mrwwandmrs said: It does die back in the cold.
I think I'd use a string trimmer or hedge trimmers (using a grass mower risks rocks or roots damaging the blade) to cut it close to the ground now-
And then immediately cover the whole patch with thick black plastic, anchored on all sides by dirt or rocks or long sticks or lumber.
In the spring, leave the plastic on in an attempt to starve out the patch- for as long as it takes. No sunlight means no photosynthesis which means no energy for the roots.
I use roundup carefully when needed, but it needs to be used when the plant is not dormant, or as it's just going dormant, so I think it might be too late this fall to have much effect with it.
If it were spring now, I'd use the same procedure, but maybe spray with something like 'burnout' right after trimming, then cover with black plastic.
Maybe others on the board have done this?