Approximate computation of the controllable abundance for the linear continuous-time systems

Similar to the definition of the unit controllable region of the linear discrete-time systems, the unit controllable region of the linear continuous-time systems under the unit energy input energy $\left(\left\Vert u_{t}\right\Vert _{\infty}\leq1,\forall t\right)$ is defined as

$R_{cx}=\left\{ z\left|z=\int_{0}^{T}e^{-At}Bu_{t}\textrm{d}t,\left\Vert u_{t}\right\Vert _{\infty}\leq1\right.\right\}$

If the time interval $[0,T]$ is divided into $N$ equal parts, the discrete time points are as

$t_{i}=i\varDelta,\quad i=0,2,\cdots,N$

where $\varDelta=T/N$ . And then, the controllable region $R_{cx}$ of the linear continuous-time systems can be approximated as

$R_{dx}=\left\{ z\left|z=\sum_{i=0}^{N-1}G^{i}Hu_{t_{i}},\left|u_{t_{i}}\right|\leq1,i=0,1,\cdots,N-1\right.\right\}$

where

$G=e^{-A\Delta},\qquad H=\int_{0}^{\Delta}e^{-At}\textrm{d}tB$

In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), the volume of the controllable region of the linear discrete-time systems is defined as the controllable abundance metering the control ability of the input variable to the state space and then the computing method for the volume are proposed and proven. Based on that, the controllable abundance for linear continuous-time systems can be computed approximately as follows

$\mathrm{Vol}\left(R_{cx}\right)\approx\mathrm{Vol}\left(R_{dx}\right)$

where the volume $\mathrm{Vol}\left(R_{cx}\right)$ and $\mathrm{Vol}\left(R_{dx}\right)$ of these controllable regions are defined as the controllable abundances for the linear continuous-time systems and the linear discrete-time systems, respectively. When the number $N$ is sufficiently large, the above approximate computation will be with the high precision.