Are the people on stage under the control of the hypnotist?
No they are not under any control but their own. In fact, you have probably seen someone during a stage show who simply didn't react to some skit, so the hypnotist moved on to others. That person might not have felt like doing what was suggested, might have objected to the content or might simply have been feeling really good at the moment and chose to ignore the hypnotist. In hypnosis, you don't relinquish control, you just remove the barriers that were previously in place that prevented you from "going all in" with what you are imagining.
What you see during a stage show is a group of people playing "make-believe" with an intensity that you would not understand unless you had experienced it. The altered state that they have entered allows them to ignore distractions, remove all traces of being self-conscious, and to "get lost" in the suggestions they are recieving. To make them real. If you have seen children playing and seen the intensity with which they can visualize the environment and components of what they are doing, you might realize it's been a while since you have imagined something that vividly.
As adults, we forget how much fun it was to imagine that the ground beneath the monkey bars is "lava" and to do so with such focus that there's no way we are going to be the first to drop. The people on stage have rediscovered how much imagination and play can be.
That is one of the greatest powers of hypnosis!