Ask Mr. Smarty Plants

Mr. Smarty Plants learns something from his visitors every day. We had never even heard of Rorippa sylvestris, Creeping yellow cress, and now we know it's not good. A member of the Mustard family, it does better in neglected moist areas. This plant is native to western Asia, the Caucasus, and Europe, and therefore falls out of our area of expertise. The goal of the Lady Bird Johnson Wildflower Center and Mr. Smarty Plant is to encourage the use of plants native not only to North America but to the areas in which those plants grow natively. However, we are always interested in controlling invasive plants, native or not. This article from B&D Lilies, Rorippa sylvestris, Noxious Imported Weed states that the plant has come into the United States as a hitchhiker on various imported bulbs. Since this is a lily nursery, they have hard words to say about improper importation, but the information about the plant itself is valuable. Pictures

There are a couple of things that puzzle us about your question. The information we mentioned above,  that this plant does better in neglected moist areas, doesn't sound right for Medina County, in the southwestern part of Central Texas, where moisture in the soil is usually somewhat scarce. Another thing, this USDA Plant Profile map does not show this plant growing in Texas at all, but certainly does in most of the rest of the United States.

Going further, we looked in our Native Plant Database for members of the genus Rorippa, and found three native to Texas, all members of the Brassicaceae, or mustard family. The mustards are well known in Texas for being invasive, and a great many of those are natives, which doesn't excuse their invasiveness. These natives are: 

Rorippa palustris (Bog yellowcress) - pictures

Rorippa palustris ssp. Fernaldiana (Fernald's yellowcress) - pictures

Rorippa teres (Southern marsh yellowcress)  According to this USDA Plant Profile map, this plant is native in the Medina County area. Flora of North America still refers to it as a wet soil plant, found in boggy areas. Pictures

It really doesn't make much difference if the plant is native to your area, it is invasive and you don't wan't it reproducing. First of all, do not allow it to go to seed. If you mow it often enough and close enough, that will be accomplished. But this plant also travels by rhizomes, and digging them out or pulling them out will just encourage spread via those underground stems. Don't spray herbicide, those underground rhizomes are not affected by sprays, which will kill the neighboring plants and give the cress a new opportunity to spread.

We sometimes recommend cutting off a plant at the base, as near the soil as possible and, within 5 minutes, painting an undiluted herbicide onto that cut root, using a disposable sponge brush. You need to do it quickly because the stem will begin to heal over to protect those underground rhizomes. If the invasion is still a small one, and you are willing to persist in finding the plants and using the painting method, as well as mowing, you can make good progress in maintaining it.

Since all these plants like wet, boggy areas, perhaps you should consider improved drainage, or maybe you are overwatering the area?

However, wherever the infection came from, the source is likely still there, and you can never consider the problem permanently solved.