Weeds under Lilac Bushes
Preen is a pre-emergent herbicide suitable for use around established plantings. It will prevent annual weed seeds from germinating.
Unfortunately, your photo shows an infestation of garlic mustard:
Garlic mustard is a noxious weed in Minnesota. Preen is not labeled for garlic mustard, and since garlic mustard is a biennial, Preen is not suitable for it. The above references give controls. I think using glyphosate (Roundup) now, before the lilac leafs out, would be a good idea. Use it carefully, however, since it will kill anything green and growing. Hand pull anything the glyphosate misses. Don't let it go to seed. Once you've got the garlic mustard under control, mulch with several inches of wood chips or shredded bark.