Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.