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.