with most groundcovers, it's not the seeds, but the runners and underground root nodes that will cause spreading...

but if it's in full sun, I would definately consider any of the creeping sedums- they're easy to pull up when they start creeping into the lawn, their flowers are usually subdued, but their foliage can be showy...

daylily varieties are the classic option for a ditch, though- they stabilize the bank, don't look too ratty out of season, and yes, they will smother the competition.

I have the same problem on a lot that I have to upkeep in a neighborhood where I have not yet been able to build a house.

The banks are too steep to mow, and their steep slant also makes the banks stay very dry since the sides face east and west and since the slope drains very quickly.

The spring before last, I planted some 6 inch tall growing ornamental blue grass plugs that where supposed to do well in dry areas. This last February I saw that the grass had not spread much at all. The plugs had just begun looking healthier. Well this spring I thought some groundcover 'Fairy' roses should fill in faster and accomplish on my banks what you expressed was need on yours.

The good thing about 'Fairy' groundcover roses it that they root as they go, but do not spread underground by their roots. As far as keeping them from becoming invasive goes, I think top spreaders are much easier to contain than those that spread by roots. For sure they are easier to contain than the ground covers and grasses that spread by roots and top spreading growth. Those multiple form speaders are almost as hard to keep up with as the groundcovers that spread by seed.

The 'Fairy' grows a nice thick mat very quickly; In the growing season it blooms continuously. it greens up much sooner after the winter freezes than most roses, and small pink flowers that fade to white in sun do not need to be deadheaded. and the canes do not have to be cut back over the dormant period.

Keeping them out of places where they are not wanted is just a matter of trimming them back every once in a while. When mine fill in that much and start to over extend the area, I just plan to use a hedge clipper to keep the perimeter line clean. The trick will be to catch the spreading canes before they root out side of the desired perimeters.

I do not know how my efforts of planting them will turn out. Hopefully all goes as planned or at least creates a situation that avoids erosion and tall weed problems.

One good thing I have noticed is that 'Fairy' roses are very good at living through dry times; even recovering if during the early spring someone cuts all the canes back to ground level. That was a thing my husband did by mistake to one of them after only 6 weeks from the time I had planted it. A month later the roots of that rose, which he had not shredded , had grown new top growth. The little vines were pretty short, but they seemed viable. I was in awe of how they could do that just like some established rose bushes do from being cut back more carefully to go through the winter.

You could try the dreaded Yelow Archangel (lamiastrum). It will eventually beat out anything with a thin, flexible stem, and even fairly sturdy plants up to 18 inches or more.

It is an attractive nonclimbing plant with white leaf markings which has a growth pattern similar to vinca, but faster. It can do well in dry shade, but I suspect it would also do well in a steep ditch.

You might try some swamp plants in the bottom of the ditch if there is much moisture.

Another possibility might be some of the larger grasses that are so popular now--clumping or spreading. There is also a small spreading bamboo that you could check into.

I think you could contain any of the above with mowing, although you might want to check further before planting any bamboo.

May I recommend Yarrow (Achillea millifolium)? There are many bloom colors and heights, and it will live just about anywhere. It is a thug in your garden, pushing out all your other plants and expanding very quickly, so it sounds like just the thing for your culvert! The foliage is lovely and fern-like, and it's easy to pull out if it starts to go where you don't want it. In your area it should be evergreen, too.