Roundup can last forever in the soil. Anyone who says it doesn't hasn't seen all the studies. Most serious gardeners I've talked to wish that roundup would be restricted more so not every home owner can use it for every little problem. Personally I wouldn't touch the stuff unless I had no choice. Also if you ever wish to list anything from your farm as organic it can have a 5-10year wait from the time you last applied roundup and other herbicides.
Goats and weed whackers would be my choice. Chop it all done a few years in a row and it will start to die. The poison ivy really wasn't so hard to kill. I just cut it down a couple times a year and it didn't really come back. For large areas and stubborn weeds I just get a sheet of black painters plastic or a tarp, throw it over the area, weigh it down with bricks, and walk away for a summer. It will cook and block the light to everything underneath it and by fall you have bare soil. Now bindweed is the only thing I haven't been able to kill. You have to keep it all cut down for 2years to kill the root and it pops back up every week. Stubborn stuff.