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We introduce a data-based approach to estimating key quantities which arisein the study of nonlinear control systems and random nonlinear dynamicalsystems. Our approach hinges on the observation that much of the existinglinear theory may be readily extended to nonlinear systems - with a reasonableexpectation of success - once the nonlinear system has been mapped into a highor infinite dimensional feature space. In particular, we develop computable,non-parametric estimators approximating controllability and observabilityenergy functions for nonlinear systems, and study the ellipsoids they induce.In all cases the relevant quantities are estimated from simulated or observeddata. It is then shown that the controllability energy estimator provides a keymeans for approximating the invariant measure of an ergodic, stochasticallyforced nonlinear system.

Kernel Methods for the Approximation of Some Key Quantities of Nonlinear Systems