This isn't being an easy question to answer for you, as the grass is not officially on the state invasive list. I will continue to research it, as perhaps it is a thug or there are concerns and I just have not found the research yet. Are they going to seed already? Is there still time to decide what to do before that happens?
Purple Lovegrass, Eragrostis spectabilis, is native to much of the USA, and is not listed as invasive. One resource says "This plant grows rhizomously and can spread and it will reseed if seeds allowed to ripen." The grass grows mostly in sandy, moist, full-sun sites. Exactly how it will behave in your situation I'm uncertain. When trialing a new planting, I suggest you deadhead before winter and remove the seeds rather than adding them to your compost or allowing them to blow around. In a year to two you'll be able to decide how to reduce any problem or if they need to be completely removed due to thuggish or weedy growth.
Here are my sources for information. The USDA Natural Resources Conservation Service plant profile page includes tabs above the map which you may find helpful: http://plants.usda.gov/core/profile?symbol=ersp. The Lady Bird Johnson Wildflower Center page includes more details and additional links: http://www.wildflower.org/plants/result.php?id_plant=ERSP .