mlsanityData to 7 Oct 2026

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

arXiv 2110.00675 ↗First posted 1 Oct 2021cs.LG cs.RO cs.SYcited by 42

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Abstract (arXiv)

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit the systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is, therefore, to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

Key facts. Cited by 42 papers in the map (data through 7 Oct 2026). Builds on 29 of its 112 references that are in the map. First posted on arXiv 1 Oct 2021.

Builds on

29 of 112 references.

References · 29

  1. 2016Understanding deep learning requires rethinking generalizationcited by 1,657
  2. 2016Generative Adversarial Imitation Learningcited by 1,576
  3. 2017Spectrally-normalized margin bounds for neural networkscited by 681
  4. 2017Exploring Generalization in Deep Learningcited by 595
  5. 2017Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Datacited by 527
  6. 2017Safe Model-based Reinforcement Learning with Stability Guaranteescited by 347
  7. 2020Neural Lyapunov Controlcited by 142
  8. 2018Motion Planning Among Dynamic, Decision-Making Agents with Deep Reinforcement Learningcited by 101
  9. 2017Imitating Driver Behavior with Generative Adversarial Networkscited by 92
  10. 2018The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systemscited by 87
  11. 2018Neural Lander: Stable Drone Landing Control using Learned Dynamicscited by 82
  12. 2015Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Designcited by 38
17 more
  1. 2019Control Regularization for Reduced Variance Reinforcement Learningcited by 34
  2. 2019Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularizationcited by 32
  3. 2019MPC-Net: A First Principles Guided Policy Searchcited by 28
  4. 2019Robust Regression for Safe Exploration in Controlcited by 24
  5. 2020Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approachcited by 22
  6. 2020GLAS: Global-to-Local Safe Autonomy Synthesis for Multi-Robot Motion Planning with End-to-End Learningcited by 21
  7. 2019Episodic Learning with Control Lyapunov Functions for Uncertain Robotic Systemscited by 19
  8. 2020Neural-Swarm: Decentralized Close-Proximity Multirotor Control Using Learned Interactionscited by 19
  9. 2020Neural-Swarm2: Planning and Control of Heterogeneous Multirotor Swarms using Learned Interactionscited by 19
  10. 2018Beyond Convexity -- Contraction and Global Convergence of Gradient Descentcited by 17
  11. 2020Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systemscited by 15
  12. 2021Learning-based Robust Motion Planning with Guaranteed Stability: A Contraction Theory Approachcited by 13
  13. 2020Safe Feedback Motion Planning: A Contraction Theory and L1-Adaptive Control Based Approachcited by 11
  14. 2020Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimizationcited by 11
  15. 2018Input-to-State Safety With Control Barrier Functionscited by 10
  16. 2018Learn Fast, Forget Slow: Safe Predictive Learning Control for Systems with Unknown and Changing Dynamics Performing Repetitive Taskscited by 7
  17. 2019Data-driven computation of invariant sets of discrete time-invariant black-box systemscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 6; 2023: 7; 2024: 6; 2025: 13; 2026: 7.

42 papers.

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  1. 2022A Survey on Generative Diffusion Modelcited by 238
  2. 2022Statistical Learning Theory for Control: A Finite Sample Perspectivecited by 52
  3. 2024Neural Contractive Dynamical Systemscited by 14
  4. 2022Robust Classification using Contractive Hamiltonian Neural ODEscited by 13
  5. 2024Learning Neural Contracting Dynamics: Extended Linearization and Global Guaranteescited by 11
  6. 2023Beyond Black-Box Advice: Learning-Augmented Algorithms for MDPs with Q-Value Predictionscited by 8
  7. 2026From Prior to Pro: Efficient Skill Mastery via Distribution Contractive RL Finetuningcited by 8
  8. 2025A roadmap for AI in roboticscited by 7
  9. 2025Predictability Enables Parallelization of Nonlinear State Space Modelscited by 7
  10. 2021A Theoretical Overview of Neural Contraction Metrics for Learning-based Control with Guaranteed Stabilitycited by 6
  11. 2023Learning Over Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems (Extended Version)cited by 6
  12. 2023Learning Control-Oriented Dynamical Structure from Datacited by 5
30 more
  1. 2024Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recoverycited by 5
  2. 2025Decision-Dependent Stochastic Optimization: The Role of Distribution Dynamicscited by 5
  3. 2025Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systemscited by 5
  4. 2024Long-term Safe Reinforcement Learning with Binary Feedbackcited by 4
  5. 2026LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layerscited by 4
  6. 2021Learning Contraction Policies from Offline Datacited by 3
  7. 2024Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overviewcited by 3
  8. 2022Neural-Rendezvous: Provably Robust Guidance and Control to Encounter Interstellar Objectscited by 2
  9. 2022Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systemscited by 2
  10. 2025Robust Stability Analysis of Positive Lure System with Neural Network Feedbackcited by 2
  11. 2025Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to Q-Learningcited by 2
  12. 2022Interstellar Object Accessibility and Mission Designcited by 1
  13. 2023Anytime-Competitive Reinforcement Learning with Policy Priorcited by 1
  14. 2023Stable Modular Control via Contraction Theory for Reinforcement Learningcited by 1
  15. 2024Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regionscited by 1
  16. 2025Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representationscited by 1
  17. 2025React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-RENcited by 1
  18. 2025Distributionally Robust Imitation Learning: Layered Control Architecture for Certifiable Autonomycited by 1
  19. 2026Contractive Diffusion Policies: Robust Action Diffusion via Contractive Score-Based Sampling with Differential Equationscited by 1
  20. 2026Controller Design for Structured State-space Models via Contraction Theorycited by 1
  21. 2021Guaranteed Trajectory Tracking under Learned Dynamics with Contraction Metrics and Disturbance Estimationcited by 0
  22. 2023Sampling-based Reactive Synthesis for Nondeterministic Hybrid Systemscited by 0
  23. 2023CaRT: Certified Safety and Robust Tracking in Learning-based Motion Planning for Multi-Agent Systemscited by 0
  24. 2025Contraction-Aware Reinforcement Learning for Nonlinear Control with Statistical Robustnesscited by 0
  25. 2025A robust and adaptive MPC formulation for Gaussian process modelscited by 0
  26. 2025Robust Convolution Neural ODEs via Contractivity-promoting regularizationcited by 0
  27. 2025Verifying Closed-Loop Contractivity of Learning-Based Controllers via Partitioningcited by 0
  28. 2026Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systemscited by 0
  29. 2026Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flowscited by 0
  30. 2026Manifold-Stable Flow Matchingcited by 0