Controllability of fractional dynamical systems with distributed delays in control using ?$$ \psi $$-Caputo fractional derivative

This research investigates the controllability of linear and non-linear fractional dynamical systems with distributed delays in control using the ?-Caputo fractional derivative. For controllability of linear systems, the positive definiteness of Grammian matrix, which is characterized by Mittag-Leffler functions, is used to provide necessary and sufficient conditions. For the controllability of non-linear systems, the iterative technique with the completeness of X is used to obtain sufficient conditions. Using the ?-Caputo fractional derivative, this study is new since it investigates the ideas of controllability. A couple of numerical results are offered to explain the theoretical results.