# Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs

arXiv: https://arxiv.org/abs/2009.01196 · HTML: https://mlsanity.com/paper/2009.01196/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2020.

## Built on by (most cited first, 22 of 22)

- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)
- [Value Iteration in Continuous Actions, States and Time](https://mlsanity.com/paper/2105.04682/) (2021)
- [Constrained Reinforcement Learning with Smoothed Log Barrier Function](https://mlsanity.com/paper/2403.14508/) (2024)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty](https://mlsanity.com/paper/2210.05989/) (2022)
- [Benchmarking Actor-Critic Deep Reinforcement Learning Algorithms for Robotics Control with Action Constraints](https://mlsanity.com/paper/2304.08743/) (2023)
- [Differentiable Control Barrier Functions for Vision-based End-to-End Autonomous Driving](https://mlsanity.com/paper/2203.02401/) (2022)
- [Robust Value Iteration for Continuous Control Tasks](https://mlsanity.com/paper/2105.12189/) (2021)
- [Differentiable Safe Controller Design through Control Barrier Functions](https://mlsanity.com/paper/2209.10034/) (2022)
- [Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control](https://mlsanity.com/paper/2502.03640/) (2025)
- [Continuous-Time Fitted Value Iteration for Robust Policies](https://mlsanity.com/paper/2110.01954/) (2021)
- [Hierarchical Multi-Agent Reinforcement Learning with Control Barrier Functions for Safety-Critical Autonomous Systems](https://mlsanity.com/paper/2507.14850/) (2025)
- [A Generalizable Physics-informed Learning Framework for Risk Probability Estimation](https://mlsanity.com/paper/2305.06432/) (2023)
- [Decentralized Safe Multi-agent Stochastic Optimal Control using Deep FBSDEs and ADMM](https://mlsanity.com/paper/2202.10658/) (2022)
- [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)
- [ABNet: Attention BarrierNet for Safe and Scalable Robot Learning](https://mlsanity.com/paper/2406.13025/) (2024)
- [PoSafeNet: Safe Learning with Poset-Structured Neural Nets](https://mlsanity.com/paper/2601.22356/) (2026)
- [Deep L1 Stochastic Optimal Control Policies for Planetary Soft-landing](https://mlsanity.com/paper/2109.00183/) (2021)
- [BarrierSteer: LLM Safety via Learning Barrier Steering](https://mlsanity.com/paper/2602.20102/) (2026)
- [Safe and Generalizable Hierarchical Multi-Agent RL via Constraint Manifold Control](https://mlsanity.com/paper/2606.24010/) (2026)

## Abstract (arXiv)

This paper introduces a new formulation for stochastic optimal control and stochastic dynamic optimization that ensures safety with respect to state and control constraints. The proposed methodology brings together concepts such as Forward-Backward Stochastic Differential Equations, Stochastic Barrier Functions, Differentiable Convex Optimization and Deep Learning. Using the aforementioned concepts, a Neural Network architecture is designed for safe trajectory optimization in which learning can be performed in an end-to-end fashion. Simulations are performed on three systems to show the efficacy of the proposed methodology.