  Hamilton-Jacobi-Isaacs (HJI) reachability analysis is a powerful tool for
analyzing the safety of autonomous systems. This analysis is computationally
intensive and typically performed offline. Online, however, the autonomous
system may experience changes in system dynamics, external disturbances, and/or
the surrounding environment, requiring updated safety guarantees. Rather than
restarting the safety analysis, we propose a method of "warm-start"
reachability, which uses a user-defined initialization (typically the
previously computed solution). By starting with an HJI function that is closer
to the solution than the standard initialization, convergence may take fewer
iterations.
  In this paper we prove that warm-starting will result in guaranteed
conservative solutions by over-approximating the states that must be avoided to
maintain safety. We additionally prove that for many common problem
formulations, warm-starting will result in exact solutions.We demonstrate our
method on several illustrative examples with a double integrator, and also on a
more practical example with a 10D quadcopter model that experiences changes in
mass and disturbances and must update its safety guarantees accordingly. We
compare our approach to standard reachability and a recently proposed
"discounted" reachability method, and find for our examples that warm-starting
is 1.6 times faster than standard and 6.2 times faster than (untuned)
discounted reachability.
