Dissipativity and Stability Analysis for Fractional Functional Differential Equations
Dissipativity and Stability Analysis for Fractional Functional Differential Equations
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DOI：
10.1515/fca-2015-0081
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发表时间：
2015
影响因子：
3
通讯作者：
Dongling Wang;A. Xiao;Hongliang Liu
中科院分区：
文献类型：
--
作者：
Dongling Wang;A. Xiao;Hongliang Liu
摘要 本文研究阶数为 0 < α < 1 的 Caputo 非线性分数阶泛函微分方程 (F-FDE) 的耗散性和稳定性。提出了 Halanay 型不等式的分数阶推广，该方程在 F-FDE 的稳定性和耗散性研究中发挥着核心作用。然后在与经典整数阶函数微分方程 (FDE) 几乎相同的假设下推导耗散率和吸收集。 F-FDE 的渐近稳定性也在单侧 Lipschitz 条件下得到证明。这些将相应的属性从整数阶 FDE 扩展到 Caputo 小数阶 FDE。结果还可以直接应用于分数阶非线性方程的一些特殊情况，例如分数阶时滞微分方程（F-DDE）、分数阶积分微分方程（F-IDE）和分数阶时滞积分微分方程（F-DIDE）。采用分数阶 Adams-Bashforth-Moulton 算法来模拟 F-FDE，并给出了几个数值例子来说明理论结果。
Abstract This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions as the classical integer-order functional differential equations (FDEs). The asymptotic stability of F-FDEs are also proved under the one-sided Lipschitz conditions. Those extend the corresponding properties from integer-order FDEs to the Caputo fractional ones. The results can also be directly applied to some special cases of fractional nonlinear equations, such as the fractional delay differential equations (F-DDEs), fractional integro-differential equations (F-IDEs) and fractional delay integro-differential equations (F-DIDEs). The fractional Adams-Bashforth-Moulton algorithm is employed to simulate the F-FDEs, and several numerical examples are given to illustrate the theoretical results.