The controllability of nonlinear fractional differential system with pure delay

In this study, we are currently investigating the controllability of nonlinear fractional differential control systems with delays in the state function. The solution representations of fractional delay differential equations have been established by using the delayed Mittag-Leffler function. Firstly we obtain the result of the controllability of a linear fractional control system with delay. Then, for the controllability criteria of nonlinear fractional delay system, we establish the set of sufficient conditions of nonlinear fractional differential systems with delay in their state function by using Schauder's fixed point theorem. In the end, a numerical example is constructed to support the results.