Trajectory stabilizing controls in hereditary linear systems

The following phenomenon is specific to a class of hereditary control systems: Trajectories of these systems can be stabilized (i.e., controlled to asymptotically approach the origin). However, it is impossible to obtain stabilization with controls whose rate of decay is the same as that of the trajectories. We call such systems trajectory stabilizable. We characterize the class of trajectory stabilizable systems and present both open and closed loop trajectory-stabilizing schemes. The controllability aspect of trajectory stabilizability is studied too. Examples illustrate the discussion.