Exponential stabilization of system of equation

Question

Let $X$ be an infinite dimensional Banach space and let $A$ be a square matrix of size $n$. Suppose that the sequence of solutions $y_n$ of some partial differential equation problem is given by $$ y_n(t,x)=A(t)y_0(x), $$ where $y_0$ is the initial state in $X$ with $t \in (n,n + 1)$. I'm loking for a sufficient and a necessary condition so that the problem decay exponentially to $0$ in the space $X$. The sufficient condition is obvious: if we let $$|a_{i,j}(t)|\le e^{-at}$$ for all $i,j$ the inequality holds. For the necessary condition I guess that it has a relation with the rank of $A$ but I can not prove it. Any ideas? Thank you in advance.