Subtangent-based approaches for dynamic set propagation

Set propagation through dynamic systems is a useful tool for reachability analysis, uncertainty analysis, optimal control, and global optimization. Several established methods for dynamic set propagation represent these sets using convex and concave relaxations. Implementing these methods typically requires evaluating relaxations at several domain points, which may be computationally costly. Subtangents of these relaxations typically provide weaker bounds, but are computationally favorable since they are readily minimized on polyhedral domains, and are fully described by a single relaxation evaluation and a single subgradient evaluation. This article provides results concerning the usefulness of subtangent relaxations in set propagation: first, two methods for computing subtangents for relaxations based on ordinary differential equations, and second, a theoretical guarantee that, under mild assumptions, subtangents are guaranteed to converge rapidly to the underlying system as the parametric domain shrinks.