具有有界时空势的一维热方程的可测集可观测性
Observability from Measurable Sets for One-Dimensional Heat Equations with Bounded Spacetime Potentials
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中文总结 AI 辅助
本文研究一维Dirichlet热方程在具有有界时空势时的可观测性，证明从任意正测度时空子集出发的可观测性不等式，并通过对偶性建立与零可控性的等价关系。
中文摘要 AI 辅助
本文研究有界区间上具有依赖于空间和时间的实有界势的Dirichlet热方程的可观测性。我们证明了从具有正测度的时空的每个可测子集出发的可观测性不等式。该常数在具有给定$L^\infty$界的势类上是一致的。通过对偶性，正时空测度等价于通过平方可积分布控制的零可控性。控制也可以选择在时间上有界且取值于$L^2$。如果观测集的几乎所有空间切片的测度都有一个固定的正下界，则可观测性常数以$C_0e^{C_1/T}$为上界，且关于切片位置一致。证明结合了在由演化传输的有限维空间上的观测估计以及到这些空间的距离的衰减估计。
英文摘要
This paper study observability for the Dirichlet heat equation on a bounded interval with a real bounded potential depending on space and time. We prove an observability inequality from every measurable subset of spacetime with positive measure. The constant is uniform over potentials with a prescribed $L^\infty$ bound. By duality, positive spacetime measure is equivalent to null controllability by square-integrable distributed controls. Controls can also be chosen bounded in time with values in $L^2$. If almost every spatial slice of the observation set has a fixed positive lower bound on its measure, the observability constant is bounded above by $C_0e^{C_1/T}$, uniformly in the locations of the slices. The proof combines observation estimates on finite-dimensional spaces transported by the evolution with decay estimates for the distance to those spaces.
发表机构
- Civil Aviation Flight University of China（中国民用航空飞行学院）
机构由 AI 辅助整理，请以论文原文为准。
