Global convergence analysis of impulsive fractional order difference systems

• Autorzy: He D. , Xu L.

Identyfikatory

DOI

10.24425/bpas.2018.124275

• Języki publikacji: EN

Abstrakty

EN

This paper is designed to deal with the convergence and stability analysis of impulsive Caputo fractional order difference systems. Using the Lyapunov functions, the Z-transforms of Caputo difference operators, and the properties of discrete Mittag-Leffler functions, some effective criteria are derived to guarantee the global convergence and the exponential stability of the addressed systems.

Słowa kluczowe

EN

impulsive fractional difference systems; discrete Mittag-Leffler function; Z-transform; convergence

PL

konwergencja; funkcja Mittag-Lefflera; transformacja Z

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2018
• Tom: Vol. 66, nr 5
• Strony: 599--604
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

He D.

• Department of Mathematics, Zhejiang International Studies University, HangZhou, 310023, PR China

autor

Xu L.

• Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, PR China

Bibliografia

• [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [2] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
• [3] J. Huo, H. Zhao, and L. Zhu, “The effect of vaccines on backward bifurcation in a fractional order HIV model”, Nonlinear Analysis: Real World Applications 26, 289–305 (2015).
• [4] J. Huo and H. Zhao, “Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks”, Physica A 448, 41–56 (2016).
• [5] I. Stamova, “On the Lyapunov theory for functional differential equations of fractional order”, Proc. Amer. Math. Soc. 144(4), 1581–1593 (2016).
• [6] T. Jankowski, “Fractional equations of Volterra type involving a Riemann-Liouville derivative”, Appl. Math. Lett. 26, 344–350 (2013).
• [7] T. Jankowski, “Boundary problems for fractional differential e-quations”, Appl. Math. Lett. 28, 14–19 (2014).
• [8] B. Ahmad, “Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations”, Appl. Math. Lett. 23, 390–394 (2010).
• [9] B. Ahmad, “Sharp estimates for the unique solution of two-point fractional-order boundary value problems”, Appl. Math. Lett. 65, 77–82 (2017).
• [10] W. Zhu, W. Li, P. Zhou, and C. Yang, “Consensus of fractionalorder multi-agent systems with linear models via observer-type protocol”, Neurocomputing 230, 60–65 (2017).
• [11] P. Zhou and W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Analysis: Real World Applications 12, 811–816 (2011).
• [12] I.K. Dassios, D.I. Baleanu, and G.I. Kalogeropoulos, “On nonhomogeneous singular systems of fractional nabla difference equations”, Appl. Math. Comput. 227, 112–131 (2014).
• [13] I.K. Dassios and D.I. Baleanu, “Duality of singular linear systems of fractional nabla difference equations”, Appl. Math. Model. 39, 4180–4195 (2015).
• [14] M. Wyrwas and D. Mozyrska, On Mittag-Leffler Stability of Fractional Order Difference Systems, Advances in Modelling and Control of Non-integer Order Systems. Springer International Publishing, 2015: 209–220.
• [15] D. Mozyrska, E. Pawłuszewicz, and M. Wyrwas, “Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation”, Int. J. Syst. Sci. 48(4), 788–794 (2017).
• [16] D. Mozyrska and M. Wyrwas, “The Z-Transform method and delta type fractional difference operators”, Discrete Dyn. Nat. Soc. ID. 852734, 2015.
• [17] D. Mozyrska and E. Pawłuszewicz, “Observability of linear q-difference fractional-order systems with finite initial memory.” Bull. Pol. Ac.: Tech. 58(4), 601–605 (2010).
• [18] T. Kaczorek, “Practical stability of positive fractional discretetime linear systems”, Bull. Pol. Ac.: Tech. 56 (4), 313–317 (2008).
• [19] T. Kaczorek, “Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers”, Bull. Pol. Ac.: Tech. 65 (3), 305–311 (2017).
• [20] P. Sopasakis and H. Sarimveis, “Stabilising model predictive control for discrete-time fractional-order systems”, Automatica 75, 24–31 (2017).
• [21] I. Stamova and J. Henderson, “Practical stability analysis of fractional-order impulsive control systems”, ISA trans. 64, 77–85 (2016).
• [22] I. Stamova, “Global stability of impulsive fractional differential equations”, Appl. Math. Comput. 237, 605–612 (2014).
• [23] I. Stamova, “Mittag-Leffler stability of impulsive differential equations of fractional order”, Q. Appl. Math. 73(3), 525–535 (2015).
• [24] L. Xu, H. Hu, and F. Qin, “Ultimate boundedness of impulsive fractional differential equations”, Appl. Math. Lett. 62, 110–117 (2016).
• [25] L. Xu, J. Li, and S.S. Ge. “Impulsive stabilization of fractional differential systems.” ISA trans. 70, 125–131 (2017).
• [26] L. Xu and W. Liu, “Ultimate boundedness of impulsive frac- tional delay differential equations”, Appl. Math. Lett. 79, 58–66 (2018).
• [27] M. Benchohra and B.A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations”, Electron. J. Differ. Eq. 10, 1–11 (2009).
• [28] S. Heidarkhani and A. Salari, “Nontrivial solutions for impulsive fractional differential systems through variational methods”, Comput. Math. Appl., (to be published).

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).