论文标题
具有设定值的边界阻尼的一维波方程：适应性良好，渐近稳定性和衰减速率
One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates
论文作者
论文摘要
本文与分析一个维波方程的分析$ z_ {tt} -z_ {xx} = 0 $ on $ [0,1] $，dirichlet条件为$ x = 0 $，在$ x = 1 $上表现为$ x = 1 $，该$（z_t（z_t（t，1），$ eq $ qe $ n $ n $ n $ n $ n $ n $ n $ wery $ ther给定$ \ mathbb r^2 $的子集。该研究是在$ l^p $功能框架中进行的，在[1， +\ infty] $中进行。我们旨在确定$σ$的条件，以确保该波方程的存在和唯一性，以及强大的稳定性和其解决方案的统一全球渐近稳定性。在后一种情况下，我们还研究了解决方案的衰减速率及其最佳性。我们首先在该波方程的解决方案与离散时间动态系统的迭代序列之间建立一对一的对应关系，我们研究了上述问题。这使我们能够在$σ$上提供一个简单的必要条件，以确保波动方程解决方案的存在和独特性，以及在$σ$验证广义扇区条件时确定最佳衰减率的有效策略。作为一种应用，我们解决了文献中指出的两个猜想，第一个寻求特定的最佳衰减率，第二种与饱和型阻尼相关的猜想。如果边界阻尼受到扰动，我们就会获得有关渐近扰动排斥和输入到状态问题的明显结果。
This paper is concerned with the analysis of a one dimensional wave equation $z_{tt}-z_{xx}=0$ on $[0,1]$ with a Dirichlet condition at $x=0$ and a damping acting at $x=1$ which takes the form $(z_t(t,1),-z_x(t,1))\inΣ$ for every $t\geq 0$, where $Σ$ is a given subset of $\mathbb R^2$. The study is performed within an $L^p$ functional framework, $p\in [1, +\infty]$. We aim at determining conditions on $Σ$ ensuring existence and uniqueness of solutions of that wave equation as well as strong stability and uniform global asymptotic stability of its solutions. In the latter case, we also study the decay rates of the solutions and their optimality. We first establish a one-to-one correspondence between the solutions of that wave equation and the iterated sequences of a discrete-time dynamical system in terms of which we investigate the above mentioned issues. This enables us to provide a simple necessary and sufficient condition on $Σ$ ensuring existence and uniqueness of solutions of the wave equation as well as an efficient strategy for determining optimal decay rates when $Σ$ verifies a generalized sector condition. As an application, we solve two conjectures stated in the literature, the first one seeking a specific optimal decay rate and the second one associated with a saturation type of damping. In case the boundary damping is subject to perturbations, we derive sharp results regarding asymptotic perturbation rejection and input-to-state issues.
