The objective of this study is to analyze controllability results for time-varying linear and nonlinear fractional dynamical systems with multiple control delays within the framework of the Caputo fractional derivative. This paper focuses on examining control problems within a finite time interval, aiming to identify a control function that steers the system’s solution from a specified initial state to a targeted final state. For linear systems, the study establishes necessary and sufficient conditions for controllability by utilizing the Grammian matrix techniques. For nonlinear systems, the existence of a solution is ensured through an iterative technique, with completeness of the space guaranteed. With the help of this technique, we establish the sufficient conditions for the controllability of time-varying nonlinear fractional dynamical systems. The results show that the controllability of fractional dynamical systems can be effectively analyzed with the given framework, along with numerical simulations and graphical representations to clarify the theoretical findings.

This article investigates controllability results for fractional dynamical systems with prescribed control, formulated using the (Formula presented.) -Caputo type fractional derivative. For linear systems, controllability is established via fractional calculus and the Gramian approach, while for nonlinear systems it is examined using Krasnoselskii’s fixed point technique. Theoretical findings are further supported with illustrative numerical examples. The study also discusses the mathematical framework required for the analysis and highlights the logical steps followed to derive the main results.