A new measure metering the control ability and efficiency -- controllable abundance

1. Definition of The Unit Controllable Region

Definition 1. The uint contrllable region $R_{c,T}}" style="font-family:'times new roman';line-height:25.2px;background-color:#ffffff;$ is constituted by all initial state $x_0$ that can be controlled to the origin of the state space of the linear time-invariant systems with the unit input energy $\left ( \left \| u(t) \right \|_\infty \leq 1 \right )$ in the finite time interval [0,T].

2. Definition of The Controllable Abundance

Definition 2. The controllable abundance is defined as the two-tuples $(r_T,v_{c,T})$ , where $r_T$ and $v_{c,T}}$ are the space dimension and the volume of the unit controllable region $R_{c,T}}$ ,  respectively.

Some results are in my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems)