In this paper, we present a model for the controlled flow of a fluid through a network of canals using a coupled system of St. Venant equations. We then generalize in a variety of ways recent results of Coron, d'Ándréa-Novel, and Bastin concerning the stabilizability around equilibrium of the flow through a single channel. This work is based on the theory of quasilinear hyperbolic systems and, in particular, on a delicate result of Li Ta-tsien concerning the existence and decay of global classical solutions.

We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval. Furthermore, we present a Lyapunov function and boundary feedback laws to stabilize a fan-shaped network around a given stationary state.

We consider the problem of exactly controlling the states of the de St. Venant equations from a given constant state to another constant state by applying nonlinear boundary controls. During this transition the solution stays in the class of C^1 -solutions. There are no restrictions on the distance between the initial state and the target state, so our result is a global controllability result for a nonlinear hyberbolic system. Résumé: On s’intéresse ici à un système décrit par les equations de St. Venant. On montre qu’on peut contrôler le système d’un état initial constant à un autre état constant avec des contrôles frontières non linéaires de façon qu’il y ait une solution C^1 . Il n’y a pas de restriction concernant la distance entre les données initiales et les données finales, donc notre résultat est un résultat de contrôlabilité globale exacte pour un système hyperbolique non linéaire.