# Verifying Closed-Loop Contractivity of Learning-Based Controllers via Partitioning

arXiv: https://arxiv.org/abs/2512.02262 · HTML: https://mlsanity.com/paper/2512.02262/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 20 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Non-Euclidean Contraction Theory for Robust Nonlinear Stability](https://mlsanity.com/paper/2103.12263/) (2021)
- [A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systems](https://mlsanity.com/paper/2306.15340/) (2023)
- [Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems](https://mlsanity.com/paper/2504.17102/) (2025)
- [Neural Exponential Stabilization of Control-affine Nonlinear Systems](https://mlsanity.com/paper/2403.17793/) (2024)
- [Perspectives on Contractivity in Control, Optimization, and Learning](https://mlsanity.com/paper/2404.11707/) (2024)

## Abstract (arXiv)

We address the problem of verifying closed-loop contraction in nonlinear control systems whose controller and contraction metric are both parameterized by neural networks. By leveraging interval analysis and interval bound propagation, we derive a tractable and scalable sufficient condition for closed-loop contractivity that reduces to checking that the dominant eigenvalue of a symmetric Metzler matrix is nonpositive. We combine this sufficient condition with a domain partitioning strategy to integrate this sufficient condition into training. The proposed approach is validated on an inverted pendulum system, demonstrating the ability to learn neural network controllers and contraction metrics that provably satisfy the contraction condition.