针对半线性波动方程的新稳定性与精确可观测性条件
New Stability and Exact Observability Conditions for Semilinear Wave Equations
AI总结：
本文基于边界测量，针对一维和多维波动方程提出基于LMI的稳定性条件和精确可观测性时间上界，通过数值例子验证了结果的有效性。
摘要：
最近，通过使用正向和反向观测器序列，解决了从有限区间边界测量估计一维具有全局Lipschitz非线性项的波动方程初始状态的问题，并通过线性矩阵不等式（LMIs）导出了精确可观测性时间的上界[5]。在本文中，我们将这一结果推广到n维波动方程在超立方体上的情况。这一扩展包括针对n维波动方程的新基于LMI的指数稳定性条件，以及以LMIs形式给出的最小精确可观测性时间上界。对于具有局部Lipschitz非线性的1维波动方程，我们找到了保证从测量中唯一恢复初始条件的初始条件区域的估计。通过数值例子展示了结果的有效性。
英文摘要：
The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of Linear Matrix Inequalities (LMIs) [5]. In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations, as well as an upper bound on the minimum exact observability time in terms of LMIs. For 1-D wave equations with locally Lipschitz nonlinearities, we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. The efficiency of the results is illustrated by numerical examples.
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