Purple Loosestrife, Lythrum, is a perennial which spreads by seeds but once established it returns year after year after year and continues to spread seed. There are a number of varieties and some have been used in perennial gardens, though it wouldn't surprise me if it were on a list of invasives that can no longer be propagated and sold in nurseries.
One way to get a small amount under control is to pull it out, roots and all, and don't let it go to seed. This may have to be repeated for several years as it is difficult to completely remove an entire root and even the smallest piece left behind can regrow next year. Nature is sure reilient but it is your will power versus the plant's! What a thought, hey?
If that open space is part of your view and the town (usually a conservation commission) could offer some support, even if only moral support, and possibly encouragement. Would they object or help if you replaced the purple loosestrife with meadow type flowers or even some plantings which could attract birds and butterflies?
Watch out, though. Gardening and bird watching become addictive and then you take it hiking and start seeing things in a whole nuther perspective.
