This manuscript focuses on investigating a class of Caputo fractional fuzzy differential system with time delay. Firstly, we understand the granular form of fuzzy numbers from a novel perspective, which contains more information than the usual membership function. Subsequently, using a successive approximation approach under the granular arithmetic context, we demonstrate the existence of the solution to this system, and the uniqueness is obtained by the completeness of fuzzy space. Furthermore, we establish a criterion for determining the finite-time stability of the system, and which is an evaluation function containing optional parameter by employing a Gronwall inequality with delay form, where is the order of the Caputo derivative. Finally, we present a numerical example to verify our primary findings and discuss the appropriate selection of parameter to optimize the evaluation function.

中文翻译：

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粒度意义上时滞的Caputo分数阶模糊微分方程的有限时间稳定性

本手稿重点研究一类时滞 Caputo 分数阶模糊微分系统。首先，我们从一个新颖的角度理解模糊数的粒度形式，它比通常的隶属函数包含更多的信息。随后，我们在粒算术环境下使用逐次逼近方法证明了该系统解的存在性，并且通过模糊空间的完备性获得了唯一性。此外，我们建立了确定系统有限时间稳定性的准则，该准则是采用具有时滞形式的Gronwall不等式的包含可选参数的评价函数，其中 是Caputo导数的阶数。最后，我们提出一个数值例子来验证我们的主要发现，并讨论适当选择参数来优化评估函数。