:) the more prep you do, the less weeding you'll do...

and there is nothing that spreads that won't spread further than you want it to.

even gravel migrates!

however, you can minimize it- having your beds framed out has to help, and setting up the 'ring' beds will, too. either desod, or start out with a killing mulch- ten layers of newspaper or two of cardboard (wet. soaking wet, in the case of cardboard. hose down the paper between layers) topped with compost and fine mulch (you can plant in this)

that Lamium looking thing (bigger leaves, longer nodes, striped instead of airbrushed?) may very well be Goutweed. which is what I was going to suggest for that hell spot you have. it WILL spread in full shade- it will spread in full shade over beech roots and under walnut trees. the only other things my parent's property will support is moss.

Pachydandra and Vina Minor both fill in nicely. Hostas, most Astilbes, and corydalis (fumewort) all look nice under trees- I have a Corydalis named 'lutea' and the deeper the shade, the better the plant looks- and it reseeds like mad.

THe lamium weed is not goutweed - I vaguely recall it may be Deadnettle? It has semi-upright purple flower spikes and green leaves that fade to pale green near the stem. It is not Ajuga either. I also have that. I have some goutweed a former owner planted - I've been pulling it up for 8 years and pitching it. Guess this year I'll throw it in the "black hole" with a little compost. Along with every other hunk of groundcover that isn't where it belongs. Surely, something will take. I was hoping to find something evergreen, but sturdy weeds are okay too. I don't need to worry about anything taking over in this area. If it spreads beyond the black hole, it will go into the woods where it has to fight brambles and honeysuckle and Virginia Creeper. That will be a battle of Titans!

How about trying a native groundcover like Sweet Woodruff? I have it growing under my trees and it's blooming these ever so tiny delicate white stars. :)

Here is a link that might be useful: Sweet Woodruff

I have some Sweet Woodruff also, but it hasn't done very well for me. A 4" square has finally expanded to about 2square feet, but that took 5 years in a spot with partial shade and good soil. My "black hole" needs something less finicky. There aren't even any weeds there!

well, if it looks LIKE lamium, and it has the upright spikes with the little weasel-faced flowers, then it IS Lamium- whose common name is Dead Nettle (lamium references the flowers, believe it or not- I don't think they look much like weasels- but then again, I dont think that toad lilies look like toads, either!)

and if it's naturalized- :) you're a lucky homeowner, since they spread and can be divided readily. I just broke one clump up into six, and moved them to my mom's house- (both my mom and my husband's mom adore them for some reason)

Good, as I pull it out of the flowerbeds, I'll just move it to the "black hole". Hopefully it will take there.