  This paper presents a theoretical overview of a Neural Contraction Metric
(NCM): a neural network model of an optimal contraction metric and
corresponding differential Lyapunov function, the existence of which is a
necessary and sufficient condition for incremental exponential stability of
non-autonomous nonlinear system trajectories. Its innovation lies in providing
formal robustness guarantees for learning-based control frameworks, utilizing
contraction theory as an analytical tool to study the nonlinear stability of
learned systems via convex optimization. In particular, we rigorously show in
this paper that, by regarding modeling errors of the learning schemes as
external disturbances, the NCM control is capable of obtaining an explicit
bound on the distance between a time-varying target trajectory and perturbed
solution trajectories, which exponentially decreases with time even under the
presence of deterministic and stochastic perturbation. These useful features
permit simultaneous synthesis of a contraction metric and associated control
law by a neural network, thereby enabling real-time computable and probably
robust learning-based control for general control-affine nonlinear systems.
