Characterising solution sets of LTI differential equations

We prove that a set of smooth trajectories is LTID (i.e., the solution set of a constant coefficient linear differential equation) if and only if it is the direct sum of a linear time-invariant closed controllable part and a linear time-invariant finite-dimensional part. This characterisation does not directly involve derivatives in its formulation. It solves a problem opened by Willems (1991). We also characterise morphisms between LTID sets as linear time-invariant maps which do not increase support.