Investigation of controllability and stability of fractional dynamical systems with delay in control

The primary objective of this research is to investigate the controllability and Hyers-Ulam stability of fractional dynamical systems represented by ψ-Caputo fractional derivative with delay in control. To establish the necessary and sufficient conditions for assessing the controllability of linear fractional systems, which are notably distinguished by the presence of Mittag-Leffler functions, we employ the positivity of the Grammian matrix. Also, we present sufficient conditions for the controllability requirements for nonlinear fractional systems utilizing the fixed-point technique. The Hyers-Ulam stability technique is used to determine the sufficient condition for the stability of fractional nonlinear systems with delay in control. Numerical instances are provided to enhance comprehension of the theoretical findings.