/1500406279
- M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publishing Corporation, New York, 2006. MR 2322133
- Jianxin Cao and Haibo Chen, Impulsive fractional differential equations with nonlinear boundary conditions, Math. Comput. Modelling 55 (2012), no. 3-4, 303–311. MR 2887377, DOI https://doi.org/10.1016/j.mcm.2011.07.037
- Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847
- Michal Fečkan, Yong Zhou, and JinRong Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 7, 3050–3060. MR 2880474, DOI https://doi.org/10.1016/j.cnsns.2011.11.017
- R. Hilfer (ed.), Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1890104
- V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, vol. 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1082551
- V. Lakshmikantham, S. Leela, and M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal. 12 (2008), no. 4, 365–376. MR 2494983
- Yan Li, YangQuan Chen, and Igor Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), no. 8, 1965–1969. MR 2879525, DOI https://doi.org/10.1016/j.automatica.2009.04.003
- Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
- Ivanka Stamova, Stability analysis of impulsive functional differential equations, De Gruyter Expositions in Mathematics, vol. 52, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. MR 2604930
- Ivanka Stamova and Gani Stamov, Lipschitz stability criteria for functional differential systems of fractional order, J. Math. Phys. 54 (2013), no. 4, 043502, 11. MR 3088804, DOI https://doi.org/10.1063/1.4798234
- Ivanka Stamova and Gani Stamov, Stability analysis of impulsive functional systems of fractional order, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 3, 702–709. MR 3111646, DOI https://doi.org/10.1016/j.cnsns.2013.07.005
References
- Bashir Ahmad and Juan J. Nieto, Existence of solutions for impulsive anti-periodic boundary value problems of fractional order, Taiwanese J. Math. 15 (2011), no. 3, 981–993. MR 2829892 (2012e:34035)
- M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, vol. 2, Hindawi Publishing Corporation, New York, 2006. MR 2322133 (2008f:34001)
- Jianxin Cao and Haibo Chen, Impulsive fractional differential equations with nonlinear boundary conditions, Math. Comput. Modelling 55 (2012), no. 3-4, 303–311. MR 2887377, DOI https://doi.org/10.1016/j.mcm.2011.07.037
- Kai Diethelm, The analysis of fractional differential equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. MR 2680847 (2011j:34005)
- Michal Fečkan, Yong Zhou, and JinRong Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 7, 3050–3060. MR 2880474, DOI https://doi.org/10.1016/j.cnsns.2011.11.017
- Applications of fractional calculus in physics, edited by R. Hilfer, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. MR 1890104 (2002j:00009)
- V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, vol. 6, World Scientific Publishing Co. Inc., Teaneck, NJ, 1989. MR 1082551 (91m:34013)
- V. Lakshmikantham, S. Leela, and M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal. 12 (2008), no. 4, 365–376. MR 2494983 (2010e:34122)
- Yan Li, YangQuan Chen, and Igor Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), no. 8, 1965–1969. MR 2879525 (2012m:34096), DOI https://doi.org/10.1016/j.automatica.2009.04.003
- Igor Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press Inc., San Diego, CA, 1999. MR 1658022 (99m:26009)
- I. M. Stamova, Stability analysis of impulsive functional differential equations, Walter de Gruyter, Berlin, New York, 2009. MR 2604930 (2011f:34004)
- Ivanka Stamova and Gani Stamov, Lipschitz stability criteria for functional differential systems of fractional order, J. Math. Phys. 54 (2013), no. 4, 043502, 11. MR 3088804, DOI https://doi.org/10.1063/1.4798234
- Ivanka Stamova and Gani Stamov, Stability analysis of impulsive functional systems of fractional order, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 3, 702–709. MR 3111646, DOI https://doi.org/10.1016/j.cnsns.2013.07.005
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Additional Information
Ivanka M. Stamova
Affiliation:
Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249
MR Author ID:
329335
Email:
ivanka.stamova@utsa.edu
Keywords:
Mittag-Leffler stability,
impulsive fractional differential equations,
Lyapunov functions,
comparison principle
Received by editor(s):
October 17, 2013
Received by editor(s) in revised form:
January 9, 2014
Published electronically:
June 12, 2015
Article copyright:
© Copyright 2015
Brown University