The controllable abundance of linear continuous-time systems

1. Computation of The Unit Controllable Region of Linear Continuous-time Systems $\Sigma (A,B)$

$R_{c,T}=\left \{ x_0 | x_0=-\int ^T_0 \textup{e}^{-As} Bu_s \textup{d}s, \forall \left \| u_t \right \| _\infty \leqslant 1,t\in [0,T]\right \}" style="font-family:'times new roman';line-height:25.2px;$

2.  The Controllable Abundance $(r_T,v_{c,T})" original="http://latex.codecogs.com/gif.latex?(r_T,v_{c,T})" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;font-family:'times new roman';line-height:25.2px;background-color:#ffffff;$  of Linear Continuous-time Systems

2.1  

    $r_T=\textup{dim}(R_{c,T}) =\textup{rank}\left [B, AB, ..., A^{n-1}B \right ]$

2.2

    $v_{c,T}=\textup{Volume}(R_{c,T})$

            $=\int _{s\in R_{c,T}} \textup{d}s$

            $=\int _{z\in \partial R_{c,T}} z\otimes \textup{d}z$

where $\partial R_{c,T}$  is the boundary of $R_{c,T}$

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