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Video reply: This does not sound like ivy. Ivy does not have purple flowers. If ivy is allowed to mature it will have little berries on it (but that would be on the upper tree level, not at the grass level).
The plant in question is probably a plant called Ajuga. If you would like to rid your lawn of Ajuga, a partial solution would be close mowing. The best thing to do would be to allow the Ajuga to spread and extend further, and not have your grass grow that far into the bed.
If you upload a photo, I can take a look and be more specific in answering.This video reply was first broadcast online during our live plant and gardening Q&A show; the video you see here has been edited from a recording of that show. Got another question? Go to the Q&A page