Yep, we have that in our vegetable garden and in the driveway which is limestone gravel. I hand-pull the stuff in the garden getting it when it's young. Purslane is a hardy plant and according to Euell Gibbons makes a fine cooked vegetable as well as tender young leaves used in salads as it has a sour flavor. To keep it out of your lawn, you'll want to make sure it doesn't go to seed each year as when the seed is at ripe stage, they shoot off in all directions and the seeds can lie dormant till next year. The seeds need lite to germinate so that tells me that one, your'e letting it go to seed and disperse, and two that you are probably mowing your lawn too short. A thick lawn will keep the sunlight from germinating the seeds. I'm not sure if a broad leaf weed killer used in the early warm months will work, as I've never had to deal with it in our lawn. I have used Ortho's Weed B Gon Chickweed Clover and Oxalis Daisy Killer on common Wood Violet that 2,4-D doesn't seem to effect, so I would suggest it, mow higher and fertilize with a lawn fertilizer spread out through the season, like about 3 or 4 times.