# A dynamic game approach to distributionally robust safety specifications for stochastic systems

## TL;DR

This paper introduces a distributionally robust safety control method for stochastic systems, using a dynamic game approach to handle uncertainty in disturbance distributions and ensuring safety with limited distributional information.

## Contribution

It develops a novel dynamic game formulation for distributionally robust safety policies and provides a practical duality-based solution to infinite-dimensional minimax problems.

## Key findings

- The proposed method is robust against distributional errors in disturbances.
- It transforms complex minimax problems into solvable semi-infinite programs.
- Numerical tests confirm improved safety robustness over standard tools.

## Abstract

This paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances. Our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times, subject to the worst possible disturbance distribution in an ambiguity set. We propose a dynamic game formulation of constructing such policies and identify conditions under which a non-randomized Markov policy is optimal. Based on this existence result, we develop a practical design approach to safety-oriented stochastic controllers with limited information about disturbance distributions. This control method can be used to minimize another cost function while ensuring safety in a probabilistic way. However, an associated Bellman equation involves infinite-dimensional minimax optimization problems since the disturbance distribution may have a continuous density. To resolve computational issues, we propose a duality-based reformulation method that converts the infinite-dimensional minimax problem into a semi-infinite program that can be solved using existing convergent algorithms. We prove that there is no duality gap, and that this approach thus preserves optimality. The results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances, while a standard stochastic safety specification tool is not.