On Controllability of Systems of the Form (x) over dot = A(t)x plus g(t, u)

In an earlier paper, the author established a sufficient condition for controllability of systems of the form (x) over dot = A(t)x + g(t, u). This condition is a growth condition which generalizes the concept of an asymptotically proper system introduced by LaSalle for linear systems. The purpose of this paper is to examine and apply this growth condition. We first show that the condition is also necessary for controllability. Then, we use these results to consider the controllability of perturbations of the above system. The main result of the paper is a class of systems which in many applications can be assumed to be controllable.