Thank you for the question. It sounds like you are planning on using lamium, switch grass and sumac for erosion control. Remember that root length isn't the only factor in determining a plant's value in erosion control. Growth rate, type of rooting structure and density all contribute.
Lamium, Lamium maculatum, is a great perennial ground cover for moist, shady areas that blooms all season and is attractive to pollinators. On a personal note, my lamium does well in 1/2 day sun. Lamium is in the mint family and spreads readily, at a moderate rate. I couldn't find research based information on root length but in my garden, it tends to root down 3-4 inches. While this doesn't seem very deep, the roots do a good job of very firmly holding the plant and the soil around it in place. The plants cover the ground, preventing weeds from taking control and rain from pounding the soil. To remove a plant, I generally have to dig it up because it doesn't pull up easily. Mature root size is achieved in 1 growing season.
Switchgrass, Panicum virgatum, is a native warm season grass. Roots can extend down 10 feet but most of the root mass is in the top 12 inches of soil. It is excellent for holding soil in place and spreads slowly. Native plants reach maturity slowly but by the 3rd year, most of the roots should be mature, gradually increasing over time.
Smooth Sumac (Rhus glabra) is a shrub native to the Eastern U.S. There are other varieties as well. Sumac has a shallow, wide, fibrous, suckering root system and a fast growth rate. It should do a good job of erosion control. Again, I couldn't find research based information on root length, but I believe sumac roots are most commonly found in the top 10-12 inches of soil and this is achieved by the 2nd year of maturity.
Thank you for contacting Extension.