Simple solution: cheatgrass is an annual. Kill it before it goes to seed and you've broken its life cycle. Ranchers control it by putting cattle and horses on the range when the cheatgrass is still green and "graze the H out of it." A lawnmower works, too.
One problem, though: it's a C4 grass (aka "warm-season grass"). It does real well in drought conditions, such as those now occurring in most of the US southwest. As the climate shifts to dryer conditions, cheatgrass is going to think it died and went to heaven.
If I have seen farther than other men, it is because I stood on the shoulders of giants. --Albert Einstein
Do not meddle in the affairs of dragons for thou art crunchy and go good with ketchup.