I fought this same battle 3 years ago. My resolution was a paint brush and a can full of Round-UP. As soon as I cut it a few inches from the ground, I painted a nice coat of full strength roundup. 3 years later, not a single new sprout.
How about cutting it all down as low as possible & then put a layer or 2 or 3 of weed barrier over it ... I assume it needs at least some sun?
Or maybe cut it all down & just keep driving over it, 2 or 3 times a week. Not too many plants like to be continuously driven over.
All I can say is good luck...
http://www.tractorbynet.com/forums/projects/177888-endless-battle.html