Controllability and Vector Potential

Kalman's fundamental notion of a controllable state space system \cite{k} has been generalised to higher order systems by Willems \cite{w}, and further to distributed systems defined by partial differential equations \cite{ps}. It turns out, that for systems defined in several important spaces of distributions, controllability is now identical to the notion of vector potential in physics, or of vanishing homology in mathematics. These notes will explain this relationship, and a few of its consequences. It will also pose an important question: does a controllable system, in any space of distributions, always admit a vector potential? In other words, is Kalman's notion of a controllable system, suitably generalised, nothing more -- nor less -- than the possibility of describing the dynamics of the system by means of a vector potential? Furthermore, it also turns out that the category of distributed systems bears many formal similarities to the category of affine algebraic sets. This raises a second important question: what is the category for which these distributed systems are `local models', just as affine algebraic sets are local models for the category of algebraic varieties? It would then be possible to extend the theory of control described in these notes to this larger category of systems.