The canonical controller for distributed systems

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-D$$\end{document} 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.