Controllability and feedback stabilizability in a nonuniform framework

We propose a new controllability property for linear time varying control systems in finite dimension: the nonuniform complete controllability, which is halfway between the classical Kalman's properties of complete controllability and uniform complete controllability. This new concept is described in terms of two gramian inequalities, which have a strong relation; as we prove in our first result; with the property of nonuniform bounded growth for the corresponding plant, also called uncontrolled part. On the other hand, the second result proves that if a control system is nonuniformly completely controllable and its plant has the property of nonuniform bounded growth, then there exist a linear feedback control leading to a nonuniformly exponentially stable closed–loop system.