I passed a guy the other day that was burning a field with broomsedge in it. He burns fields every few years. He always has broomsedge. Like others have said, if you'll get your pH and soil nutrition levels up, you'll have less Broomsedge. Also, if you can increase your grazing pressure, you can force the cows to eat the broomsedge (when it's green) along with the more desirable grasses. It really only grows so well because pastures aren't amended properly and rotational grazing isn't utilized as it should be.