  Many existing tools in nonlinear control theory for establishing stability or
safety of a dynamical system can be distilled to the construction of a
certificate function that guarantees a desired property. However, algorithms
for synthesizing certificate functions typically require a closed-form
analytical expression of the underlying dynamics, which rules out their use on
many modern robotic platforms. To circumvent this issue, we develop algorithms
for learning certificate functions only from trajectory data. We establish
bounds on the generalization error - the probability that a certificate will
not certify a new, unseen trajectory - when learning from trajectories, and we
convert such generalization error bounds into global stability guarantees. We
demonstrate empirically that certificates for complex dynamics can be
efficiently learned, and that the learned certificates can be used for
downstream tasks such as adaptive control.
