Stabilization of the linearized water tank system

Jean-Michel Coron (1, 2) , Amaury Hayat (3, 2) , Shengquan Xiang (4) , Christophe Zhang (5, 2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 Département de Mathématiques - EPFL
5 Department Mathematik [Erlangen]

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.

Mots clés: Saint-Venant equations; controllability; feedback equivalence; backstepping; rapid stabilization AMS Subject Classification. 35L04; 93D20; 93D15

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jean-Michel Coron, Amaury Hayat, Shengquan Xiang, Christophe Zhang. Stabilization of the linearized water tank system. Archive for Rational Mechanics and Analysis, 2022, 244 (3), pp.1019-1097. ⟨hal-03161523v1⟩