# Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers

arXiv: https://arxiv.org/abs/2006.07579 · HTML: https://mlsanity.com/paper/2006.07579/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2020.

## Built on by (most cited first, 30 of 44)

- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear System](https://mlsanity.com/paper/2112.04219/) (2021)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity](https://mlsanity.com/paper/2111.12906/) (2021)
- [Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations](https://mlsanity.com/paper/2401.14033/) (2024)
- [Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers](https://mlsanity.com/paper/2103.17106/) (2021)
- [Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systems](https://mlsanity.com/paper/2210.08339/) (2022)
- [Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansion](https://mlsanity.com/paper/2506.01356/) (2025)
- [Convolutional Neural Networks as 2-D systems](https://mlsanity.com/paper/2303.03042/) (2023)
- [Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guarantees](https://mlsanity.com/paper/2404.07373/) (2024)
- [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)
- [Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutipliers](https://mlsanity.com/paper/2202.04592/) (2022)
- [Forward Invariance in Neural Network Controlled Systems](https://mlsanity.com/paper/2309.09043/) (2023)
- [Certified Robust Invariant Polytope Training in Neural Controlled ODEs](https://mlsanity.com/paper/2408.01273/) (2024)
- [Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systems](https://mlsanity.com/paper/2208.13006/) (2022)
- [Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verification](https://mlsanity.com/paper/2310.11104/) (2023)
- [A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizations](https://mlsanity.com/paper/2407.06888/) (2024)
- [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)
- [Robust Stability of Neural Network-controlled Nonlinear Systems with Parametric Variability](https://mlsanity.com/paper/2109.05710/) (2021)
- [Identification For Control Based on Neural Networks: Approximately Linearizable Models](https://mlsanity.com/paper/2409.15858/) (2024)
- [A New Perspective On AI Safety Through Control Theory Methodologies](https://mlsanity.com/paper/2506.23703/) (2025)
- [ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networks](https://mlsanity.com/paper/2510.05261/) (2025)
- [A Unified Representation of Neural Networks Architectures](https://mlsanity.com/paper/2512.17593/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)

## Abstract (arXiv)

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal inner-approximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. This in turn allows for off-by-one IQCs to refine the description of activation functions by capturing their slope restrictions. Both results rely on semidefinite programming to approximate the ROA. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty.