This manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

中文翻译：

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具有非瞬时脉冲的分数阶微分动力系统的可控性准则

该手稿展望了非瞬时脉冲Volterra型分数阶微分系统的可控性标准。通过引入通常由Mittag-Leffler函数定义的适当的Gramian矩阵并借助Laplace变换，使用代数方法推导了非瞬时脉冲Volterra型分数阶微分方程的可控性的充要条件。 Cayley–Hamilton定理。本文存在的重要特征是，我们已将非瞬时脉冲纳入分数阶动力学系统并研究了可控性分析，因为现有文献中不存在这种情况。包括在内 我们提供了两个说明性的例子，说明存在于分数动力系统中的非瞬时脉冲。因此，这证明了我们所获得的主要部分标准的有效性和有效性。