Controllability notions: equivalency and sets of controllable points

The authors present various notions of controllability of any point p in R/sup n/, for linear systems with admissible controls in a compact set U containing the origin. They prove the equivalency of some of these notions and characterize the sets of controllable points for the system. They also prove that a necessary condition for p to be a local constrained controllable point of a linear system is that Ap in U. Thus, it is possible to characterize the set of local controllable points for the system. C, the set of complete constrained controllable points of the system. is shown to be a convex nonvoid and symmetric neighborhood of the origin. C is also a connected set since its interior is the union of closed trajectories through the origin. The set C as the intersection of the reachable and attainable set for any point p in R/sup n/ is completely characterized. Thus, the size and shape of C is invariant of the choice of p. The shape of C is related to the location of the spectrum of the system in the complex plane.