Lespedeza is an open branched annual weed which is present between late spring and the end of summer in Louisiana. There are other varieties of lespedeza, but common lespedeza is what most people have a problem with in their yards. The stems harden and become woody the longer this weeds stays in your lawn. It grows outward into prostrate mats mixed in with your grass.
Good cultural practices help. The best deterrent to lespedeza is a healthy lawn. Deep and infrequent watering helps to develop strong grass roots. Mowing your lawn at the correct height allows it to photosynthesize enough food. Taller grass provides more shade which denies new weeds sunlight. Cutting your grass too low weakens your lawn while allowing more sunlight to fresh weeds.
Beyond cultural changes, pre-emergent herbicides can prevent lespedeza weeds from appearing in your lawn. The main caveat here is you have to do it before the weed is growing each year. Plan to do before next spring. A few pre-emergent herbicides work well against lespedeza. Products with the active ingredient atrazine, simazine, or dithiopyr work very well at prevention. Read the label to make sure you are applying it correctly.
Once you have a lespedeza weed problem, your go to solution is selective post-emergent herbicides. Luckily, there are a number of post-emergent herbicides available that take care of lespedeza. Products with the active ingredient atrazine, dicambia, metsulfuron, or simazine to name a few. There are plenty more herbicide choices. Read the label before purchasing and be certain it is labeled for your grass type, else you may end up damaging the plants you want to keep.