Hamilton-Jacobi Multi-Time Reachability

Abstract

For the analysis of dynamical systems, it is fundamental to determine all states that can be reached at any given time. In this work, we obtain and apply new governing equations for reachability analysis over multiple start and terminal times all at once, and for systems operating in time-varying environments with dynamic obstacles and any other relevant dynamic fields. The theory and schemes are developed for both backward and forward reachable tubes with time-varying target and start sets. The resulting value functions elegantly capture not only the reachable tubes but also time-to-reach and time-to-leave maps as well as start time vs. duration plots and other useful secondary quantities for optimal control. We discuss the numerical schemes and computational efficiency. We first verify our results in an environment with a moving target and obstacle where reachability tubes can be analytically computed. We then consider the Dubin's car problem extended with a moving target and obstacle. Finally, we showcase our multi-time reachability in a non-hydrostatic bottom gravity current system. Results highlight the novel capabilities of exact multi-time reachability in dynamic environments.