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).