Consider a system $\dot x = A(t)x$, where $A(t)$ is a matrix satisfying $A(t)\rightarrow A$ exponentially fast. For each fixed $t$, $A(t)$ is Hurwitz (all the eigenvalues in the open left half plane). However $A$ has two eigenvalues on the imaginary axis and all the others in the open left half plane. I have a few questions:
What is the stability of $\dot x = A(t)x$?
Consider another system $\dot y = Ay$, where $y(0)=x(0)$. Will $x(t)$ converge to $y(t)$?
What is the steady state solution to $\dot x = A(t)x + Bu(t)$ where $u(t)$ is a sinusoid? Will it converge to a sinusoid?
I have been searching for the answers to the three questions. Most of the control books assume that $A(t)$ is uniformly Hurwitz, i.e., the real parts of the eigenvalues of $A(t)$ are smaller than a negative number, so it won't apply to my problem. I think for 1 the right answer should be the system is only marginally stable. However, some references confirming that would be great.
For the second question, I have mainly looked into continuous dependence of ODE on the right hand side. I don't know whether that is the right direction. My guess of the second question is no, however, I would like to know how to analyze this kind of problem.
The third question is my main interest. Will $x(t)$ converge to $y(t)$ in $\dot y=Ay+Bu(t)$? If not, is the difference bounded?
Any references and suggestions will be appreciated!