Don't know the name, but my neighbor grows it as ground cover in her raspberry patch. Very pretty stuff, since the patch is in my side yard, I see it all the time.

Perennial for sure, comes back every year, each time sneaking more and more onto my property. Oh well, prettier than my grass...

Not sure what you mean by "right way to do with it".

Thank you very much for your answer. If you are sure they are perennial, I will keep them there.

It's very hard to tell from the picture, especially with the print obscuring much of it, but compare it with Iberis sempervirens.