实直线上Airy方程的Hausdorff型时间迹可观性及环面上的点可观性
Hausdorff type Time-Trace Observability for Airy Equations on the Line and Point Observability on the Torus
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中文总结 AI 辅助
研究实直线和环面上Airy方程的可观性，通过引入新思想，证明了从Hausdorff厚集的时间迹可观性不等式、周期集观测的可观性不等式，给出环面上有限点可观性充要条件及相关推论，得到不同条件下的可观性结果。
中文摘要 AI 辅助
本文主要有三个结果。其一，证明了实直线上Airy方程从Hausdorff厚集在时间迹意义下对每个观测时间\(T>0\)的可观性不等式，观测泛函是Hausdorff厚集上\(L^2(0,T)\)时间迹的块上确界。其二，证明了实直线上Airy方程在某些周期集上观测的可观性不等式，特别给出了一类空间点序列的可观性。其三，给出了环面上带实值有界势的Airy方程有限点可观性的充要条件，找到有限维不变子空间上的卡尔曼秩条件，作为推论得到环面上Airy和线性KdV方程的精确点可观性结果。对于有有限阶正则性假设的势，证明了每个有聚点的观测集，特别是任何正Hausdorff维数的集，对每个观测时间\(T>0\)都给出可观性不等式。由于Airy方程既无高频指数衰减也无逐点平滑效应，本文引入了适应Airy情形的新思想。
英文摘要
The main results of this paper are threefold. First, we prove an observability inequality for the Airy equation on the real line from Hausdorff-thick sets in a time-trace sense for every observation time $T>0$. The observation functional is a block supremum of $L^2(0,T)$ time traces over the Hausdorff-thick set. Second, we prove observability inequalities for the Airy equation on the real line with observations on some periodic sets, which in particular yields observability on a class of spatial point sequences. Third, we give a necessary and sufficient condition for finite point observability for the Airy equation on the torus with a bounded real-valued potential. Indeed, for a finite observation set $F$, a Kalman rank condition on a finite-dimensional invariant subspace is found. As a corollary, we obtain sharp point observability results for the Airy and linear KdV equations on the torus. Moreover, for potentials with finite order regularity assumptions we prove that every observation set with an accumulation point, in particular any set of positive Hausdorff dimension, gives an observability inequality for each observation time $T>0$. Since the Airy equation has neither high frequency exponential decay nor pointwise smoothing effects, which are essential in recent works on Hausdorff type observation results on heat equations, we introduce several new ideas adapted to the Airy case.
