Null controllability of an ODE-heat system coupled at boundary and internal term

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. •We construct a piecewise constant feedback controller for the null controllability of coupled ODE-heat system coupled at boundary and internal term.•We obtain the crucial norm estimates for the kernels, states and control. Especially, we obtain the kernels estimate by using suitable induction hypothesis expression in successive approximation.•We obtain the uniform estimate of the decay rate β for the target system, which is independent of the time intervals.•The procedure of this paper avoiding some technical skills for constructing continuous feedback control in references, which can be extended to similar problems for some other PDE-ODE system.