The main concern of this article is to investigate the boundary controllability of some \(2\times 2\) one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.

本文的主要关注点是研究某些\（2 \ times 2 \）的边界可控性具有内部和边界耦合的一维抛物线系统：内部耦合被选择为具有恒定系数的线性，而边界一则通过在域的一端采用某些Kirchhoff型条件来考虑。我们在这里考虑Dirichlet边界控制仅作用于域另一端的两个状态分量之一。特别是，我们显示出可控制性属性随控件所应用系统的哪个组件而变化。关于这一点，我们指出，内部耦合系数和基尔霍夫参数的选择对于推论正或负的可控性结果起着至关重要的作用。除此之外，我们基于著名的惩罚性HUM方法进行数值研究。我们对一般的内边界耦合抛物线系统进行了离散化，主要是将边界耦合的影响合并到离散环境中。这使我们能够说明我们的理论结果，并进行更多适用于通用框架的示例的实验，例如，在两个组件中的一个组件上都具有Neumann边界控制的类似系统。