Exponential map/ Lie derivative in variation for constant formula for ODE

Question

In short: The question is how to go from the first equation on page 8, of this paper to the second equation.

Some background I'm working in optimization and I am currently reading a paper see page 8,Proof of Proposition 2.3. where the author uses a somewhat geometric notation which makes it a bit hard for me to grasp:

We are studying an equation $$f'(t) = Tf(t) + V(f(t))=H(f(t))$$ with initial value $f(0).$ Thus, the equation is linear in $T$ and in general nonlinear in $V.$

The idea is to split this up into $$f'_1(t) = Tf_1$$ and $$f'_2(t) = V(f_2(t))$$ because the variation of constant formula teaches us that (now I use the exponential map $e^{tH}(F)f(0) := F(f(t))$, and analogously for $e^{tT}(F)f_1(0) := F(f_1(t))$)

$$e^{tH}(id)f(0) = e^{tT}(id)f(0) + \int_0^t e^{(t-s)H}(D_V e^{sT}(id))(f(0)) ds$$ where $D_V$ is the Lie-derivative with respect to vector field $V$. You find this variation of constant formula in the above paper. The author then says, that since this equation determines $exp(tH)$ we can apply it again for the exponential $exp((t-s)H)$ appearing in the integrand. You see this step carried out by the author in the above paper.

However, I do not see why this is true, as this equation above only determines $e^{tH}(id)$ and not $e^{(t-s)H}(D_V e^{sT}(id))$ which appears in the integrand.

Can anybody shed some light on the arguments of the author of this paper?

Answer 1

I think the key is that the variation of constants formula holds for any vector field $G$, i.e. the vector $G_{\varphi_H^t(v)}$ can be written as $$G_{\varphi_H^t(v)}=\big(e^{tD_H}G\big)_v=\left(e^{tD_T}G+\int_0^te^{(t-s)D_H}D_Ve^{sD_T}G\,ds \right)_v. \tag1$$ Indeed, use the fact from page 7 that $$\frac{d}{dt}\big(e^{tD_F}G\big)_v=\big(D_Fe^{tD_F}G\big)_v$$ in order to verify that $$\frac{d}{dt}\left(e^{tD_T}G+\int_0^te^{(t-s)D_H}D_Ve^{sD_T}G\,ds \right)_v=\left(D_H \left[e^{tD_T}G+\int_0^te^{(t-s)D_H}D_Ve^{sD_T}G\,ds \right]\right)_v.$$

This equation (1) is applied in the first equation on page 8 of the cited paper by setting $G=\text{Id}$ and in the second equation by setting $G=D_Ve^{sD_T}\text{Id}$.