  We introduce a data-based approach to estimating key quantities which arise
in the study of nonlinear control systems and random nonlinear dynamical
systems. Our approach hinges on the observation that much of the existing
linear theory may be readily extended to nonlinear systems - with a reasonable
expectation of success - once the nonlinear system has been mapped into a high
or infinite dimensional feature space. In particular, we develop computable,
non-parametric estimators approximating controllability and observability
energy functions for nonlinear systems, and study the ellipsoids they induce.
In all cases the relevant quantities are estimated from simulated or observed
data. It is then shown that the controllability energy estimator provides a key
means for approximating the invariant measure of an ergodic, stochastically
forced nonlinear system.
