# Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems using Fourier Transforms

## TL;DR

This paper introduces a scalable Fourier transform-based method to compute underapproximations of stochastic reach-avoid probabilities in high-dimensional linear systems, enabling verification of larger systems than previously possible.

## Contribution

It presents a novel Fourier transform approach for underapproximating reach-avoid probabilities in high-dimensional stochastic LTI systems, handling arbitrary disturbance densities.

## Key findings

- Successfully applied to a 40-state chain of integrators.
- Provides non-trivial lower bounds for reach-avoid probabilities.
- Efficiently handles Gaussian disturbances with convex optimization.

## Abstract

We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques are limited and cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value functions. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a chain of integrators with 40 states.