Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping

In this paper, we investigate the approximate controllability of mild solutions for second-order differential systems. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential systems under the assumption that the associated linear part of the system is approximately controllable. In addition, we extend our system with nonlocal conditions. Our research on approximate controllability was also extended by utilizing impulse systems. To demonstrate the theory of the primary outcomes, an application is shown.