# Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints

arXiv: https://arxiv.org/abs/2305.10620 · HTML: https://mlsanity.com/paper/2305.10620/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2023.

## Built on by (most cited first, 6 of 6)

- [Accurate synthesis of Dysarthric Speech for ASR data augmentation](https://mlsanity.com/paper/2308.08438/) (2023)
- [Learning Performance-Oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation](https://mlsanity.com/paper/2401.05629/) (2024)
- [Domain Adaptive Safety Filters via Deep Operator Learning](https://mlsanity.com/paper/2410.14528/) (2024)
- [Bilingual Streaming ASR with Grapheme units and Auxiliary Monolingual Loss](https://mlsanity.com/paper/2308.06327/) (2023)
- [Feasibility-aware Learning of Robust Temporal Logic Controllers using BarrierNet](https://mlsanity.com/paper/2512.06973/) (2025)
- [BarrierSteer: LLM Safety via Learning Barrier Steering](https://mlsanity.com/paper/2602.20102/) (2026)

## Abstract (arXiv)

This paper presents two new control approaches for guaranteed safety (remaining in a safe set) subject to actuator constraints (the control is in a convex polytope). The control signals are computed using real-time optimization, including linear and quadratic programs subject to affine constraints, which are shown to be feasible. The first control method relies on a soft-minimum barrier function that is constructed using a finite-time-horizon prediction of the system trajectories under a known backup control. The main result shows that the control is continuous and satisfies the actuator constraints, and a subset of the safe set is forward invariant under the control. Next, we extend this method to allow from multiple backup controls. This second approach relies on a combined soft-maximum/soft-minimum barrier function, and it has properties similar to the first. We demonstrate these controls on numerical simulations of an inverted pendulum and a nonholonomic ground robot.