This study examines the controllability criteria for linear and semilinear fractional dynamical systems with delays in both state and control variables in the framework of the Caputo fractional derivative. To establish the controllability criteria for linear fractional dynamical systems, the study derives necessary and sufficient conditions by employing the positive definiteness of the Grammian matrix. Extending this analysis to semilinear fractional dynamical systems, Krasnoselskii’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. Furthermore, in addressing semilinear fractional dynamical systems with delays in both state and control, Banach’s fixed point theorem is employed to derive sufficient conditions for the existence of a solution. In order to enhance the comprehension of the theoretical results, the study presents three specific examples along with appropriate graphical representations.

本研究在 Caputo 分数阶导数的框架中检查了线性和半线性分数阶动力系统的可控性标准，这些系统在状态和控制变量中都有延迟。为了建立线性分数阶动力系统的可控性标准，该研究通过采用 Grammian 矩阵的正定性推导出必要和充分的条件。将此分析扩展到半线性分数阶动力学系统，采用 Krasnoselskii 的不动点定理来推导出解存在的充分条件。此外，在处理状态和控制都有延迟的半线性分数阶动力系统时，采用 Banach 不动点定理来推导出解存在的充分条件。为了增强对理论结果的理解，本研究提供了三个具体示例以及适当的图形表示。