Stabilization of the Linearized Water Tank System

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