Can anybody give me a good reference which under suitable assumptions discusses exponential stability of $0$ for the equation
$\dot{u}_t = A(t)u(t) + b(t)$
Here $u_t\in\mathbb R^n$ is the unknown, $b_t\in\mathbb R^n$, $A(t)\in \mathbb R^{n\times n}$ and all functions are smooth as you want. In fact I'm interested in the case in which also $u$ and $b$ are matrix valued, but this shouldn't make much difference I think.
I didn't find anything on this problem in Arnold's classic ODE book.
I know that $0$ is exponentially stable when $A$ is constant and $b=0$ under the assumption that the eigenvalues of $A$ all have strictly negative real part.
What happens in this more general setting?