# Harmonic Control Lyapunov Barrier Functions for Constrained Optimal Control with Reach-Avoid Specifications

arXiv: https://arxiv.org/abs/2310.02869 · HTML: https://mlsanity.com/paper/2310.02869/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 9 of 9)

- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Emergence of Locomotion Behaviours in Rich Environments](https://mlsanity.com/paper/1707.02286/) (2017)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Reinforcement Learning for Safety-Critical Control under Model Uncertainty, using Control Lyapunov Functions and Control Barrier Functions](https://mlsanity.com/paper/2004.07584/) (2020)
- [safe-control-gym: a Unified Benchmark Suite for Safe Learning-based Control and Reinforcement Learning in Robotics](https://mlsanity.com/paper/2109.06325/) (2021)
- [Gradient Descent Converges to Minimizers](https://mlsanity.com/paper/1602.04915/) (2016)
- [Reinforcement Learning for Safe Robot Control using Control Lyapunov Barrier Functions](https://mlsanity.com/paper/2305.09793/) (2023)
- [Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications](https://mlsanity.com/paper/2009.04432/) (2020)
- [Provable Reset-free Reinforcement Learning by No-Regret Reduction](https://mlsanity.com/paper/2301.02389/) (2023)

## Abstract (arXiv)

This paper introduces harmonic control Lyapunov barrier functions (harmonic CLBF) that aid in constrained control problems such as reach-avoid problems. Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy to encode the properties of control Lyapunov barrier functions (CLBFs). As a result, they can be initiated at the start of an experiment rather than trained based on sample trajectories. The control inputs are selected to maximize the inner product of the system dynamics with the steepest descent direction of the harmonic CLBF. Numerical results are presented with four different systems under different reach-avoid environments. Harmonic CLBFs show a significantly low risk of entering unsafe regions and a high probability of entering the goal region.