I posted this question Mathematics SE and got no answer so I have posted it here.
I learned in my signal processing class that an LTI system can be defined using a linear constant coefficient differential equation. Whenever we have 'initial rest' condition, the LTI system is causal. Initial rest condition is basically the output $y(t)$ is $0$ till $x(t)$ becomes non-zero for the first time.
$$\sum_{i=0}^N a_i y^{(i)} = \sum_{i=0}^M b_i x^{(i)} \hspace{1cm} initial \hspace{0.25cm} rest \hspace{0.25cm} condition $$
We get the Laplace transform of its transfer function as
$$H(s) = \frac{\displaystyle\sum_{i=0}^M b_i s^{i}}{\displaystyle\sum_{i=0}^N a_i s^{i}} $$
and more importantly, its ROC is the half plane right of the rightmost pole (condition for causality).
My question is how can I modify the initial condition to get any ROC I desire? For example, what should be the initial condition if I want my ROC to be left half plane, left of the leftmost pole? What if I want my ROC to be a vertical strip between two adjacent poles having real parts as $\alpha$ and $\beta$, ROC $= \{s| \alpha < \textrm{Re}\{s\} < \beta\}$?
I have no idea how to approach this problem.
