  In this article we study the so-called water tank system. In this system, the
behavior of water contained in a 1-D tank is modelled by Saint-Venant
equations, with a scalar distributed control. It is well-known that the
linearized systems around uniform steady-states are not controllable, the
uncontrollable part being of infinite dimension. Here we will focus on the
linearized systems around non-uniform steady states, corresponding to a
constant acceleration of the tank. We prove that these systems are controllable
in Sobolev spaces, using the moments method and perturbative spectral
estimates. Then, for steady states corresponding to small enough accelerations,
we design an explicit Proportional Integral feedback law (obtained thanks to a
well-chosen dynamic extension of the system) that stabilizes these systems
exponentially with arbitrarily large decay rate. Our design relies on feedback
equivalence/backstepping.
