闭流形上齐次wave equation的可观测性
Observability properties of the homogeneous wave equation on a closed manifold
摘要：
考虑闭黎曼流形上的波动方程。我们观察到的限制的解决方案，可测量的子集$ω$沿时间间隔$[0, T]$与$T>0$。众所周知，如果$ω$是开的，并且如果$(ω,T)$对满足几何控制条件，则满足可观测性不等式，将解的总能量与它们在$ω\times (0, T)$中的能量进行比较。可观测性常数$C\_T(ω)$被定义为波动方程的所有非平凡解的局部化能量与其总能量之比的下确界。在本文中，我们提供了一个低/高频率分裂过程的基础上，使我们能够获得一般的几何条件，保证波动方程是可观测的可测量子集$ω$的可观测常数的估计。我们还建立，作为$T\rightarrow+\infty$，比率$C\_T(ω)/T$收敛到两个量的最小值：第一个是光谱性质，涉及拉普拉斯特征函数，第二个是几何性质，涉及平均时间花费在$ω$由黎曼测地线。
英文摘要：
We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $ω$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $ω$ is open and if the pair $(ω,T)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $ω\times (0, T)$. The observability constant $C\_T(ω)$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a measurable subset $ω$. We also establish that, as $T\rightarrow+\infty$, the ratio $C\_T(ω)/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions, the second one is of a geometric nature and involves the average time spent in $ω$ by Riemannian geodesics.
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