Remark 4.1
The use of
in place of
is a valid way to obtain an approximation of
in view of the results in [
40] which also provide a link between the purely abstract algebraic formalism of Section
3 and the actual solution of (
15).
More precisely, if
is a Lebesgue integrable control function on
, then the image of the Chen-Fliess series
, under the evaluation map
,
, is a formal series of partial differential operators acting on smooth functions defined on the manifold
. If
then application of
to
yields a formal series of
functions on
denoted
. In [
40, Prop. 4.3, p. 698], this series is actually shown to converge to
, the composition of
with the solution of system (
15) corresponding to
. Specifically, it was shown that: for analytic, complete vector fields
, any compact set
and, any compact set
, there exists a time horizon
such that the formal power series
(evaluated at
) actually converges uniformly to
for
, where
is the solution of (
15), with
, for any
, and any integrable
. Furthermore, a precise upper bound was obtained in the same reference for the difference between
and the
-
partial sum of the series
for
:
|
|
|
(18) |
for all
,
,
as defined above, and all
, where
denotes the
truncated series obtained by considering terms only up to order
in the Chen-Fliess series
, and
is a constant.