  We consider a control system describing the interaction of water waves with a
partially immersed rigid body constraint to move only in the vertical
direction. The fluid is modeled by the shallow water equations. The control
signal is a vertical force acting on the floating body. We first derive the
full governing equations of the fluid-body system in a water tank and
reformulate them as an initial boundary value problem of a first-order
evolution system. We then linearize the equations around the equilibrium state
and we study its well-posedness. Finally we focus on the reachability and
stabilizability of the linear system. Our main result asserts that, provided
that the floating body is situated in the middle of the tank, any symmetric
waves with appropriate regularity can be obtained from the equilibrium state by
an appropriate control force. This implies, in particular, that we can project
this system on the subspace of states with appropriate symmetry properties to
obtain a reduced system which is approximately controllable and strongly
stabilizable. Note that, in general, this system is not controllable (even
approximately).
