In this paper, we are interested in the stabilization of the flow modeled by the Saint-Venant equations. We have solved two problems in this study. The first, we have proved that the operator associated to the Saint-Venant system has a finite number of unstable eigenvalues. Consequently, the system is not exponentially stable on the space L 2 ( Ω ) × L 2 ( Ω ) , but is exponentially stable on a subspace of the space L 2 ( Ω ) × L 2 ( Ω ) , ( Ω is a given domain). The second problem, if the advection is dominant, the natural stabilization is very slow. To solve these problems, we have used an extension method due to Russel (1974) and Fursikov (2002). Thanks to this method, we have determined a boundary Dirichlet control able to accelerate the stabilization of the flow. Also, the boundary Dirichlet control is able to kill all the unstable eigenvalues to get an exponentially stable solution on the space L 2 ( Ω ) × L 2 ( Ω ) . Then, we extend this method to the finite difference equations analog of the continuous Saint-Venant equations. Also, in this case, we obtained similar results of stabilization. A finite difference scheme is used to compute the control and several numerical experiments are performed to illustrate the efficiency of the control.

中文翻译：

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基于边界控制的 Saint-Venant 方程的镇定方法

在本文中，我们对由 Saint-Venant 方程建模的流动的稳定性感兴趣。我们在这项研究中解决了两个问题。首先，我们已经证明了与 Saint-Venant 系统相关的算子具有有限数量的不稳定特征值。因此，系统在空间 L 2 ( Ω ) × L 2 ( Ω ) 上不是指数稳定的，而是在空间 L 2 ( Ω ) × L 2 ( Ω ) 的子空间上指数稳定，（ Ω 是给定的领域）。第二个问题，如果平流占优势，自然稳定很慢。为了解决这些问题，我们使用了 Russel (1974) 和 Fursikov (2002) 提出的扩展方法。由于这种方法，我们确定了能够加速流动稳定的边界狄利克雷控制。还，边界狄利克雷控制能够消除所有不稳定的特征值以获得空间 L 2 (Ω) × L 2 (Ω) 上的指数稳定解。然后，我们将此方法扩展到连续圣维南方程的有限差分方程模拟。此外，在这种情况下，我们获得了类似的稳定结果。有限差分方案用于计算控制，并进行了几个数值实验来说明控制的效率。