Yes, lucid dreams have been scientifically proven to exist. Stephen LaBerge PhD, one of the scientific researchers at Stanford university and one of the fathers of the field, used a special machine to track eye movements during a dream (these are associated to your rapid eye movement while you’re dreaming). He asked dreamers to point their eyes left and right in quick succession and this activity was recorded on the machine. He found that in rem sleep people have no control over moving their eyes, but it does suggest the participant is dreaming. More information on this and other experiments of Stephen Laberge are in his book Exploring the World of Lucid Dreaming.