187.4.a. If a player controls Units at a Battlefield, outside of Combat, they maintain Control of that Battlefield for as long as they have Units at that Battlefield.
187.4.b. While a Combat or Showdown is ongoing at a Battlefield, Control of that Battlefield cannot change until instructed by steps of the Combat or Showdown.
187.4.c. If a player has no Units at a Battlefield and the turn is in an Open state, they lose Control of that Battlefield in the following cleanup, unless there is a Combat or Showdown ongoing there.