# Robust Stability Analysis of Positive Lure System with Neural Network Feedback

arXiv: https://arxiv.org/abs/2505.18912 · HTML: https://mlsanity.com/paper/2505.18912/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 25 May 2025.

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

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Provably Bounding Neural Network Preimages](https://mlsanity.com/paper/2302.01404/) (2023)
- [Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers](https://mlsanity.com/paper/2406.12744/) (2024)

## Built on by (most cited first, 2 of 2)

- [Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints](https://mlsanity.com/paper/2505.22889/) (2025)
- [Delay Independent Safe Control with Neural Networks: Positive Lur'e Certificates for Risk Aware Autonomy](https://mlsanity.com/paper/2510.06661/) (2025)

## Abstract (arXiv)

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and robustness properties of Metzler matrices. Specifically, we consider a control system of Lur'e type in which not only the linear part includes parametric uncertainty but also the nonlinear sector bound is unknown. We investigate tools from positive linear systems to effectively solve the problems in complicated and uncertain nonlinear systems. By leveraging the positivity characteristic of the system, we derive an explicit formula for the stability radius of Lur'e systems. Furthermore, we extend our analysis to systems with neural network (NN) feedback loops. Building on this approach, we also propose a refinement method for sector bounds of NNs. This study introduces a scalable and efficient approach for robustness analysis of both Lur'e and NN-controlled systems. Finally, the proposed results are supported by illustrative examples.