Title: Underapproximation of Reach-Avoid Sets for Discrete-Time Stochastic Systems via Lagrangian Methods,
Abstract: We examine Lagrangian techniques for computing underapproximations of
finite-time horizon, stochastic reach-avoid level-sets for discrete-time,
nonlinear systems. We use the concept of reachability of a target tube in the
control literature to define robust reach-avoid sets which are parameterized by
the target set, safe set, and the set in which the disturbance is drawn from.
We unify two existing Lagrangian approaches to compute these sets and establish
that there exists an optimal control policy of the robust reach-avoid sets
which is a Markov policy. Based on these results, we characterize the subset of
the disturbance space whose corresponding robust reach-avoid set for the given
target and safe set is a guaranteed underapproximation of the stochastic
reach-avoid level-set of interest. The proposed approach dramatically improves
the computational efficiency for obtaining an underapproximation of stochastic
reach-avoid level-sets when compared to the traditional approaches based on
gridding. Our method, while conservative, does not rely on a grid, implying
scalability as permitted by the known computational geometry constraints. We
demonstrate the method on two examples: a simple two-dimensional integrator,
and a space vehicle rendezvous-docking problem.