# Non-Euclidean Contraction Theory for Robust Nonlinear Stability

arXiv: https://arxiv.org/abs/2103.12263 · HTML: https://mlsanity.com/paper/2103.12263/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2021.

## Built on by (most cited first, 9 of 9)

- [Robust Implicit Networks via Non-Euclidean Contractions](https://mlsanity.com/paper/2106.03194/) (2021)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [A Contraction Theory Approach to Optimization Algorithms from Acceleration Flows](https://mlsanity.com/paper/2105.08832/) (2021)
- [Contraction-Guided Adaptive Partitioning for Reachability Analysis of Neural Network Controlled Systems](https://mlsanity.com/paper/2304.03671/) (2023)
- [RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustness](https://mlsanity.com/paper/2306.12612/) (2023)
- [Distributionally Robust Imitation Learning: Layered Control Architecture for Certifiable Autonomy](https://mlsanity.com/paper/2512.17899/) (2025)
- [Verifying Closed-Loop Contractivity of Learning-Based Controllers via Partitioning](https://mlsanity.com/paper/2512.02262/) (2025)
- [A KKL Observer Perspective on Reservoir Computing](https://mlsanity.com/paper/2610.04343/) (2026)

## Abstract (arXiv)

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the ℓ1 and ℓ_∞ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as matrix measure and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.