常矩阵耦合的热方程采样控制系统能控性

Controllability of the Sampled-Data Controlled System of Heat Equations Coupled by Constant Matrices

摘要: 本文研究由常矩阵耦合的热方程采样控制系统的零能控和近似能控问题，所考虑的是给定采样周期情况下系统的两种能控性。我们通过热方程的唯一延拓性，Kalman能控性秩条件和采样周期的选取对系统的能控性进行了刻画。提出在给定采样周期下此类系统近似能控的一个充要条件；并说明当控制施加在整体区域时，这一条件也是在给定采样周期下此类系统零能控的一个充要条件，而当控制只施加在局部区域时，此类系统对任意周期都不是零能控的。

Abstract: This paper studies the null controllability and approximate controllability for sampled-data controlled systems of heat equations coupled by constant matrices. We consider two types of controllability under a given sampled-data period and characterize the controllability of the system through the unique continuation property of the heat equation, the Kalman rank condition and the selection of the sampled-data period. We provide a necessary and sufficient condition for the approximate controllability of such systems under a given sampled-data period; and show that when the control domain is equal to the entire space, this condition is also a necessary and sufficient condition for the null controllability of such systems under a given sampled-data period. However, when the control domain is a proper subset of the entire space, such systems are not null controllable for any period.