Underapproximation of reach-avoid sets for discrete-time stochastic systems via Lagrangian methods

We examine Lagrangian techniques for computing under approximations of finite-time horizon, stochastic reach-avoid level sets for discrete-time, nonlinear systems. We use the concept of reachability of a target tube to define robust reach-avoid sets which are parameterized by the target set, safe set, and the set 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 for 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 a given target and safe set is a guaranteed under approximation of the stochastic reach-avoid level set of interest. The proposed approach dramatically improves the computational efficiency for obtaining an under approximation 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 constraints due to computational geometry. We demonstrate the method on two examples: a simple two-dimensional integrator, and a space vehicle rendezvous-docking problem.