Can control tell them to stay on duty?
In the signalling grades, no, but I can think of two occasions where I was asked to stay on for a short time.
Once during severe snowfall, when my relief couldn't get in and no one else was available, so I stayed on for an hour and was out up in a local hotel so I could work the next night shift. Doing that meant I didn't have to drive 45 miles home, so the 12 hour between shifts was relaxed.
The other because my relief had a puncture on the way in but estimated he would be no longer than 15minutes late, which proved correct.
However I also know of a signaller who refused point blank to work more than his 12 hour shiftshift, refused to accept Is Line clear? from the preceding Box, sent the closing bells and walked out! And there was nothing the LOM could do about it.