# Learning Min-norm Stabilizing Control Laws for Systems with Unknown Dynamics

arXiv: https://arxiv.org/abs/2004.10331 · HTML: https://mlsanity.com/paper/2004.10331/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2020.

## Built on by (most cited first, 5 of 5)

- [Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty](https://mlsanity.com/paper/2106.07108/) (2021)
- [Bridging Model-based Safety and Model-free Reinforcement Learning through System Identification of Low Dimensional Linear Models](https://mlsanity.com/paper/2205.05787/) (2022)
- [A Review On Safe Reinforcement Learning Using Lyapunov and Barrier Functions](https://mlsanity.com/paper/2508.09128/) (2025)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Enabling Efficient, Reliable Real-World Reinforcement Learning with Approximate Physics-Based Models](https://mlsanity.com/paper/2307.08168/) (2023)

## Abstract (arXiv)

This paper introduces a framework for learning a minimum-norm stabilizing controller for a system with unknown dynamics using model-free policy optimization methods. The approach begins by first designing a Control Lyapunov Function (CLF) for a (possibly inaccurate) dynamics model for the system, along with a function which specifies a minimum acceptable rate of energy dissipation for the CLF at different points in the state-space. Treating the energy dissipation condition as a constraint on the desired closed-loop behavior of the real-world system, we use penalty methods to formulate an unconstrained optimization problem over the parameters of a learned controller, which can be solved using model-free policy optimization algorithms using data collected from the plant. We discuss when the optimization learns a stabilizing controller for the real world system and derive conditions on the structure of the learned controller which ensure that the optimization is strongly convex, meaning the globally optimal solution can be found reliably. We validate the approach in simulation, first for a double pendulum, and then generalize the framework to learn stable walking controllers for underactuated bipedal robots using the Hybrid Zero Dynamics framework. By encoding a large amount of structure into the learning problem, we are able to learn stabilizing controllers for both systems with only minutes or even seconds of training data.