Impulsive fractional integro-delay differential equation-controllability through delayed Mittag-Leffler function perturbation

The research is focused on establishing the existence and uniqueness of solutions for a specific set of Caputo fractional integro-differential equations with delay under impulsive conditions. To achieve this, the study employs the fractional derivative within the Caputo framework and utilizes the delayed perturbation of the Mittag-Leffler matrix. The research findings are rooted in the fixed points of the Banach fixed point theorems to prove the existence of the semilinear fractional integro-differential delay system. Furthermore, the study delves into controllability analysis using the delayed Grammian matrix. Theoretical examples are included to illustrate and demonstrate the implications of the established theorems.