Exponential stability of density-velocity systems with boundary conditions and source term for the H-2 norm
Exponential stability of density-velocity systems with boundary conditions and source term for the H-2 norm
复制标题
具有边界条件和 H-2 范数源项的密度-速度系统的指数稳定性
DOI：
10.1016/j.matpur.2021.07.001
复制
发表时间：
2021
影响因子：
2.3
通讯作者：
Shang Peipei
中科院分区：
文献类型：
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作者：
Hayat Amaury;Shang Peipei
本文讨论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系统，在物理学中无处不在，因为它们包含了所有存在于通量守恒和动量方程中的系统。在本文中，我们证明了任何这样的系统都可以用简单的局部边界反馈在h2范数上快速地指数稳定，只要源项的条件成立。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，得到的反馈律只依赖于边界处的目标值，这意味着它们不依赖于源项的表达式或作用在系统上的力。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任何关于摩擦模型、坡度或所考虑的渠道形状的信息。这是通过证明一个基本h2李雅普诺夫函数的存在性而得到的。将其应用于一般的圣维南方程、等熵欧拉方程、水在刚性管道中的运动、渗透现象、交通流等系统。
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.