Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^t e^{A(t-s)}b(s)ds=\vec{0}$.
Well, it's clear with $\lim\limits_{t\to\infty} e^{At}x_0 = \vec{0}$. But I'm struggling with $\lim\limits_{t\to\infty} \int\limits_0^t e^{A(t-s)}b(s)ds$. Can someone please give me a hint?
I feel like I don't know some property which should be used here, such as a property of integrals.