In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

在这项研究中，我们着重于分数阶阻尼系统在线性和非线性情况下具有多个时变控制延迟的可控制性。对于基于Mittag-Leffler矩阵函数的线性系统，我们定义了可控性Gramian矩阵，这对于判断系统是否可控很有用。此外，在两种特殊情况下，我们提出了易于验证的服务等效可控条件。对于非线性系统，在其相应线性系统的可控制性下，我们在非线性项上获得了充分的条件以确保系统是可控制的。最后，给出两个例子来说明这一理论。