Asymptotically reachable states and related symmetry in systems theory

In executing state-to-state maneuvers, end states which are stabilizable states provide for robust maneuver control. In the normal circumstance that a maneuver only takes the system to a neighborhood of a stabilizable state, feedback control can be used to regulate to a neighborhood of the desired end state. In contrast, if the end state is not a stabilizable state, large try-again maneuvers which cannot be bounded by the terminal tracking error of the prior maneuver are often required in the event of a near miss. In this paper it is shown that the subspace of stabilizable states for a linear stabilizable system is the intersection of the reachable subspace and a particular controlled invariant subspace we call the constant state subspace. The stabilizable states are also the states which can be approached asymptotically with appropriate choice of control, and we use this characterization as our definition of stabilizable states.