Exponential stability of density-velocity systems with boundary conditions and source term for the H2 norm

Abstract

In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are typical hyperbolic systems that are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the H2 norm using simple local boundary feedbacks, provided a condition on the source term is valid. This condition holds for most physical systems, even when the source term is not dissipative. Besides, the feedback laws obtained only depend on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system. This makes them both very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback law does not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic H2 Lyapunov function. We apply it to several systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, the traffic flow, etc.