In this article, the null controllability of a coupled ODE-heat system is considered by backstepping method. First, according to a backstepping transformation, the system is transferred into an intermediate system which has an equivalence of null controllability with the original system. Second, we divide the finite time interval into a sequence of time intervals, and on each time interval, transform the intermediate system into a target system through another backstepping transformation. Consequently, we obtain the crucial norm estimates of states and control laws. Finally, based on the invertibility of the target system, we prove null controllability of the intermediate system and ultimately, show the null controllability of the original coupled ODE-heat system. In addition, numerical experiment results are presented to examine the validity for the presented controller.

中文翻译：

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边界和内项耦合的 ODE 热系统的零可控性

本文采用反推法考虑了耦合常微分方程-热系统的零可控性。首先，根据反推变换，将系统转移到与原系统具有等价的零可控性的中间系统。其次，我们将有限时间间隔划分为一系列时间间隔，并在每个时间间隔上，通过另一个反推变换将中间系统变换为目标系统。因此，我们获得了状态和控制律的关键范数估计。最后，基于目标系统的可逆性，证明了中间系统的零可控性，最终证明了原始ODE-热耦合系统的零可控性。此外，还提供了数值实验结果来检验所提出的控制器的有效性。