When does control of an outer region exchange hands?

As soon as you have warped a ship into an uncontrolled outer region, you gain control of that outer region. If your outer region becomes attacked, you still control it for as long as you have at least on ship in it. The moment your last ship leaves the outer region, you lose control of the outer region and it immediately becomes either a) uncontrolled or b) controlled by an opponent who has at least one ship in it.