Suppose I have a infinite dimensional dynamical system as
$\dot{x_n}=Ax$
where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find about the stability of this system based on the properties of matrix $A$ and sequence $\{x_n\}$.
More specifically, I was searching for stability properties of dynamical system that is arranged as:
$\dot{x_n}=a_n x_{n+1}$
where $a_n$ is a constant that means each state integrates the next step, but the process does not stop any where.
Thank You!