is it necessary to have control system always stable?

Answer 1

if it is not stable then there is not much to control?
so by definition : yes control systems need to be stabilised first, then we look at performance

Answer 2

Yes, would you want to ride in an airplane that had a control system that oscillated and the plane rode like a roller coaster?

Answer 3

General speaking is that it is a stable state nearby dumping,oscillation,stop-start and reset condition also occurs.