  This paper studies an infinite time horizon LQR optimal control problem for a
system describing, within a linear approximation, the vertical oscillations of
a floating solid, coupled to the motion of the free boundary fluid on which it
floats. The fluid flow is described by a viscous version of the linearized
Saint-Venant equations (shallow water regime). The major difficulty we are
facing is that the domain occupied by the fluid is unbounded so that the system
is not exponentially stable. This fact firstly raises challenges in proving the
wellposedness, requiring the combined use of analytic semigroup theory and of
an interpolation technique. The main contribution of the paper is that we show
that, in spite of the lack of exponential stabilizability, we can define a
wellposed LQR problem for which a Riccati based approach to design feedback
controls can be implemented.
