Axiomatic characterization of linear differential systems (and operators)

In his IEEE Trans. Aut. Contr. paper in 1991, Willems posed the following question: given a set of smooth trajectories, when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set? We show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a “kernel” representation are: linearity, time-invariance, jet-closedness and jet-determinedness. (It is interesting to note that the properties of jet-closedness and jet-determinedness were introduced by Willems himself in his Automatica paper in 1986.)