# Stochastic Reachability of Discrete-Time Stochastic Systems via Probability Measures

**Authors:** Karthik Sivaramakrishnan, Vignesh Sivaramakrishnan, Meeko M. K. Oishi  
**arXiv:** [2304.00598](https://arxiv.org/abs/2304.00598)  
**Date:** 2023-04-02  
**Summary page:** https://www.summarizepaper.com/en/paper/2304.00598v1/

## Key points

- The paper presents a framework for stochastic reachability in discrete-time systems using probability measures.
- The authors address the problem of reaching and avoiding known probability measures.
- They propose a Markov feedback policy to ensure reachability and avoidance properties.
- A backward recursion method is established for propagating state probability measures.
- Sufficient conditions for the existence of a Markov feedback policy are provided using the 1-Wasserstein distance.
- The set of initial distributions that satisfy reachability specifications is characterized.
- An analytical example is presented to demonstrate the approach.
- The paper contributes significantly to the field of stochastic reachability by providing insights into finding suitable Markov feedback policies.

## Summary

The paper titled "Stochastic Reachability of Discrete-Time Stochastic Systems via Probability Measures" presents a framework for stochastic reachability in discrete-time systems using probability measures. The authors aim to address the problem of reaching and avoiding known probability measures by characterizing the initial set of distributions that satisfy reachability specifications and finding a Markov feedback control that ensures these properties for a given initial distribution. To achieve this, they propose a Markov feedback policy that guarantees the state probability measure reaches a target probability measure while avoiding an undesirable probability measure over a finite time horizon. They establish a backward recursion method for propagating state probability measures and provide sufficient conditions for the existence of such a Markov feedback policy using the 1-Wasserstein distance. Additionally, they characterize the set of initial distributions that satisfy the reachability specifications. To demonstrate their approach, an analytical example is presented. Overall, this paper contributes significantly to the field of stochastic reachability by providing a framework based on probability measures and offering insights into finding suitable Markov feedback policies. The proposed methods and results have potential applications in various domains where stochastic systems need to satisfy specific reachability requirements.