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Applications of Lyapunov Functions to Caputo Fractional Differential Equations

Author

Listed:
  • Ravi Agarwal

    (Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
    Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA)

  • Snezhana Hristova

    (Department of Applied Mathematics and Modeling, University of Plovdiv, Tzar Asen 24, 4000 Plovdiv, Bulgaria)

  • Donal O’Regan

    (School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 CF50 Galway, Ireland)

Abstract

One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

Suggested Citation

  • Ravi Agarwal & Snezhana Hristova & Donal O’Regan, 2018. "Applications of Lyapunov Functions to Caputo Fractional Differential Equations," Mathematics, MDPI, vol. 6(11), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:11:p:229-:d:179212
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    References listed on IDEAS

    as
    1. Fengrong Zhang & Changpin Li & YangQuan Chen, 2011. "Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-12, August.
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    4. Li, C.P. & Deng, W.H. & Xu, D., 2006. "Chaos synchronization of the Chua system with a fractional order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 171-185.
    5. Tavazoei, Mohammad Saleh & Haeri, Mohammad, 2009. "A note on the stability of fractional order systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1566-1576.
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