We have an invasive plant called Creeping Charlie that is taking over whose root system gives it a superior advantage over most remedies to deal with this sort of thing. Anyone tried anything that works?
All the goodness and the heroisms will rise up again, then be cut down again and rise up. ~Steinbeck
Creeping Charlie is a common name given to several plants. The most common one is a relative of mint, that has square stems and a strong aroma. It is a shade lover, exposing it to more sun would probably kill it.
Sheet mulching smothers out some plants that spread from rhizomes and runners, might be worth trying (put "sheet mulch" into the search here on Permies, lots of threads about it.) Put "Johnson grass" into the search bar here too, for ideas, it's a rhizome and seed thrower plant that is also an utter PITA to eliminate.
Might try taking it down to ground level with a mower set so low it's throwing dirt, with a bag on it (throwing the cut parts is spreading it more) many times, very often, keeping the plant from having any chance to recover. That will leave a bare dirt spot every time you cut, and it will erode until you can reseed it, but you can't reseed until the Charlie is dead, so you will probably lose soil to erosion.
It's also throwing seeds, so if you want it out, you will need to keep it from setting seed too.
Crowding it out with things you like more may work. Personally, I'd go for a plant that makes a dense root mat, like monkey grass (which is not a lawn grass) that might take up all the space the Charlie is exploiting. Most opportunistic plants like this tend grow best on disturbed soils, with no competing growth, but not thrive if there's competition. (look up Pioneer Species, I'd put this in that category.)
The most important question is why do you want it gone? It tends to grow in shaded areas as a ground cover, if it's the same species I have, I like it, keeps the dirt from exposure and erosion where most things won't grow. The bees adore it. Is it really worth the fight to eradicate it? Saying "I have a shade tolerant, sweet smelling, bee attracting ground cover!" might sit easier in your head than "I have a noxious weed." It's certainly less effort! :)
And I'll happily trade you Johnson Grass for it, I learned most of what I just wrote by learning the hard way when I moved that in this area any disturbed soil turns into Johnson Grass, and that stuff will get 12 feet tall behind my barn when I am not looking, I can't even get a brushcutter through it. Whee. I've learned to seed every piece of soil I touch with at least clover, it germinates faster than JG.
The best way to kill a plant is with itself: the principle of ju jitsu. Use its strengths against itself. If it is a creeping mat forming plant then mow it regularly and leave the cuttings where they fall. it will eventually smother under the sheet mulch. If you are in a hurry, throw more mulch on top, preferably leaves, from another strong weed or from a another patch of the same weed. Make sure you police the parameter regularly, pulling up all the stolons and throwing them back onto the offending weed. This method takes a year or two of constant vigilance but it works.
Also ask yourself what are the conditions in the ecosystem which allows this weed to thrive? I have waged an unending battle against invasive Kikuyu grass in all the years I have farmed here, the place was covered with it when I moved here. Eventually I realized it is the habit of regular mowing which gives it its strength. People like to mow so it looks 'nice', and indeed it has to be kept short in summer because of fire hazard and because it provides a habitat for snakes. So now what I do is let is grow without mowing all winter, it chokes itself in its eagerness to grow. Then mow once in spring, letting it bake in summer and otherwise leave it be. It is much less invasive with this treatment.
Another thing I do is work with edge plants which are strong enough to resist invasion. A new piece of land gets cleared it is dug out to knee depth with the top layer of kikuyu laid at the bottom of a swale which directs water into the bed. It gets smothered by the soil on top of it. Then I literally rebuild the soil, adding lots of biomass and covering with the subsoil. A season of green manure (usually duff peas) which gets cut on flowering and left to rot, and we are good to go. The swale gets planted with chop and drop shrubs, the edge gets planted with wild garlic (mole control) and outside of that a circle of Kikuyu resisting plants. Vetiver is good for this if you have enough water and no frost. But my favourite is silver wormwood 'Powis Castle'. It roots easily, needs no water once rooted, suckers well and keeps the Kikuyu out, but is not so invasive that I have to fight to keep it out of the bed. Oh and did I mention it is pretty? When the wormwood gets too happy, it is lovely to chop the top handsbreadth and use for an insect repellent fine mulch which breaks down into the most beautiful soil. This might sound like a lot of work but is a lot easier than spending my life trying to keep Kikuyu out of the bed.