grass is grass- if it's healthy, yes, it controls erosion just fine...but it's dull, isn't it?

Crown vetch is the classic 'erosion control' plant where I'm from (of course, I'm from pennsylvania, where it was first promoted) and it's nitrogen-fixing as well, so it's ultimately good for the soil itself...

bugleweed has the advantage of not being too tall, and making pretty drifts of purple flowers that last well into the summer.

there's also the native daylillies- we call them 'ditch lillies' around here for just that reason- they're usually found holding the bank of a culvert or run-off stream together by sheer force of will :)

goutweed does just fine in full sun up here, not sure if it would be a zone warmer, though. the same with spiderwort, both of which are tough, fast growing monters that deserve their 'invasive' reputations.

I've got a 40 foot long full sun slope approximately 3 feet wide that sloped on a 45 degree angle that was a pain to cut grass along so I smothered the grass early last spring and slowly added in last fall these things:
cutleaf stephenandra (which is a low growing shrub & great for irrigation control and spreads nicely by self layering/rooting)
clumps of sedums
clumps of lamium
clumps of dwarf iris' (again a great irrigation control)
clumps of sweet woodruff
clumps of vinca

My slope looks so much better now than it did with terrible looking tough to cut grass.

Good luck and watch the ajuga - bugleweed - even up here, it escaped out of one garden spot and got into the lawn where I did want grass and not the ajuga - I'm still digging the ajuga out - 4 years later in spots I don't want it.