Boundary stabilization of the focusing NLKG equation near unstable equilibria: radial case

J. Krieger, Shengquan Xiang
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引用次数: 5

Abstract

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball of radius L in $\mathbb{R}^3$. First we show that the system is linearly unstable near the static solution $u\equiv 1$ for any dissipative boundary condition $u_t+ au_{\nu}=0, a\in (0, 1)$. Then by means of boundary controls (both open-loop and closed-loop) we stabilize the system around this equilibrium exponentially under the condition $\sqrt{2}L\neq \tan \sqrt{2}L$. Furthermore, we show that the equilibrium can be stabilized with any rate less than $ \frac{\sqrt{2}}{2L} \log{\frac{1+a}{1-a}}$, provided $(a,L)$ does not belong to a certain zero set. This rate is sharp.
不稳定平衡附近聚焦 NLKG 方程的边界稳定:径向情况
我们研究了立方聚焦克莱因-戈登方程在 $\mathbb{R}^3$ 中半径为 L 的闭球上静态解附近的稳定性和稳定性。首先,我们证明在任何耗散边界条件 $u_t+ au_{\nu}=0, a\in (0, 1)$ 下,系统在静态解 $u\equiv 1$ 附近是线性不稳定的。然后,通过边界控制(开环和闭环),我们可以在 $\sqrt{2}L\neq \tan \sqrt{2}L$ 的条件下,使系统指数式地稳定在这个平衡点附近。此外,我们还证明,该平衡可以以小于 $\frac\sqrt{2}}{2L} 的速率稳定下来。\log{frac{1+a}{1-a}}$,条件是 $(a,L)$ 不属于某个零集。这个比率非常尖锐。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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CiteScore
2.30
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