  A non-autonomous evolution semi-linear differential system under
non-instantaneous impulses, delays, and perturbed by non-local conditions is
studied. Its piece-wise continuous solutions belong to a finite-dimensional
Banach space. The existence and uniqueness of solutions on the interval
$[-r,\tau]$ are obtained by applying Karakostas fixed-point theorem. Further
results concerning solution prolongation are developed. An example is
presented, and several remarks on the infinite-dimensional case are included.

Optimization and Control;Dynamical Systems