# A Barrier Function Approach to Finite-Time Stochastic System Verification and Control

arXiv: https://arxiv.org/abs/1909.05109 · HTML: https://mlsanity.com/paper/1909.05109/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2019.

## Built on by (most cited first, 14 of 14)

- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [Compositional Policy Learning in Stochastic Control Systems with Formal Guarantees](https://mlsanity.com/paper/2312.01456/) (2023)
- [Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification](https://mlsanity.com/paper/2408.09033/) (2024)
- [A Generalizable Physics-informed Learning Framework for Risk Probability Estimation](https://mlsanity.com/paper/2305.06432/) (2023)
- [Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates](https://mlsanity.com/paper/2406.00826/) (2024)
- [Decentralized Safe Multi-agent Stochastic Optimal Control using Deep FBSDEs and ADMM](https://mlsanity.com/paper/2202.10658/) (2022)
- [Data-Driven Permissible Safe Control with Barrier Certificates](https://mlsanity.com/paper/2405.00136/) (2024)
- [Learning-Based Shielding for Safe Autonomy under Unknown Dynamics](https://mlsanity.com/paper/2410.07359/) (2024)
- [Distributionally Robust Policy and Lyapunov-Certificate Learning](https://mlsanity.com/paper/2404.03017/) (2024)
- [Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems](https://mlsanity.com/paper/2407.08868/) (2024)
- [Myopically Verifiable Probabilistic Certificates for Safe Control and Learning](https://mlsanity.com/paper/2404.16883/) (2024)
- [Set Invariance with Probability One for Controlled Diffusion: Score-based Approach](https://mlsanity.com/paper/2507.22385/) (2025)
- [Kernel-Based Learning of Safety Barriers](https://mlsanity.com/paper/2601.12002/) (2026)

## Abstract (arXiv)

This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite-time horizon. We use stochastic control barrier functions as a means to quantify the probability that a system exits a given safe region of the state space in finite time. A barrier certificate condition that bounds the expected value of the barrier function over the time horizon is recast as a sum-of-squares optimization problem for efficient numerical computation. Unlike prior works, the proposed certificate condition includes a state-dependent upper bound on the evolution of the expectation. We present formulations for both continuous-time and discrete-time systems. Moreover, for systems for which the drift dynamics are affine-in-control, we propose a method for synthesizing polynomial state feedback controllers that achieve a specified probability of safety. Several case studies are presented which benchmark and illustrate the performance of our verification and control method in the continuous-time and discrete-time domains.