  This paper studies the relationship between the positive limit sets of
continuous semiflows and the newly introduced definition of persistently
excited (PE) sets and associated subspaces of reproducing kernel Hilbert (RKH)
spaces. It is shown that if the RKH space contains a rich collection of cut-off
functions, persistently excited sets are contained as subsets of the positive
limit set of the semiflow. The paper demonstrates how the new PE condition can
be used to guarantee convergence of function estimates in the RKH space
embedding method for adaptive estimation. In particular, the paper is applied
to uncertain ODE systems with positive limit sets given by certain types of
smooth manifolds, and it establishes convergence of adaptive function estimates
over the manifolds.
