mlsanityData to 7 Oct 2026

Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers

arXiv 2006.07579 ↗First posted 13 Jun 2020eess.SY cs.SYcited by 42

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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.

Key facts. Cited by 42 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Jun 2020.

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Citing papers per year: 2020: 1; 2021: 6; 2022: 6; 2023: 5; 2024: 9; 2025: 10; 2026: 5.

42 papers.

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  1. 2020Imitation Learning with Stability and Safety Guaranteescited by 34
  2. 2024Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulationcited by 23
  3. 2021Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systemscited by 15
  4. 2021Neural Network Verification in Controlcited by 14
  5. 2021Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear Systemcited by 13
  6. 2022Neural network training under semidefinite constraintscited by 11
  7. 2024ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networkscited by 10
  8. 2021Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativitycited by 7
  9. 2021Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multiplierscited by 6
  10. 2022Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systemscited by 6
  11. 2024Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activationscited by 6
  12. 2025Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansioncited by 6
30 more
  1. 2023Convolutional Neural Networks as 2-D systemscited by 5
  2. 2024Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guaranteescited by 4
  3. 2024Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networkscited by 4
  4. 2024Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllerscited by 4
  5. 2022Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutiplierscited by 3
  6. 2024Certified Robust Invariant Polytope Training in Neural Controlled ODEscited by 3
  7. 2022Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systemscited by 2
  8. 2023Forward Invariance in Neural Network Controlled Systemscited by 2
  9. 2023Local Lipschitz Constant Computation of ReLU-FNNs: Upper Bound Computation with Exactness Verificationcited by 2
  10. 2024A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizationscited by 2
  11. 2025Robust Stability Analysis of Positive Lure System with Neural Network Feedbackcited by 2
  12. 2025Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraintscited by 2
  13. 2021Robust Stability of Neural Network-controlled Nonlinear Systems with Parametric Variabilitycited by 1
  14. 2024Identification For Control Based on Neural Networks: Approximately Linearizable Modelscited by 1
  15. 2025A New Perspective On AI Safety Through Control Theory Methodologiescited by 1
  16. 2025ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networkscited by 1
  17. 2026Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorialcited by 1
  18. 2022Revisiting PGD Attacks for Stability Analysis of Large-Scale Nonlinear Systems and Perception-Based Controlcited by 0
  19. 2022Robust Longitudinal Control for Vehicular Autonomous Platoons Using Deep Reinforcement Learningcited by 0
  20. 2023PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEscited by 0
  21. 2023Rational Neural Network Controllerscited by 0
  22. 2025Graph Neural Network-Based Distributed Optimal Control for Linear Networked Systems: An Online Distributed Training Approachcited by 0
  23. 2025Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groupscited by 0
  24. 2025Frequency Response Identification of Low-Order Systems: Finite-Sample Analysiscited by 0
  25. 2025Delay Independent Safe Control with Neural Networks: Positive Lur'e Certificates for Risk Aware Autonomycited by 0
  26. 2025A Unified Representation of Neural Networks Architecturescited by 0
  27. 2026Quadratic Characterizations for Reachability Analysis of Neural Networkscited by 0
  28. 2026Implicit Neural Networks as Static Controllers: Certificates and Performance Separationcited by 0
  29. 2026Scalable Incremental Robustness Analysis of Neural Network Feedback Systemscited by 0
  30. 2026Learning Provable Neural Network Observer for Uncertain Dynamical Systemscited by 0