This paper generalises results of Willems–Trentelman, and van der Schaft, on achievable behaviours, to the case of linear distributed systems defined by partial differential or difference equations. It shows that the ‘minimal’ controller which achieves a particular subsystem is the canonical controller of van der Schaft, thereby answering the ‘open problem’ of van der Schaft (Syst Control Lett 49:141–149, 2003) in the setting of infinite dimensional and $$n-D$$ systems. This result is used to describe the collection of all linear subsystems of the electro-magnetic field, containing the vacuum solutions, that can be achieved by suitable choices of electric charge and current density.

中文翻译：

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分布式系统的规范控制器

本文将 Willems-Trentelman 和 van der Schaft 关于可实现行为的结果推广到由偏微分或差分方程定义的线性分布式系统的情况。它表明实现特定子系统的“最小”控制器是 van der Schaft 的规范控制器，从而回答了 van der Schaft (Syst Control Lett 49:141–149, 2003) 在无限设置下的“开放问题”维和 $$nD$$ 系统。该结果用于描述包含真空溶液的电磁场的所有线性子系统的集合，其可以通过适当的电荷和电流密度来实现。