Approximate controllability of a hydro-elastic coupled system

We analyze the controllability of the motion of a fluid by means of the action of a vibrating shell coupled at the boundary of the fluid. The model considered is linear. We study its approximate controllability, i.e. whether the fluid may reach a dense set of final configurations at a given time. We show that this problem can be reduced to a unique continuation question for the Stokes system. We prove that this unique continuation property holds generically among analytic domains and therefore, that there is approximate controllability generically. We also prove that this result fails when Ω is a ball showing that the analyticity assumption on the domain is not sufficient