Impulsive solutions, inadmissible initial conditions and pole/zero structure at infinity

In this paper we study solutions to linear ordinary constant coefficient differential equations on the half-line and relate impulsive solutions to the pole/zero structure at infinity of an associated polynomial matrix. While this relation has been thoroughly studied for first order systems, and through first order analysis also for higher order systems partially, the use of the `state map´, in particular the shift and cut map, makes it very straightforward to characterize the inadmissible initial condition space and the smooth solution space. This paper contains results about the space of initial conditions that have impulsive solutions and those having smooth solutions, and the relation with the zero structure at infinity. We show, amongst other results, that the rank over the reals of the shift and cut map is precisely the dimension of the space of smooth and impulsive solutions for a linear differential system.