In this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new \(\alpha \)-Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples.

在本文中，我们的重点是探索由包含状态延迟的线性分数阶微分方程控制的系统的相对可控性。我们引入了凯莱-汉密尔顿定理的一个新颖的对应物。利用 Mittag-Leffler 函数的延迟扰动以及确定函数和 Cayley-Hamilton 定理的类似物，我们建立了代数卡尔曼型秩准则，用于评估具有状态延迟的分数阶微分方程的相对可控性。此外，我们阐明了有关线性分数时滞系统的相对可控性准则的必要和充分条件，以新的\(\alpha \) -Gramian 矩阵表示，并定义了一个将系统从任何初始状态转移到任何最终状态的控制。给定时间内的状态。理论研究结果通过说明性例子的呈现来举例说明。