Why do you want to close your borders, after all your space will soon become my space
Is "Your space is my space, and only my space, so get out now, or I will bomb you" your national theme song?
I tend to think that a better way of defining territorial borders than the influence boundary would be to have a United Planets standard range-from-owned planet boundary line (and if two planets owned by different groups would produce an overlapping region of 'national space', the international standard would be to have the border at the midpoint), and a nationally-claimed boundary that you can set, in the same way that the United Nations recognizes territorial waters to be those within 12 nautical miles of your coast, with overlaps resolved by placing the border at the middle or by treaty, and several nations claiming various other distances from the coastline as their territorial waters. So perhaps the internationally recognized boundary is, say, 3 parsecs from a planet, while you claim everything within 5 parsecs of a planet.
I say this because I really don't like the Influence borders as representations of imperial borders - no rational person is ever going to claim that a system that I never colonized or controlled and which was originally well outside of my area of influence is within my territory if someone else has well-established colonies there, and yet I can make the influence borders shift so that, based on the influence borders, that area is "mine" even if that area happens to be the home system of some other faction and I don't have a colony within 10 parsecs (note that this is a relatively extreme example - by no means do I expect this to be a common situation in any game where the human and computer players are reasonably well balanced against one another).
Additionally, having an internationally recognized standard border gives an interesting issue for United Planet votes - where should the border be drawn - and also provides a potentially interesting point of contention between players by allowing you to lay claim to areas of space in some official fashion, and it provides a way for the players to clearly communicate to others just how much of a safe zone they want to maintain around their worlds (presumably, the national claim would be presented somewhere in the diplomatic relations menu, and the internationally recognized limit would be presented either there or within the United Planets overview, or in both locations). Preferably there would also be some form of indication in the main map of where these borders happen to be, so that you don't need look into the diplomacy menu and then start counting tiles, and hopefully with a way to see both the national claim and the internationally recognized boundary (either at the same time, or switching between the two).