Mr. Kefauver:
Goats seem to be the most oft recommended "natural" control I have encountered in my literature.
Grubbing it out is next most often mentioned. (Sizable project at 1/4 A!)
My Uncle Jim used to just pull it out. Of course, then everything he touched (doorknobs, chairs, etc) passed a little of the oil on to the rest of us -- doggone it, but it's hard not to scratch.
Realistically, RoundUp(TM) ranks high on my list -- but, like most things, it's had some bad press.
Maybe others can jump in here with better info for you, John.
Seems like "Safer" (as in Safer's Insecticidal Oil Soap(TM)) used to market an herbicide, but I don't find a reference for it now. Of course, though they called those salts & oils "natural", they really are "chemicals", like most things -- maybe they ran into controversy there, too!?!?
Good luck with your project, John. Let us know what works!
((By the way, Administration here might break this out to a new thread of its own since the original topic of this thread was about planting.))