I am looking for a generic estimate to the following problem coming from biology:

I am solving the ODE

$$y'(t)=Ay(t)+zf(t), y(0)=0.$$

where $f$ is an external force determined by us and $z$ a constant vector.

Now $A$ and $z$ come from some measurement, so in general they will be perturbed $\widetilde{A}$ and $\widetilde{z}.$

That is, I am actually solving

$$x'(t)=\widetilde{A}x(t)+\widetilde{z}f(t), x(0)=0$$

on my computer. Obviously, if any matrix $A$ or $\widetilde{A}$ had positive eigenvalues one could not say anything about the long-term dynamics, because there could be exponentially growing modes. Recall that by the Laplace transform the solution to these equations is then actually very simple:

$$\widehat{y}(t) = (t-A)^{-1}z\widehat{f}(t)$$ $$\widehat{x}(t) = (t-\widetilde{A})^{-1}\widetilde{z}\widehat{f}(t).$$

For small times, we can actually measure how close our model is to the true solution, that is by applying arbitrary forces in $L^1$ to the system we find for $t \in [0,T]$

$$\left\lVert y(t)-x(t) \right\rVert \le C \left\lVert f \right\rVert_{L^1[0,T]}$$

So we assume that both $A$ and $\widetilde{A}$ have only strictly negative eigenvalues.

Given that the error is known to satisfy a Lipschitz estimate for small times $t \in [0,T]$ and arbitrary controls in $L^1$.

Can we obtain any sharp ab-initio long-time estimates $t \in [0,\infty]$ of the form

$$\left\lVert y(t)-x(t) \right\rVert \le \widehat{C} \left\lVert f \right\rVert_{L^1[0,\infty]}$$ on this problem?

By ab-initio I mean estimates only depending on $C,\widetilde{A}$ and $\widetilde{z}$?

EDIT: If something similar would hold in any other $L^p$ space, I'd be interested as well. Or if there are any other global time estimates we can draw from this, please let me know.