Existence and controllability results to second order neutral differential equation with non-instantaneous impulses

In this paper, existence and uniqueness of the solutions of the second order neutral differential equation with non-instantaneous impulses in a Hilbert space X are studied. This is followed by a brief discussion on the exact and trajectory controllability. Further, the strongly continuous cosine family of linear operators and Banach fixed point theorem are used to establish the results. Finally, an example is given to illustrate the analytical findings.