# Underapproximation of Reach-Avoid Sets for Discrete-Time Stochastic Systems via Lagrangian Methods

## TL;DR

This paper introduces a grid-free, Lagrangian-based method to efficiently compute conservative underapproximations of stochastic reach-avoid sets for discrete-time nonlinear systems, improving scalability and computational efficiency.

## Contribution

It unifies existing Lagrangian approaches, characterizes the disturbance subset for guaranteed underapproximation, and demonstrates significant efficiency gains over traditional gridding methods.

## Key findings

- Method provides scalable, grid-free underapproximations.
- Approach is conservative but computationally efficient.
- Validated on space vehicle rendezvous and 2D integrator examples.

## 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.