Answer: Sprays that would kill such weeds are either not labeled for garden use or would be harmful to vegetable plants they contacted. There are however, several options for controlling such persistent weeds, none of which is fast or easy.
First, remember that weeds, no matter how tough, are plants. When a plant is chopped off, it uses stored energy to regrow, and then restores its reserves when the leafy tops start to collect sunlight and make carbohydrates. Anytime you destroy a weed's top and force it to regrow from stored reserves you weaken it. Repeated destruction will eventually kill it, IF you don't allow it time to replenish it's reserves between your attacks!
When you mulch, try using newspaper 8-10 sheets thick covered with some leaves. This will eliminate most of your weed problems. Few weeds can punch through, although they will find tears or gaps in the paper mulch. When they do, lay 8-10 sheets of paper over them and then some leaves to cover the paper. Remember, they can't hold their breath forever. Just stay with it once you start!
Repeated rototilling in the summer is very damaging to even persistent weeds. Allow them to resprout between tillings (sometimes a sprinkling is needed to get them to resprout) and then rototill again. Pick out nutgrass tubers or Johnsongrass runners that surface in the process and throw them away.
If you stay will it you will win the war. Good luck!
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