Eradication can be accomplished by pulling the weeds or herbicides. I know trying to eradicate weeds in a woodland is very challenging. If I have to kill plants in a setting where there are plants I don't want harmed I cut off
the plant to kill, enclose the stem in a cylinder made out of a grocery bag with the bottom cut off, spray the cut stem with Roundup. If you can keep the plants you can see from going to seed control will be much better.