SIAM Journal on Control and Optimization, Volume 58, Issue 3, Page 1491-1518, January 2020.
This paper analyses fractional-order systems with time-varying delays. First, we present some results on the existence, uniqueness, exponential boundedness, and convergence rate of solutions to an equilibrium point of mixed fractional-order systems with time-varying delays. In particular, we show that in general the convergence rate of solutions to an equilibrium point is subpolynomial. This is a significant difference from ordinary differential equations. Then, we investigate the Mittag--Leffler stability of time delay fractional-order systems. To do this, we use the linearization method combined with a new weighted type norm which is compatible with the dependence on history and the hereditary property of these models. Based on an integral presentation of solutions and some special properties of Mittag--Leffler functions, we also obtain a criterion on the asymptotic stability of fractional-order systems with unbounded time-varying delays. Finally, some examples with simulations are given to illustrate the effectiveness of the theoretical results.

中文翻译：

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时滞非线性分数阶系统的定性理论

《 SIAM控制与优化杂志》，第58卷，第3期，第1491-1518页，2020年1月。
本文分析了具有时变时滞的分数阶系统。首先，我们给出了具有时变时滞的混合分数阶系统平衡点解的存在性，唯一性，指数有界性和收敛速度的一些结果。特别是，我们表明，一般而言，平衡点解的收敛速度是次多项式。这与常微分方程有很大的不同。然后，我们研究了时滞分数阶系统的Mittag-Leffler稳定性。为此，我们将线性化方法与新的加权类型范数结合使用，该范式与这些模型对历史的依赖和遗传特性兼容。根据解决方案的完整介绍和Mittag-Leffler函数的一些特殊属性，我们还获得了具有无穷时变时滞的分数阶系统的渐近稳定性的判据。最后，通过仿真实例说明了理论结果的有效性。