Exponential stability and relative controllability of first-order delayed integro-differential systems with impulses

We establish sufficient conditions for the exponential stability of nonsingular impulsive delayed integro-differential systems. Our approach to addressing nonsingular differential problems involves the application of permutable matrices and their associated delayed exponential. Furthermore, we investigate the controllability of a nonlinear impulsive and delayed problem by employing the corresponding Gramian matrix. Finally, to illustrate the theoretical outcomes, we provide examples and graphical representations for each situation.