论文标题
通过非线性Dirichlet致动，稳定波方程
Stabilization of the wave equation through nonlinear Dirichlet actuation
论文作者
论文摘要
在本文中，我们考虑了具有Dirichlet边界条件的高维波方程的非线性（特别是饱和）稳定的问题。波动力学受到耗散非线性速度反馈的约束，并在最佳能量空间$ l^2（ω）\ times h^{ - 1}（ω）上产生强烈连续的收缩。首先证明，在上述拓扑结构中，对闭环方程的任何解决方案都会收敛到零。其次，在反馈非线性的线性增长左右的条件下，为具有光滑初始数据的溶液建立了多项式能量衰减速率。这构成了与$ H^1（ω）\ times l^2（ω）$在具有Neumann边界条件的$ H^1（ω）\ times l^2（ω）$中的非线性稳定有关的众所周知的结果。
In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space $L^2(Ω) \times H^{-1}(Ω)$. It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in $H^1(Ω)\times L^2(Ω)$ of the wave equation with Neumann boundary conditions.
