I have usually managed to keep it under control by constantly hand weeding but I was unable to do so during Autumn and early Spring and now I'm suffering for it. Yes, it's particularly bad this year - I'm not in UK but we had heat then rain - so had to make the decision to just dig everything up and move it bare rooted to a weed free site. When we've done that we will spray strong weedkiller (the type used for paths) leave it a month or so then lay grass.
We did try selective weedkiller on the grass to eradicate it from there but it didn't work. In fact, Hubs swears it grew stronger.
My thoughts are with you Asperge