We consider the control problem of the heat equation on bounded and unbounded domains, and more generally the corresponding inhomogeneous equation for the Schrödinger semigroup. We show that if the sequence of null-controls associated to an exhaustion of an unbounded domain converges, then the solutions do in the same way, and that the control cost estimate carries over to the limiting problem on the unbounded domain. This allows to infer the controllability on unbounded domains by studying the control problem on a sequence of bounded domains.

中文翻译：

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无界域上热或薛定谔半群控制问题的穷竭逼近

我们考虑有界域和无界域上的热方程的控制问题，更一般地考虑薛定谔半群的相应非齐次方程。我们表明，如果与无界域耗尽相关的空控制序列收敛，则解决方案以相同的方式进行，并且控制成本估计会延续到无界域上的限制问题。这允许通过研究一系列有界域上的控制问题来推断无界域的可控性。