含反平方势的热方程的时间最优控制
Time optimal control for the heat equation with inverse-square potentials
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对Hardy临界条件下含反平方势的热方程，通过建立观测不等式证明了零能控性，并进一步证明了时间最优控制的唯一性及其bang-bang性质。
AI中文摘要：
本文研究了在Hardy临界条件下，具有奇异反平方势的热方程的时间最优控制问题。我们首先针对具有正Lebesgue测度的一般时空可测集，建立了奇异抛物方程的观测不等式。与依赖于尚不明确的底层奇异算子的Lebeau-Robbiano谱不等式不同，我们的证明结合了基于Carleman的开圆柱上的观测结果、远离奇异原点的解的实解析性估计、实解析函数的小性传播估计以及 telescoping 级数技术。利用这一观测结果，我们推导出了控制支撑在可测子集上的零能控性。最后，我们证明了相应的时间最优控制是唯一的，并且在控制域上几乎处处满足bang-bang性质。
英文摘要：
This paper studies the time optimal control problem for the heat equation with singular inverse-square potentials under the Hardy critical condition. We first establish an observability inequality for the singular parabolic equation from general space-time measurable sets of positive Lebesgue measure. Rather than depending on the still-unknown Lebeau-Robbiano spectral inequality for the underlying singular operator, our proof combines Carleman-based observability results over open cylinders, real-analyticity estimates of solutions away from the singular origin, propagation of smallness estimates for real-analytic functions, and a telescoping-series technique. Using this observability result, we derive the null-controllability with controls supported on measurable subsets. Finally, we prove that the corresponding time optimal control is unique and obeys the bang-bang property almost everywhere over the control domain.
