Abstract
Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable and then under the action of small (either linear or nonlinear) nonautonomous perturbations, the trivial solution of the perturbed system is also asymptotically stable.
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References
Adrianova, L.Ya: Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs, vol. 46. American Mathematical Society, Providence (1995)
Băleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010)
Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)
Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. RWA 13, 287–298 (2012)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McCrow-Hill, New York (1955)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226, 157–168 (2014)
Cong, N.D., Doan, T.S., Tuan, H.T.: On fractional Lyapunov exponent for solutions of linear fractional differential equations. Fract. Calc. Appl. Anal. 17, 285–306 (2014)
Cong, N.D., Doan, T.S., Tuan, H.T., Siegmund, S.: Structure of the fractional Lyapunov spectrum for linear fractional differential equations. Adv. Dyn. Syst. Appl. 9, 149–159 (2014)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Linearized asymptotic stability for fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 39, 1–13 (2016)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn. 86, 1885–1894 (2016)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Erratum to: On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn. 86, 1895–1895 (2016)
Deng, W.H., Li, C.P., Lü, J.H.: Stability analysis of linear fractional differential systems with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)
Diethelm, K.: 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 Heidelberg (2010)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin Heidelberg (2014)
Graef, J.R., Grace, S.R., Tunç, E.: Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivatives. Fract. Calc. Appl. Anal. 20, 71–87 (2017)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)
Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn with Applications. Academic Press, San Diego (1985)
Losada, J., Nieto, J.J., Puorhadi, E.: On the attractivity of solutions for a class of multi-term fractional functional differential equations. J. Comput. Appl. Math. 312, 2–12 (2017)
Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Special Top. 193, 27–47 (2011)
Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp 963–968, Lille (1996)
Podlubny, I.: 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 (1999)
Qian, D., Li, C., Agarwal, R.P., Wong, P.J.Y.: Stability analysis of fractional differential systems with Riemann–Liouville derivative. Math. Comput. Model. 52, 862–874 (2010)
Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, 1594–1609 (2010)
Tisdell, C.C.: On the application of sequential and fixed-point method to fractional differential equations of arbitrary order. J. Integral Equ. Appl. 24, 283–319 (2012)
Tuan, H.T.: On some special properties of Mittag-Leffler functions. arxiv:http://arXiv.org/abs/1708.02277 (2017)
Wen, X. -J., Wu, Z. -M., Lu, J. -G.: Stability analysis of a class of nonlinear fractional–order systems. IEEE Trans. Circ. Syst. II Express Briefs 55, 1178–1182 (2008)
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The authors are grateful to the referees for reading this paper and his/her comments and suggestions which helped to improve its content.
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This research of the authors is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2017.01. The authors are grateful to the referees for reading this paper and his/her comments and suggestions which helped to improve its content.
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Cong, N.D., Doan, T.S. & Tuan, H.T. Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations. Vietnam J. Math. 46, 665–680 (2018). https://doi.org/10.1007/s10013-018-0272-4
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DOI: https://doi.org/10.1007/s10013-018-0272-4