We are doing pretty well controlling Japanese Knotweed by depriving it of photosynthesis. I have a small area of the yard that became overrun when the neighbor's patch got tarred over. The stuff made its way to my yard.
My son and I slice it off at ground level as if it were a pesky maple seedling. The roots are more like trees, so I treat the plant like a tree. Keep mowing it down and don't let it go to seed. It will get smaller and smaller, weaker and weaker.
I had success spraying Ortho Brush-Be-Gone on some Knotweed near the fence. Nothing came back there. But i decided not to poison my yard, so I haven't done any more.
The stuff is edible when it is young. It is related to buckwheat and rhubarb. The small stalks in early spring make a great "rhubarb" pie or can be cut up in salads.
The roots are used in Traditional Chinese Medicine for baldness, graying and a number of other imbalances.