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It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference where this result is extended to the case of a differential equation in a finite-dimensional Lie group. In particular, I need the parameter space be merely a metrisable topological vector space.

(If this question and its answer are folklore and thus not appropriate for Mathoverflow, I apologize in advance)


In more detail, one finds the following result (slightly paraphrased) in Dieudonne's Treatise on Analysis Vol.1 (10.7.2):

Let $ I \subseteq \mathbb R $ be an open interval and $P $ be a metric space (the parameter space). Consider the linear differential equation in the finite-dimensional vector space $ E $ \begin{equation} \dot{x}(t) = A(t, p) \cdot x(t) + b(t, p), \end{equation}
where $ A: I \times P \to L(E, E) $ and $ b: I \times P \to E $ are continuous. If $ t \mapsto u(t,p ) $ is a solution to the initial value problem $ u(t_0, \cdot ) = x_0 $, then $ u $ is continuous in $I \times P $.

I'm interested in the case where $E$ is not a vector space but a finite-dimensional Lie group and the differential equation is adapted accordingly.

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See appendix B of the book Lie groups by Duistermaat and Kolk, 2000.

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    $\begingroup$ I think you refer to Theorem B3, which however only handles the case of a scalar parameter $\epsilon \in \mathbb{R}$. Thanks nonetheless! $\endgroup$ – Tobias Diez Dec 7 '13 at 17:21
  • $\begingroup$ Indeed, but this result extends straigthforwardly at least to normed vector spaces (not necessarily complete), see for example Theorem A6 in my book (or also PhD thesis). Note that there I prove a bit more abstractly that the flow depends smoothly on the vector field in a uniform context, but the same argument can be used to prove that it depends $C^k$ on parameters in a normed space. I don't know if this implies metrisable TVS's. $\endgroup$ – Jaap Eldering Dec 8 '13 at 12:47

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