Epsilon-Tubes and Generalized Skorokhod Metrics for Hybrid Paths Spaces

Abstract

We develop several generalized Skorokhod pseudo-metrics for hybrid path spaces, cast in a quite general setting, where the basic open sets are epsilontubes around paths that, intuitively, allow for some "wiggle room" in both time and space via set-valued retiming maps between the time domains of paths. We then determine necessary and sufficient conditions under which these topologies are Hausdorff and their distance functions are metrics. On spaces of paths with closed time domains, our metric topology of generalized Skorokhod uniform convergence on finite prefixes is equivalent to the implicit topology of graphical convergence of hybrid paths, currently used extensively by Teel and co-worke..