Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces

This paper is concerned with time-varying systems with non-necessarily bounded everywhere continuous time-differentiable time-varying point delays. The delay-free and delayed dynamics are assumed to be time-varying and impulsive, in general, and the external input may also be impulsive. For given bounded initial conditions, the (unique) homogeneous state-trajectory and output trajectory are equivalently constructed from three different auxiliary homogeneous systems, the first one being delay-free and time-invariant, the second one possessing the delay-free dynamics of the current delayed system and the third one being the homogeneous part of the system under study. In this way, the constructed solution trajectories of both the unforced and forced systems are obtained from different (input-state space/output space and state space to output space) operators. The system stability and the compactness of the operators describing the solution trajectories are investigated.