# Input-to-State Safety With Control Barrier Functions

arXiv: https://arxiv.org/abs/1803.03035 · HTML: https://mlsanity.com/paper/1803.03035/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Mar 2018.

## Built on by (most cited first, 15 of 15)

- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Learning Robust Hybrid Control Barrier Functions for Uncertain Systems](https://mlsanity.com/paper/2101.06492/) (2021)
- [Learning-based Robust Motion Planning with Guaranteed Stability: A Contraction Theory Approach](https://mlsanity.com/paper/2102.12668/) (2021)
- [End-to-End Imitation Learning with Safety Guarantees using Control Barrier Functions](https://mlsanity.com/paper/2212.11365/) (2022)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Back to Base: Towards Hands-Off Learning via Safe Resets with Reach-Avoid Safety Filters](https://mlsanity.com/paper/2501.02620/) (2025)
- [Meta-Reinforcement Learning for Robust and Non-greedy Control Barrier Functions in Spacecraft Proximity Operations](https://mlsanity.com/paper/2602.07335/) (2026)
- [Safe Reinforcement Learning Filter for Multicopter Collision-Free Tracking under disturbances](https://mlsanity.com/paper/2410.06852/) (2024)
- [Safe Uncertainty-Aware Learning of Robotic Suturing](https://mlsanity.com/paper/2505.16596/) (2025)
- [Safe Execution of RL Policies Via Acceleration-Based CBF-QP Constraint Enforcement for Real-World Robotic Deployments](https://mlsanity.com/paper/2607.14488/) (2026)
- [Safe Learning of Uncertain Environments](https://mlsanity.com/paper/2103.01413/) (2021)
- [Learning Disturbances Online for Risk-Aware Control: Risk-Aware Flight with Less Than One Minute of Data](https://mlsanity.com/paper/2212.06253/) (2022)
- [LEAP-CBF: A Safety Filter for Uncertain Systems with Least-Effort Adversarial Potentials](https://mlsanity.com/paper/2609.28364/) (2026)
- [Latent Safety Filters: When a Lossy Encoder Admits a Transferable Certificate](https://mlsanity.com/paper/2610.04297/) (2026)

## Abstract (arXiv)

This letter presents a new notion of input-to-state safe control barrier functions (ISSf-CBFs), which ensure safety of nonlinear dynamical systems under input disturbances. Similar to how safety conditions are specified in terms of forward invariance of a set, input-to-state safety (ISSf) conditions are specified in terms of forward invariance of a slightly larger set. In this context, invariance of the larger set implies that the states stay either inside or very close to the smaller safe set; and this closeness is bounded by the magnitude of the disturbances. The main contribution of the letter is the methodology used for obtaining a valid ISSf-CBF, given a control barrier function (CBF). The associated universal control law will also be provided. Towards the end, we will study unified quadratic programs (QPs) that combine control Lyapunov functions (CLFs) and ISSf-CBFs in order to obtain a single control law that ensures both safety and stability in systems with input disturbances.