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 $H^2$ 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 $H^2$ 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.

在本文中，我们解决了具有边界条件的密度-速度系统的指数稳定性问题。密度-速度系统是典型的双曲线系统，它们在物理学中无处不在，因为它们包含所有包含通量守恒和动量方程的系统。在本文中，我们表明任何这样的系统都可以在$H^2$使用简单局部边界反馈的范数，前提是源项的条件有效。这个条件适用于大多数物理系统，即使源项不是耗散的。此外，获得的反馈规律仅取决于边界处的目标值，这意味着它们不取决于源项的表达或施加在系统上的力。这使得它们在实践中非常容易实现，并且对模型错误具有鲁棒性。例如，对于由 Saint-Venant 方程建模的河流，这意味着反馈定律不需要有关摩擦模型、坡度或所考虑通道形状的任何信息。这一壮举是通过显示一个基本的存在而获得的$H^2$李雅普诺夫函数 我们将其应用于几个系统：一般的圣维南方程、等熵欧拉方程、刚性管道中的水运动、渗透现象、交通流等。