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Hybrid systems with memory refer to dynamical systems exhibiting both hybridand delay phenomena. While systems of this type are frequently encountered inmany physical and engineering systems, particularly in control applications,various issues centered around the robustness of hybrid delay systems have notbeen adequately dealt with. In this paper, we establish some basic results on aframework that allows to study hybrid systems with memory through generalizedconcepts of solutions. In particular, we develop the basic existence ofgeneralized solutions using regularity conditions on the hybrid data, which areformulated in a phase space of hybrid trajectories equipped with the graphicalconvergence topology. In contrast with the uniform convergence topology thathas been often used, adopting the graphical convergence topology allows us toestablish well-posedness of hybrid systems with memory. We then show that, as aconsequence of well-posedness, pre-asymptotic stability of well-posed hybridsystems with memory is robust.

Hybrid systems with memory: Existence and well-posedness of generalized solutions