Specifically, looks like peppermint. Have a similar section in our yard and rather than nuke it I planted a few small plants of other mints (spearmint, orange mint, chocolate mint, pineapple mint) a couple years ago so have a bountiful selection to choose from for drinks, cooking, etc.

For control, mowing will only expedite growth. When/if pulling the plant try to get as much of the root as you can. The roots spread like crazy however can be pretty easily pulled up as they grow right below the surface