Exp. Stability of perturbed system with temporally vanishing perturbation I have a perturbation problem for which I can't find a fitting theorem in Khalil's Nonlinear Systems. Maybe someone can point me in the right direction: Given a nominal system $\dot x(t) = A(t)x(t)$ which is exponentially stable and a perturbed system $\dot x(t) = A(t)x(t)+g(t)$ where $g(t)$ is bounded and as $t\rightarrow \infty$ converges exponentially to zero. I imagine that the perturbed system is exponentially stable as well. Any pointers appreciated.

Ok, it is not true.

Consider $\dot x = -x +e^{-3t}$.

The solution is $x=c_1e^{-t}-0.5e^{-3t}$ with $c_1=x_0+0.5$.

So $\|x\|$ has a part that is independent from $x_0$, which means that for arbitrarily small $\|x_0\|$, the exponential stability condition breaks.