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SCIENCE CHINA Information Sciences, Volume 63 , Issue 3 : 132201(2020) https://doi.org/10.1007/S11432-019-9946-6

Global Mittag-Leffler stability for fractional-order coupled systems on network without strong connectedness

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  • ReceivedJan 26, 2019
  • AcceptedMay 30, 2019
  • PublishedFeb 10, 2020

Abstract


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61873071) and Shandong Provincial Natural Science Foundation (Grant No. ZR2019MF027).


References

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  • Figure 3

    (Color online) Response curves of (17) with the initial values $\textbf{\textit{Z}}_{1},\textbf{\textit{Z}}_{2}$, and $\textbf{\textit{Z}}_{3}$. (a) and (b) show the curves of states and control inputs of system (17).

  • Table 1  

    Table 1Parameters used in the coupled system (16)

    Parameter ValueParameterValueParameterValueParameter ValueParameter ValueParameter Value
    $\alpha_{1}$0.17 $\beta_{1,1}$0 $\beta_{2,1}$0.1$\beta_{3,1}$0.4$\beta_{4,1}$0$\beta_{5,1}$0
    $\alpha_{2}$0.30$\beta_{1,2}$0.1 $\beta_{2,2}$0 $\beta_{3,2}$0$\beta_{4,2}$0$\beta_{5,2}$0
    $\alpha_{3}$0.46$\beta_{1,3}$0 $\beta_{2,3}$0 $\beta_{3,3}$0$\beta_{4,3}$0.2$\beta_{5,3}$0.4
    $\alpha_{4}$0.59$\beta_{1,4}$0$\beta_{2,4}$0$\beta_{3,4}$0.2$\beta_{4,4}$0$\beta_{5,4}$0
    $\alpha_{5}$0.78$\beta_{1,5}$0$\beta_{2,5}$0$\beta_{3,5}$0$\beta_{4,5}$0$\beta_{5,5}$0
    $\alpha_{6}$0.91$\beta_{1,6}$0$\beta_{2,6}$0$\beta_{3,6}$0$\beta_{4,6}$0$\beta_{5,6}$0.4
    $\alpha_{7}$0.98 $\beta_{1,7}$0$\beta_{2,7}$0$\beta_{3,7}$0$\beta_{4,7}$0$\beta_{5,7}$0.4
    $\alpha_{8}$1.11$\beta_{1,8}$0$\beta_{2,8}$0$\beta_{3,8}$0$\beta_{4,8}$0$\beta_{5,8}$0
    $\alpha_{9}$1.21$\beta_{1,9}$0$\beta_{2,9}$ 0$\beta_{3,9}$0$\beta_{4,9}$0.4$\beta_{5,9}$0
    $\alpha_{10}$1.34$\beta_{1,10}$0$\beta_{2,10}$0$\beta_{3,10}$0$\beta_{4,10}$0$\beta_{5,10}$0
    $\beta_{6,1}$0$\beta_{7,1}$0$\beta_{8,1}$0$\beta_{9,1}$0$\beta_{10,1}$0
    $\beta_{6,2}$ 0$\beta_{7,2}$0.4$\beta_{8,2}$0$\beta_{9,2}$0 $\beta_{10,2}$0
    $\beta_{6,3}$0$\beta_{7,3}$0$\beta_{8,3}$0$\beta_{9,3}$0$\beta_{10,3}$0
    $\beta_{6,4}$0.4$\beta_{7,4}$0$\beta_{8,4}$0$\beta_{9,4}$0$\beta_{10,4}$0
    $\beta_{6,5}$0.4$\beta_{7,5}$0 $\beta_{8,5}$0$\beta_{9,5}$0$\beta_{10,5}$0
    $\beta_{6,6}$0$\beta_{7,6}$0$\beta_{8,6}$0$\beta_{9,6}$0$\beta_{10,6}$0
    $\beta_{6,7}$0$\beta_{7,7}$0$\beta_{8,7}$0.2$\beta_{9,7}$0 $\beta_{10,7}$0
    $\beta_{6,8}$0.4$\beta_{7,8}$0.2$\beta_{8,8}$0$\beta_{9,8}$0$\beta_{10,8}$0
    $\beta_{6,9}$0$\beta_{7,9}$0 $\beta_{8,9}$0$\beta_{9,9}$0$\beta_{10,9}$0.1
    $\beta_{6,10}$0$\beta_{7,10}$0$\beta_{8,10}$0.4$\beta_{9,10}$0.1$\beta_{10,10}$ 0
  • Table 2  

    Table 2Equilibrium values of $\textbf{\textit{Z}}^{\ast}$ of (17) for different values of $m_{k}$

    Parameter $\textit{m}_{k}=1$ $\textit{m}_{k}=2$ $\textit{m}_{k}=3$ $\textit{m}_{k}=4$ Parameter $\textit{m}_{k}=1$ $\textit{m}_{k}=2$ $\textit{m}_{k}=3$ $\textit{m}_{k}=4$
    $\textit{x}^{\ast}_{1}$ 0.1755 0.0982 0.0680 0.0520 $\textit{u}^{\ast}_{1}$ 0.1595 0.1785 0.1854 0.1890
    $\textit{x}^{\ast}_{2}$ 0.3199 0.1959 0.1401 0.1088 $\textit{u}^{\ast}_{2}$ 0.2666 0.3265 0.3503 0.3627
    $\textit{x}^{\ast}_{3}$ 0.4110 0.2765 0.2066 0.1644 $\textit{u}^{\ast}_{3}$ 0.3161 0.4254 0.4769 0.5058
    $\textit{x}^{\ast}_{4}$ 0.4691 0.3371 0.2618 0.2133 $\textit{u}^{\ast}_{4}$ 0.3351 0.4816 0.5609 0.6095
    $\textit{x}^{\ast}_{5}$ 0.5045 0.3870 0.3124 0.2611 $\textit{u}^{\ast}_{5}$ 0.3363 0.5159 0.6248 0.6963
    $\textit{x}^{\ast}_{6}$ 0.5560 0.4424 0.3660 0.3112 $\textit{u}^{\ast}_{6}$ 0.3475 0.5530 0.6862 0.7779
    $\textit{x}^{\ast}_{7}$ 0.6302 0.5115 0.4293 0.3689 $\textit{u}^{\ast}_{7}$ 0.3707 0.6018 0.7575 0.8680
    $\textit{x}^{\ast}_{8}$ 0.6496 0.5407 0.4628 0.4040 $\textit{u}^{\ast}_{8}$ 0.3609 0.6007 0.7713 0.8978
    $\textit{x}^{\ast}_{9}$ 0.6949 0.5882 0.5101 0.4500 $\textit{u}^{\ast}_{9}$ 0.3657 0.6192 0.8054 0.9473
    $\textit{x}^{\ast}_{10}$ 0.7136 0.6153 0.5413 0.4830 $\textit{u}^{\ast}_{10}$ 0.3568 0.6153 0.8119 0.9660