Poa annua grass, also known as annual bluegrass, is a very invasive species and extremely difficult to control. There is no herbicide that is poa annua-specific, and the only herbicides that currently control this weed when full-grown kill all other grasses as well. But because poa annua is a cold-weather grass and dies back during the hot summer months, there is one type of herbicide that can be effective against this weed without killing everything else. It is a pre-emergence herbicide that is applied in the fall, just before the sleeping poa annua seeds sprout for the winter.
Mow the summer grass low, down to 1 inch high in late fall, just before the weather starts turning cold for the winter in your area. Do this before the first frost.
Spray the area with a pre-emergence herbicide containing prodiamine. Consult your local nursery for the best brand for your area and your particular needs. Follow all manufacturer recommendations. Never spray any herbicide in windy weather and don't spray if rain is forecast for the next 24 hours. Always wear gloves and eye goggles when spraying any herbicide.
Repeat the pre-emergence spraying after 30 days.
Wait seven more days and then rake the area to score the top of the soil. Overseed with a cold-weather rye grass that will compete with any remaining poa annua that might survive the herbicide. Consult with a local nursery for the best type of cold-weather grass for your area. Follow the manufacturer's recommendations for how much grass seed to use.
Cover the grass seed with 1/4 inch of composted manure, and water well. The pre-emergence herbicide should dissipate before the new grass seed sprouts, allowing the new grass time to crowd out any remaining poa annua.
If your poa annua returns, then repeat the entire process, including reseeding with a winter grass, the following fall.