Abstract
In this paper, Lyapunov direct method is employed to study the stability problem of Caputo-type fractional nonautonomous systems with order between 1 and 2. By utilizing Riemann–Liouville fractional integral, some sufficient conditions on stability are derived. In the proof of the obtained results, Bellman–Gronwall’s inequality, the generalized Bihari inequality and estimates of Mittag-Leffler functions are employed. Besides, two examples and corresponding numerical simulations are provided to show the validity and feasibility of the proposed stability criterion.
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Valério, D., Trujillo, J.J., Rivero, M., Machado, J.T., Baleanu, D.: Fractional calculus: a survey of useful formulas. Eur. Phys. J. Spec. Top. 222, 1827–1846 (2013)
Machado, J.T.: Fractional order description of DNA. Appl. Math. Model. 39, 4095–4102 (2015)
Valério, D., Machado, J., Kiryakova, V.: Some pioneers of the applications of fractional calculus. Fract. Calc. Appl. Anal. 17, 552–578 (2014)
Machado, J.T.: Fractional derivatives: probability interpretation and frequency response of rational approximations. Commun. Nonlinear Sci. Numer. Simul. 14, 3492–3497 (2009)
Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of IMACS-SMC, 2, 963–968, Lille France (1996)
Qian, D., Li, C.P., Agarwal, R.P., Wong, P.J.: Stability analysis of fractional differential system with Riemann–Liouville derivative. Math. Comput. Model. 52, 862–874 (2010)
Tavazoei, M.S., Haeri, M.: A note on the stability of fractional order systems. Math. Comput. Simul. 79, 1566–1576 (2009)
Deng, W.H., Li, C.P., Lü, J.H.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)
Rivero, M., Rogosin, S.V., Tenreiro Machado, J.A., Trujillo, J.J.: Stability of fractional order systems. Math. Probl. Eng. 2013, 356215 (2013). doi:10.1155/2013/356215
Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)
Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)
Li, Y., Chen, Y.Q., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)
Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)
Zhang, F.R., Li, C.P., Chen, Y.Q.: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ. 2011, 635165 (2011). doi:10.1155/2011/635165
Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Fractional diffusion: probability distributions and random walk models. Phys. A 305, 106–112 (2002)
Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)
Gafiychuk, V., Datsko, B.: Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 59, 1101–1107 (2010)
Beghin, L., Orsingher, E.: The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calc. Appl. Anal. 6, 187–204 (2003)
Zhang, W., Cai, X., Holm, S.: Time-fractional heat equations and negative absolute temperatures. Comput. Math. Appl. 67, 164–171 (2014)
Chen, L.P., Chai, Y., Wu, R.C., Yang, J.: Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Trans. Circuits Syst. II Express Briefs 59, 602–606 (2012)
Chen, L.P., He, Y.G., Chai, Y., Wu, R.C.: New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dyn. 75, 633–641 (2014)
Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, New York (2006)
Ye, H.P., Gao, J.M., Ding, Y.S.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)
Bainov, D.D., Simeonov, P.S.: Integral Inequalities and Applications. Springer, Berlin (1992)
Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New Jewsey (2002)
Shahri, E.S.A., Alfi, A., Machado, J.T.: An extension of estimation of domain of attraction for fractional order linear system subject to saturation control. Appl. Math. Lett. 47, 26–34 (2015)
De la Sen, M.: About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory. Fixed Point Theory Appl. 2011, 867932 (2011). doi:10.1155/2011/867932
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful suggestions and comments. This work was supported by the National Science and Technology Major Project (No. 2012CB821202), the National Nature Science Foundation ( No. 61327807) and Beijing Natural Science Foundation (No. 4122043).
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Guo, Y., Ma, B. Extension of Lyapunov direct method about the fractional nonautonomous systems with order lying in \(\mathbf{(1,2)}\) . Nonlinear Dyn 84, 1353–1361 (2016). https://doi.org/10.1007/s11071-015-2573-4
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DOI: https://doi.org/10.1007/s11071-015-2573-4