Polygonum aviculare (s.l.)

Prostrate Knotweed, "Polygonum aviculture", Most easily controlled by yanking out of the soil even though it grows best in compacted soils. Few "weed" killers have any affect on it.

Here is a link that might be useful: Prostrate Knotweed

It does mow down nicely and looks fine during the summer, but you will really notice it later in the fall when the grass is still greenish and the knotweed is rusty brown.

You can easily take it out with glyphosate, however this will take out any grass also and open you up to a big weed invasion, unless you plan to reseed or otherwise re-landscape the area . Probably better to try to aerate the lawn and add some manure/organic matter. Remove the dried weeds in early spring, then reseed grass and get it started. The knotweed will probably come up pretty thick next year. when it does, it will not look like the grass. While small, you shouldn't have any trouble taking it out with lawn care herbicides containing 2,4-D or Dicamba. These won't hurt the grass and will control the knotweed to give the grass time to establish and out-compete the knotweed. You can always spray it with the same products later in the year and get results, but to eliminate the problem you need to get the grass re-established.

As is indicated by Ron Calhoun in the link I provided from Michigan State University few "weed" killers will work on this, and spraying any glyphosate product is not very environmentally sould any more then spraying any other poison is.