Feedback-reversibility and reachability of linear impulsive systems

For a linear impulsive system, the set of states that are reachable from the origin when the initial time, impulse times, and final time are fixed is contained in an invariant subspace determined by the system data. It is known that reversibility of the system is sufficient to yield, for a specified initial time, the existence of some impulse time set and final time for which the reachable set equals the invariant subspace. In this paper, we relax the reversibility requirement and present a condition that is necessary as well as sufficient under which this property holds. This new condition involves the property of achieving reversibility via feedback and admits an explicit geometric characterization. Moreover, this feedback-reversibility property only needs to hold for the subsystem defined as the full system restricted to the invariant subspace. We further show that feedback-reversibility of the restricted system ensures that the reachable set equals the invariant subspace for almost any impulse time set and final time for which the number of impulse times contained in the underlying time interval exceeds a lower bound.