黎曼-刘维尔意义上带衍生的中性分数系统解的基本矩阵、积分表示和稳定性分析

H. Kiskinov, Mariyan Milev, Slav I. Cholakov, A. Zahariev
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摘要

本文研究了一类具有黎曼-刘维尔意义上的导数和分布延迟的非线性扰动中性线性分数系统。首先,本文证明了在一些自然假设条件下,这些具有不连续初始函数的系统的初始问题具有唯一解。证明中使用的假设与一阶导数系统中使用的假设类似。然后,根据所得到的结果,我们推导出均相系统的基本矩阵和广义基本矩阵的存在性和唯一性。在线性情况下,通过这些基本矩阵,我们得到了均相系统和相应非均相系统解的积分表示。此外,对于具有黎曼-黎乌韦尔导数的分数系统,我们引入了 Lyapunov、Ulam-Hyers 和 Ulam-Hyers-Rassias 意义上的加权稳定性新概念,这与整数阶或卡普托类型导数情况下的经典稳定性概念不谋而合。研究证明,当且仅当同质系统的所有解都是加权有界解时,该系统的零解才是加权稳定的。此外,对于均相系统,我们还证明了当且仅当 Ulam-Hyers 定义中出现的不等式只具有有界解时,Lyapunov 和 Ulam-Hyers 意义上的加权稳定性是等价的。最后,我们导出了自然充分条件,在这些条件下,同质系统零解的加权全局渐近稳定性特性在非线性扰动下得以保留。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense
The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances.
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