  Hybrid systems with memory refer to dynamical systems exhibiting both hybrid
and delay phenomena. While systems of this type are frequently encountered in
many physical and engineering systems, particularly in control applications,
various issues centered around the robustness of hybrid delay systems have not
been adequately dealt with. In this paper, we establish some basic results on a
framework that allows to study hybrid systems with memory through generalized
concepts of solutions. In particular, we develop the basic existence of
generalized solutions using regularity conditions on the hybrid data, which are
formulated in a phase space of hybrid trajectories equipped with the graphical
convergence topology. In contrast with the uniform convergence topology that
has been often used, adopting the graphical convergence topology allows us to
establish well-posedness of hybrid systems with memory. We then show that, as a
consequence of well-posedness, pre-asymptotic stability of well-posed hybrid
systems with memory is robust.
