Some questions in algebra that arise in the control of systems described by PDE

This talk will explain a few questions that arise in the control of systems described by PDE, and how the algebraic structure (injectivity, flatness) of the space in which the solutions of the PDE are located help answer these questions in the case of linear constant coefficient PDE. It will describe some partial results in the setting of periodic solutions/Sobolev spaces, and finally pose some 'open problems'.