Migration =/= exploration.
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As far as enforcing population controls on Earth goes, well, the jury is still out on that. Every first world nation currently has birthrates far below replacement. It appears that as countries become wealthier, their birthrates drop. The only western countries currently with a positive population growth are ones with fairly high immigration rates - the US, Canada, and a couple of others.
Japan, on the other hand, is due to lose half its population in the next thirty years. As are many European nations. And, population growth world wide is slowing as countries are becoming wealthier overall.
IOW, the system tends to balance itself without the need for enforcement. Mostly because we have such a massive abundance of resources.
Again, the problem isn't instituting controls. It's that you are instituting controls for the next thousand years. Pointing to real world cultures doesn't apply. No one chooses to move to repressive regimes and then chooses to force their decedents to live in those repressive regimes while choosing to force their decedents to never, ever change those repressive regimes.
And, as far as "we'll die when the sun expands", well, that's 5 BILLION years in the future. If we survive as a species for 5 billion years (that's about 10 times as long as pretty much any other species) which is wildly optimistic, I'm pretty sure that we have other options by that time. Heck, that far into the future, moving the Earth further out into the Solar System is an option.