An Approach to the Stability of Systems of Differential Equations of Fractional Order with Time Delay
Abstract
In this paper, a stability study is carried out for systems of fractional order differential equations with time delay. These systems of differential equations of fractional order with time delay are of current interest due to their applicability since several models from different areas of knowledge, where the current development depends to a great extent on the history that has elapsed, are faithful reflections of systems of differential equations with time delay. The article initially presents and describes the fundamental concepts of the study of stability of systems of ordinary differential equations of fractional order, both linear and nonlinear, and then these concepts are presented for differential equations with time delay. The results of this research provide fundamental tools in the study of the stability of systems of differential equations of fractional order with time delay that can be applied to a variety of models from the applied sciences.
Keywords: fractional analysis, fractional differential equations, stability of fractional differential equations, fractional differential equations with delay.
Full Text:
PDFReferences
BERTRAM R. Fractional Calculus and Its Applications. Springer, Berlin, Heidelberg, 1975. https://doi.org/10.1007/BFb0067095
SÁNCHEZ J. M. Génesis y desarrollo del Cálculo Fraccional. Revista “Pensamiento Matemático”, 2011, 1: 1–15. http://www2.caminos.upm.es/Departamentos/matematicas/revistapm/revista_impresa/numero_1/genesis_y_desarrollo_del_calculo_fraccional.pdf
PODLUBNY I. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, 2001. https://arxiv.org/abs/math/0110241
HILFER R. Applications of Fractional Calculus in Physics. World Scientific Publishing, 2000. https://doi.org/10.1142/3779
LOVERRO A. Fractional Calculus: History, Definitions and Applications for the Engineer. Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana, 2004.
PODLUBNY I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, 1998. https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9
KILBAS A., BLANCA B., and TRUJILLO J. J. Cálculo fraccionario y ecuaciones diferenciales fraccionarias. Universidad Nacional de Educación a Distancia, 2003. https://dialnet.unirioja.es/servlet/libro?codigo=97917
MATIGNON D. Stability Results for Fractional Differential Equations with Applications to Control Processing. Computational Engineering in System Application, 1996, 2: 963–968.
TARASOV V. E. Fractional derivative as fractional power of derivative. International Journal of Mathematics, 2007, 18(3): 281–299. https://doi.org/10.1142/S0129167X07004102
LI Y., CHEN Y. Q., and PODLUBNY I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8): 1965–1969. https://doi.org/10.1016/j.automatica.2009.04.003
LI Y., CHEN Y. Q., and PODLUBNY I. Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications, 2010, 59(5): 1810–1821. https://doi.org/10.1016/j.camwa.2009.08.019
LI C. P., & ZHANG F. R. A survey on the stability of fractional differential equations. European Physical Journal: Special Topics, 2011, 193(1): 27–47. https://doi.org/10.1140/epjst/e2011-01379-1
AGARWAL R., HRISTOVA S., and O’REGAN D. Lyapunov functions and strict stability of Caputo fractional differential equations. Advances in Difference Equations, 2015, 2015: 346. https://doi.org/10.1186/s13662-015-0674-5
AGARWAL R., HRISTOVA S., and O’REGAN D. Practical Stability of Caputo Fractional Differential Equations by Lyapunov Functions. Differential Equations & Applications, 2016, 8(1): 53-68. https://doi.org/10.7153/dea-08-04
AGARWAL R. P., O’REGAN D., and HRISTOVA S. Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions. Georgian Mathematical Journal, 2017, 24(1): 1-13. https://doi.org/10.1515/gmj-2016-0080
AGARWAL R., HRISTOVA S., and O’REGAN D. Some stability properties related to initial time difference for Caputo fractional differential equations. Fractional Calculus and Applied Analysis, 2018, 21(1): 72–93. https://doi.org/10.1515/fca-2018-0005
AGARWAL R., HRISTOVA S. and O’REGAN D. Applications of Lyapunov Functions to Caputo Fractional Differential Equations. Mathematics, 2018, 6(11): 229. https://doi.org/10.3390/math6110229
KILBAS A., SRIVASTAVA H., and TRUJILLO J. J. Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam, 2006. https://www.elsevier.com/books/theory-and-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3
SHANTANU D. Functional Fractional Calculus for System Identification and Controls. 2nd ed. Springer-Verlag, Berlin, Heidelberg, 2008. https://doi.org/10.1007/978-3-540-72703-3
KENNETH M., & BERTRAM R. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
LI C., & ZENG F. Numerical methods for fractional calculus. Chapman & Hall, 2015. https://www.routledge.com/Numerical-Methods-for-Fractional-Calculus/Li-Zeng/p/book/9780367658793
LI C., & CAI M. Theory and Numerical Approximations of Fractional Integrals and Derivatives. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2019. https://doi.org/10.1137/1.9781611975888
LAZAREVIĆ M. P. & DEBELJKOVIĆ D. L. Finite time stability analysis of linear autonomous fractional order systems with delayed state. Asian Journal of Control, 2005, 7(4): 440–447. https://doi.org/10.1111/j.1934-6093.2005.tb00407.x
LAZAREVIĆ M. P. Finite time stability analysis of PDα fractional control of robotic time-delay systems. Mechanics Research Communications, 2006, 33(2): 269–279. https://doi.org/10.1016/j.mechrescom.2005.08.010
LAZAREVIĆ M. P. & SPASIĆ A. M. Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Mathematical and Computer Modelling, 2009, 49(3–4): 475–481. https://doi.org/10.1016/j.mcm.2008.09.011
ZHANG X. Some results of linear fractional order time-delay system. Applied Mathematics and Computation, 2008, 197(1): 407–411. https://doi.org/10.1016/j.amc.2007.07.069
CHEN Y. Q. & MOORE K. L. Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dynamics, 2002, 29(1–4): 191–200. https://doi.org/10.1023/A:1016591006562
HWANG C. & CHENG Y. C. A note on the use of the Lambert W function in the stability analysis of time-delay systems. Automatica, 2005, 41(11): 1979–1985. https://doi.org/10.1016/j.automatica.2005.05.020
DENG W., LI C., and LÜ J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 2007, 48(4): 409–416. https://doi.org/10.1007/s11071-006-9094-0
HU J. B., LU G. P., ZHANG S. B., and ZHAO L. D. Lyapunov stability theorem about fractional system without and with delay. Communications in Nonlinear Science and Numerical Simulation, 2015, 20(3): 905–913. https://doi.org/10.1016/j.cnsns.2014.05.013
LIU S., WU X., ZHOU X. F., and JIANG W. Asymptotical stability of Riemann–Liouville fractional nonlinear systems. Nonlinear Dynamics, 2016, 86(1): 65–71. https://doi.org/10.1007/s11071-016-2872-4
LIU S., ZHOU X. F., LI X., and JIANG W. Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays. Applied Mathematics Letters, 2017, 65: 32–39. https://doi.org/10.1016/j.aml.2016.10.002
ČERMÁK J., HORNÍČEK J., and KISELA T. Stability regions for fractional differential systems with a time delay. Communications in Nonlinear Science and Numerical Simulation, 2016, 31(1–3): 108-123. https://doi.org/10.1016/j.cnsns.2015.07.008
AGARWAL R., HRISTOVA S., and O’REGAN D. Lyapunov functions to Caputo fractional neural networks with time-varying delays. Axioms, 2018, 7(2): 30. https://doi.org/10.3390/axioms7020030
HE B. B., ZHOU H. C., CHEN Y. Q., and KOU C. H. Asymptotical stability of fractional order systems with time delay via an integral inequality. IET Control Theory and Applications, 2018, 12(12): 1748–1754. https://doi.org/10.1049/iet-cta.2017.1144
AGARWAL R., HRISTOVA S., and O’REGAN D. Lyapunov Functions and Stability of Caputo Fractional Differential Equations with Delays. Differential Equations and Dynamical Systems, 2022, 30: 513–534. https://doi.org/10.1007/s12591-018-0434-6
SHER M., SHAH K., and RASSIAS J. On qualitative theory of fractional order delay evolution equation via the prior estimate method. Mathematical Methods in the Applied Sciences, 2020, 43(10): 6464–6475. https://doi.org/10.1002/mma.6390
Refbacks
- There are currently no refbacks.
