In the formal verification of dynamical systems, one often looks at a trajectory through a state space as a sample behavior of the system. Thus, metrics on trajectories give important information about the different behavior of the system given different starting states. In the important special case of linear dynamical systems, the set of trajectories forms a finite-dimensional vector space. In this paper, we exploit this vector space structure to define (semi)norms on the trajectories, give an isometric embedding from the trajectory metric into low-dimensional Euclidean space, and bound the Lipschitz constant on the map from start states to trajectories as measured in one of several different metrics. These results show that for an interesting class of trajectories, one can treat the trajectories as points while losing little or no information.

@inproceedings{duggirala18when,
  Author = {Parasara Sridhar Duggirala and Donald R. Sheehy},
  Booktitle = {Proceedings of the Canadian Conference on Computational Geometry},
  Title = {When Can We Treat Trajectories as Points?},
  Pages = {340--345},
  Year = {2018}}