# Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers

arXiv: https://arxiv.org/abs/2406.12744 · HTML: https://mlsanity.com/paper/2406.12744/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 28 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 15 of 15)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (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)
- [Stability-certified reinforcement learning: A control-theoretic perspective](https://mlsanity.com/paper/1810.11505/) (2018)
- [Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimization](https://mlsanity.com/paper/2012.05981/) (2020)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [DeepSplit: Scalable Verification of Deep Neural Networks via Operator Splitting](https://mlsanity.com/paper/2106.09117/) (2021)
- [Stability of Non-linear Neural Feedback Loops using Sum of Squares](https://mlsanity.com/paper/2204.03913/) (2022)
- [Structured Neural-PI Control with End-to-End Stability and Output Tracking Guarantees](https://mlsanity.com/paper/2305.17777/) (2023)
- [Scalable Forward Reachability Analysis of Multi-Agent Systems with Neural Network Controllers](https://mlsanity.com/paper/2309.03846/) (2023)
- [On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations](https://mlsanity.com/paper/2305.09904/) (2023)

## Built on by (most cited first, 4 of 4)

- [Robust Stability Analysis of Positive Lure System with Neural Network Feedback](https://mlsanity.com/paper/2505.18912/) (2025)
- [Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints](https://mlsanity.com/paper/2505.22889/) (2025)
- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (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 introduces a novel method for the stability analysis of positive feedback systems with a class of fully connected feedforward neural networks (FFNN) controllers. By establishing sector bounds for fully connected FFNNs without biases, we present a stability theorem that demonstrates the global exponential stability of linear systems under fully connected FFNN control. Utilizing principles from positive Lur'e systems and the positive Aizerman conjecture, our approach effectively addresses the challenge of ensuring stability in highly nonlinear systems. The crux of our method lies in maintaining sector bounds that preserve the positivity and Hurwitz property of the overall Lur'e system. We showcase the practical applicability of our methodology through its implementation in a linear system managed by a FFNN trained on output feedback controller data, highlighting its potential for enhancing stability in dynamic systems.