Can I get full custody of my one month old and can I make sure the father has no rights?

His father is wanting to get rights to him due to the fact that I will not allow my son to stay with him alone, his father does have 3 more kids who he doesn't care to see as he does have rights to those kids. He has to take Anti-psychotics and has been locked up many times in the mental hospitals. He also has many felonies and has been known to use and make meth. He hasn't tried to be there for his son until today when I decided I may move out of state for a better life for my son and I. We are not married but he did sign the birth certificate. I would like for him to have no rights if possible or visits unless I am there. I do not feel my son would be safe around him alone. While pregnant, I found out he was using meth and left him, since then he has been back and forth on it. He does not pay for anything when it comes to his kids, not even child support. Is it possible to make sure he has no rights at all?