Controllability of time-varying systems with impulses, delays and nonlocal conditions

In this paper, we prove the controllability of time-varying semilinear systems with impulses, delay, and nonlocal Conditions, where some ideas are taking from previous works for this kind of systems with impulses and nonlocal conditions only, this is done by using new techniques avoiding fixed point theorems employed by A.E. Bashirov et al. In this case the delay helps us to prove the approximate controllability of this system by pulling back the control solution to a fixed curve in a short time interval, and from this position, we are able to reach a neighborhood of the final state in time tau by assuming that the corresponding linear control system is exactly controllable on any interval [t(0), tau], 0 < t(0) < tau.