论文标题
落后和控制热方程问题的公式
A formula for backward and control problems of the heat equation
论文作者
论文摘要
（一个）。使用时间分析结果，我们解决了平滑域和紧凑型歧管的非均匀向后热方程（确切的控制问题）的基本问题，即：何时可能是时间独立控制？即，一个时间间隔的控制函数为0，另一个时间间隔为0。对于一般$ l^2 $初始值，答案是：当且仅当完整空间域用于控制功能时。还以无限序列的形式找到了对控制功能的明确公式，该序列涉及热核，该套件迅速收敛。 （b）。在适当子域中支持的时间依赖控制功能的形式精确公式也是通过拉普拉斯式的特征函数获得的。该公式在特征函数跨越的任何有限维空间上都是严格的，并且在整个域上没有平滑度的假设，从而在P74 \ cite {Zu：1}上的问题上进行了部分进展。 （c）。副产品是热核的反转公式。
(a). Using time analyticity result, we address a basic question for a nonhomogeneous backward heat equation (exact control problem) in the setting of smooth domains and compact manifolds, namely: when is essentially time independent control possible? i.e. The control function is 0 on one time interval and stationary on the other. For general $L^2$ initial values, the answer is: if and only if the full space domain is used for the control function. Also an explicit formula for the control function is found in the form of an infinite series involving the heat kernel, which converges rapidly. (b). A formal exact formula for a time dependent control function supported in a proper subdomain is also obtained via eigenfunctions of the Laplacian. The formula is rigorous on any finite dimensional space spanned by the eigenfunctions and there is no smoothness assumption on the whole domain, making partial progress on a problem on p74 \cite{Zu:1}. (c). A byproduct is an inversion formula for the heat kernel.
