I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi\_t x$ be the value of the solution at time $t$. 


I assume that my ODE has a unique attractor that is exponentially stable: $\|\Phi\_t x - x^\*\| \le \beta e^{-\alpha t}$, with $\alpha>0$ and I define the function $G$ by:
$$ G x = \int\_0^\infty (\Phi\_t x - x^\*) dt$$


I believe that the exponential stability of the ODE plus $f$ being $C^k$ suffices to show that $G$ is $C^k$ but I am not 100% sure and I am not able to find a reference. 


Would anyone know a reference? (my main interest is the $C^2$ case). 



Smooth dependence on the initial condition of the integral of an ODE
I don't know a reference, but here's the rough proof that $G \in C^1$.


First, from the integral equality
$$
\Phi\_t(x+h) - \Phi\_t(x) = \int\_0^1 D\Phi\_t(x+\eta h) \cdot h \,\mathrm{d}\eta
$$
follows the mean-value estimate
$$
\| \Phi\_t(x+h) - \Phi\_t(x) - D\Phi\_t(x) \cdot h \|
\le \sup\_{\eta \in [0,1]} \| D\Phi\_t(x+\eta h) - D\Phi\_t(x) \|\,\|h\|.
$$
This allows us to estimate
$$
\|G(x+h) - G(x) - \int\_0^\infty D\Phi\_t(x) \cdot h \,\mathrm{d}t\| \\
\le \int\_0^\infty \| \sup\_{\eta\in[0,1]} D\Phi\_t(x+\eta h) - D\Phi\_t(x) \|\,\mathrm{d}t\,\|h\|.
$$
To show that this is $o(h)$, we split up the integral into intervals $[0,T]$ and $[T,\infty)$. For any finite $T$, the first part is small since $t \mapsto D\Phi\_t(x)$ is uniformly continuous on $[0,T]$. The second part is exponentially small when $T \to \infty$ due to the exponential stability. Hence $G$ is differentiable and
$$
DG(x) = \int\_0^\infty D\Phi\_t(x) \,\mathrm{d}t.
$$
Similar estimates can be repeated to prove that $G \in C^1$ and by induction that $G \in C^k$.


**Extra details in reply to Willy Wong's comment:**


Since $x^\*$ is exponentially stable and $f \in C^1$, it follows that $x^\*$ is linearly stable, hence $\|D\Phi\_t(x^\*)\| \le C e^{-\alpha t}$. Then for $x$ close to $x^\*$, $D\Phi\_t(x)$ is a small perturbation, hence the exponential estimates are approximately preserved. Since $\Phi\_t(x) \to x^\*$ and $D\Phi\_t(x) = D\Phi\_{t-T}(\Phi\_T(x)) \cdot D\Phi\_T(x)$, we see that when $T$ is large enough, essentially the same estimate holds for any initial $x$. Thus $\|D\Phi\_t(x^\*)\| \le \tilde{C} e^{-\tilde{\alpha} t}$ with $\tilde{\alpha} \approx \alpha$. This implies that the integral of $D\Phi\_t(x^\*)$ is well-defined.


Note that the fact that $f \in C^1$ at $x^\*$ is also implicitly used to argue that $t \mapsto D\Phi\_t(x)$ is uniformly (equi)continuous on any bounded interval.


