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

• URL: http://arxiv.org/abs/2110.00675v7
• Date: Mon, 06 Oct 2025 20:27:34 GMT
• Title: Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview
• Authors: Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques E. Slotine,
• Abstract summary: Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system.<n>It takes advantage of a superior property of exponential stability used in conjunction with the comparison lemma.<n>This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes.
• Score: 13.228663415967624
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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