Stabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control
Stabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control
复制标题
边界反馈控制二维纳维-斯托克斯方程的稳定性
DOI：
10.1007/pl00000972
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发表时间：
2001
影响因子：
1.3
通讯作者：
A. Fursikov
中科院分区：
文献类型：
--
作者：
A. Fursikov
抽象的。对于有界域中定义的 2D Navier-Stokes 方程
$ \Omega $ 我们研究给定稳态流附近溶液的稳定性
$ \hat v(x) $ 通过在零件上定义的反馈控制
$ \Gamma $ 边界
$ \partial\Omega $.提出了反馈概念的新数学形式化。在它的帮助下达到规定的数量
$ \sigma > 0 $ 且初始条件 v0(x) 位于
$ \hat v(x) $ 一个控制 u(t,x'),
$ x' \in \Gamma $ 的构造使得所获得的二维纳维-斯托克斯方程边值问题的解 v(t,x) 满足不等式：
$ \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 $。为了证明这个结果，我们首先获得二维 Oseen 方程稳定性的类似结果。
Abstract. For 2D Navier—Stokes equations defined in a bounded domain
$ \Omega $ we study stabilization of solution near a given steady-state flow
$ \hat v(x) $ by means of feedback control defined on a part
$ \Gamma $ of boundary
$ \partial\Omega $. New mathematical formalization of feedback notion is proposed. With its help for a prescribed number
$ \sigma > 0 $ and for an initial condition v0(x) placed in a small neighbourhood of
$ \hat v(x) $ a control u(t,x'),
$ x' \in \Gamma $, is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier—Stokes equations satisfies the inequality:
$ \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 $. To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.