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Impulsive fractional differential equations with Riemann–Liouville derivative and iterative learning control

Author

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  • Chen, Qian
  • Debbouche, Amar
  • Luo, Zijian
  • Wang, JinRong

Abstract

We try to seek a representation of solution to an initial value problem for impulsive fractional differential equations (IFDEs for short) involving Riemann–Liouvill (RL for short) fractional derivative, then prove an interesting existence result, and introduce Ulam type stability concepts of solution for this kind of equations by introducing some differential inequalities. In addition, we study iterative learning control (ILC for short) problem for system governed by IFDEs via a varying iterative state that does not coincide with a given initial state and apply proportional type learning principle involving the original learning condition to generate each output to following the final path in a finite time interval, then give a convergence result. Numerical examples are reported to check existence and stability of solutions and display the error for different iterative times.

Suggested Citation

  • Chen, Qian & Debbouche, Amar & Luo, Zijian & Wang, JinRong, 2017. "Impulsive fractional differential equations with Riemann–Liouville derivative and iterative learning control," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 111-118.
    • Handle: RePEc:eee:chsofr:v:102:y:2017:i:c:p:111-118
    DOI: 10.1016/j.chaos.2017.03.024
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    References listed on IDEAS

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    1. Wen-Xue Zhou & Ji-Gen Peng & Yan-Dong Chu, 2012. "Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2012, pages 1-10, February.
    2. Yan Li & YangQuan Chen & Hyo-Sung Ahn & Guohui Tian, 2013. "A Survey on Fractional-Order Iterative Learning Control," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 127-140, January.
    3. Wen-Xue Zhou & Ying-Xiang Chang & Hai-Zhong Liu, 2012. "Weak Solutions for Nonlinear Fractional Differential Equations in Banach Spaces," Discrete Dynamics in Nature and Society, Hindawi, vol. 2012, pages 1-13, July.
    4. Wang, JinRong & Ibrahim, Ahmed Gamal & Fečkan, Michal, 2015. "Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 103-118.
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    Keywords

    IFDEs; Solution; Stability; ILC;
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