I concur with @Cedrus in general. You'll need a barrier at some point.
That lawn looks like Bermuda Grass (I call it Metastasis Grass and Grass out the A--) and it spreads. And spreads. Like diseases.
If that's what you have, then the first order of business will be to shoot the edge where the Ajuga and G out the A meet with some roundup (glyphosate) to create a dead spot. Cordon that off, and keep it cordoned, and the grass will stay where it belongs; likewise for the ajuga. With a bit of time, and attention, both will neatly grow up to the edge, so pretty.
There's some bad news, though, and that is that the Metastasis grass might have really invaded the ajuga more than appears from the picture. In that case, you might want to get a chemical that's formulated to kill Bermuda grass only. (Oh yes, there is such a thing. I have some, in my toxic waste collection somewhere.)
Therefore, it's super important to know what kind of grass you have. If you're not sure, tell us where you are. If you're in Michigan, that's almost certain to be something else. If it's California, the Carolinas, or the like, that might be what you have.
Does the grass turn brown in the winter? If that's the case, it's Bermuda Grass.
I actually don't know what kind of grass I have. I'm in northern Illinois. When I started reseeding my lawn my gardener--who inspected the lawn--did sell me a bluegrass mix, so perhaps that's what he IDed.
If I don't want to use glyphosate (lots of toddlers and pets here), is there any reason I couldn't just snip the ajugas and snip the grass with pruning sheers to restore the edge? From what I can see, the grass has not actually invaded the bed, it just looks like it because the spears of grass are branching in that direction.
You don't have Bermuda grass. Much much too cold where you are.
That might be bentgrass, which also spreads.
In any case, you can certainly try the snip and edge treatment. Be warned that spreading grass can get really aggressive sometimes. I wouldn't worry about roundup, but if you disagree, I can respect your decision. Your ajuga is pretty thick, so that will help.
Snip, clip, edge and see if it works!
And please tell us in a bit, one way or another. You'll help us understand better too, in order to help the next people. :vs_cool:
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