# Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints

arXiv: https://arxiv.org/abs/2505.22889 · HTML: https://mlsanity.com/paper/2505.22889/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 29 references that are in the map.
- First posted on arXiv 28 May 2025.

## Builds on (most cited first, 17 of 17)

- [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)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [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)
- [RecurJac: An Efficient Recursive Algorithm for Bounding Jacobian Matrix of Neural Networks and Its Applications](https://mlsanity.com/paper/1810.11783/) (2018)
- [Can a Transformer Represent a Kalman Filter?](https://mlsanity.com/paper/2312.06937/) (2023)
- [Stability of Non-linear Neural Feedback Loops using Sum of Squares](https://mlsanity.com/paper/2204.03913/) (2022)
- [Generalization Error Analysis for Selective State-Space Models Through the Lens of Attention](https://mlsanity.com/paper/2502.01473/) (2025)
- [Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networks](https://mlsanity.com/paper/2405.05236/) (2024)
- [Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers](https://mlsanity.com/paper/2406.12744/) (2024)
- [A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizations](https://mlsanity.com/paper/2407.06888/) (2024)
- [Safe Control for Pursuit-Evasion with Density Functions](https://mlsanity.com/paper/2505.15718/) (2025)
- [Robust Stability Analysis of Positive Lure System with Neural Network Feedback](https://mlsanity.com/paper/2505.18912/) (2025)

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

- [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)

We study the local stability of nonlinear systems in the Lur'e form with static nonlinear feedback realized by feedforward neural networks (FFNNs). By leveraging positivity system constraints, we employ a localized variant of the Aizerman conjecture, which provides sufficient conditions for exponential stability of trajectories confined to a compact set. Using this foundation, we develop two distinct methods for estimating the Region of Attraction (ROA): (i) a less conservative Lyapunov-based approach that constructs invariant sublevel sets of a quadratic function satisfying a linear matrix inequality (LMI), and (ii) a novel technique for computing tight local sector bounds for FFNNs via layer-wise propagation of linear relaxations. These bounds are integrated into the localized Aizerman framework to certify local exponential stability. Numerical results demonstrate substantial improvements over existing integral quadratic constraint-based approaches in both ROA size and scalability.