# A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems

## TL;DR

This paper introduces a risk-sensitive reachability framework for stochastic systems that quantifies and reduces the probability of constraint violations over finite time horizons using CVaR, with practical algorithms and an illustrative example.

## Contribution

It develops a novel risk-sensitive safe set concept, reduces its computation to a CVaR-based MDP, and provides a tractable algorithm with theoretical guarantees for stochastic system safety.

## Key findings

- The approach allows tuning risk sensitivity from worst-case to risk-neutral.
- A tractable algorithm for approximating risk-sensitive safe sets is proposed.
- Demonstration on stormwater catchment design shows practical utility.

## Abstract

A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage existing theory of reachability analysis and risk measures to devise a risk-sensitive reachability approach for safety of stochastic dynamic systems under non-adversarial disturbances over a finite time horizon. Specifically, we first introduce the notion of a risk-sensitive safe set as a set of initial states from which the risk of large constraint violations can be reduced to a required level via a control policy, where risk is quantified using the Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation of a risk-sensitive safe set can be reduced to the solution to a Markov Decision Process (MDP), where cost is assessed according to CVaR. Third, leveraging this reduction, we devise a tractable algorithm to approximate a risk-sensitive safe set, and provide theoretical arguments about its correctness. Finally, we present a realistic example inspired from stormwater catchment design to demonstrate the utility of risk-sensitive reachability analysis. In particular, our approach allows a practitioner to tune the level of risk sensitivity from worst-case (which is typical for Hamilton-Jacobi reachability analysis) to risk-neutral (which is the case for stochastic reachability analysis).