  In this paper, an adaptive non-parametric method is proposed to estimate the
scalar-valued nonlinear function that appears in uncertain systems governed by
ordinary differential equations (ODEs). By employing an infinite-dimensional
reproducing kernel Hilbert space (RKHS) as the hypothesis space, the nonlinear
estimation problem in finite-dimensional Euclidean space is recast into that of
constructing a linear observer in the infinite-dimensional RKHS. The analysis
of convergence is facilitated by the introduction of a novel condition of
partial persistent excitation (partial PE), which is defined for a subspace of
the RKHS. Using this condition, we prove that the projection of the function
estimation error onto the PE subspace converges in norm asymptotically to zero.
While this is an abstract notion of convergence that depends implicitly on the
kernel used to define the RKHS, we derive conditions that ensure the pointwise
convergence of the function estimates over the PE subset. This paper
additionally introduces a weaker but geometrically intuitive notion of a
partial PE condition, one that resembles PE conditions as they have been
formulated historically in Euclidean spaces. Sufficient conditions are derived
that describe when the two conditions are equivalent. Finally, qualitative
properties of the convergence proofs derived in the paper are illustrated with
numerical simulations.
