# Risk-Aware Stability, Ultimate Boundedness, and Positive Invariance

arXiv: https://arxiv.org/abs/2204.07329 · HTML: https://mlsanity.com/paper/2204.07329/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2022.

## Built on by (most cited first, 2 of 2)

- [Fine-Tuning Generative Models for Extreme Events via CVaR-Penalized Wasserstein Gradient Flows](https://mlsanity.com/paper/2608.11544/) (2026)
- [Distributionally Robust Safety Verification of Neural Networks via Worst-Case CVaR](https://mlsanity.com/paper/2509.17413/) (2025)

## Abstract (arXiv)

This paper introduces the notions of stability, ultimate boundedness, and positive invariance for stochastic systems in the view of risk. More specifically, those notions are defined in terms of the worst-case Conditional Value-at-Risk (CVaR), which quantifies the worst-case conditional expectation of losses exceeding a certain threshold over a set of possible uncertainties. Those notions allow us to focus our attention on the tail behavior of stochastic systems in the analysis of dynamical systems and the design of controllers. Furthermore, some event-triggered control strategies that guarantee ultimate boundedness and positive invariance with specified bounds are derived using the obtained results and illustrated using numerical examples.