Answer: I am so sorry you have this problem, it is a huge job to eradicate where it is not wanted. You may be able to weaken it by repeated close cutting, however I doubt this would exhaust it enough to kill it -- although it may give the lawn some opportunity to take better hold and outgrow it. Perhaps vinegar in the super concentrated form would help you keep the foliage in check as one measure in your arsenal toward eliminating it? I do not know if this would be considered an herbicide where you are, however. Here is some information about it. It would only be expected to kill the foliage, not the roots, unfortunately.
Also be sure you do not allow it to go to seed as this is another means for it to spread. You might also consider if there is a way to allow it to grow in some areas as a ground cover which you barricade off with root barriers.
Finally, perhaps there is a provision for a commercial applicator to use chemical controls in certain exceptional cases. This plant is considered an aggressive invasive in some areas. Here is some information about that you may find interesting -- and some chemical and manual control recommendations.
I'm sorry I don't have a magic answer for you.
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