Reachable abundance of linear discrete systems with Jordan system matrix

         When the linear discrete systems are with some repeated roots, the system matrices can be transformed as the upper Jordan matrices, that is, the matrices of the system models can be represented as

$A=\left[\begin{array}{ccccc} \lambda & 0 & 0 & \cdots & 0\\ 1 & \lambda & 0 & \cdots & 0\\ 0 & 1 & \lambda & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \lambda \end{array}\right],\quad B=\left[\begin{array}{c} b_{1}\\ b_{2}\\ b_{3}\\ \vdots\\ b_{n} \end{array}\right]$

Let

$\zeta_{n,k}=A^{k}B=\left[\begin{array}{c} \lambda^{k}b_{1}\\ k\lambda^{k-1}b_{1}+\lambda^{k}b_{2}\\ \frac{k(k-1)}{2}\lambda^{k-2}b_{1}+k\lambda^{k-1}b_{2}+\lambda^{k}b_{3}\\ \vdots\\ \sum_{i=1}^{n}\frac{k!}{(i-1)!(k-i+1)!}\lambda^{k-i+1}b_{i} \end{array}\right]$

where

$\frac{k!}{(i-1)!(k-i+1)!}\lambda^{k-i+1}=0\quad\textrm{if}\;k-i+1<0$

It can be proven that there exists the transformation matrices

$P=\left[\begin{array}{ccccc} 1 & & & & 0\\ 0 & 1\\ 0 & 0 & 1\\ & & & \ddots\\ -\frac{b_{n}}{b_{1}} & -\frac{b_{n-1}}{b_{1}} & -\frac{b_{n-2}}{b_{1}} & & 1 \end{array}\right]\times\cdots\times\left[\begin{array}{ccccc} 1 & & & & 0\\ 0 & 1\\ -\frac{b_{3}}{b_{1}} & -\frac{b_{2}}{b_{1}} & 1\\ & & & \ddots\\ 0 & 0 & 0 & & 1 \end{array}\right]\times\left[\begin{array}{ccccc} 1 & & & & 0\\ -\frac{b_{2}}{b_{1}} & 1\\ 0 & 0 & 1\\ & & & \ddots\\ 0 & 0 & 0 & & 1 \end{array}\right]$

to make that the following equation holds,

$P\zeta_{n,k}=\beta_{n,k}=b_{1}\left[\begin{array}{c} \lambda^{k}\\ k\lambda^{k-1}\\ \frac{k(k-1)}{2}\lambda^{k-2}\\ \vdots\\ \frac{k!}{(n-1)!(k-n+1)!}\lambda^{k-n+1} \end{array}\right]$

that is, $\beta_{n,k}$ has sth. to do with $b_{1}$ but not $b_{i}$ $(i>1)$ . Therefore, the reachable abundance can be computed as

$\textrm{Vol}(R_{r,N})=V_{n}\left(C_{n}\left([B,AB,...,A^{N-1}B]\right)\right)$

          $=V_{n}\left(C_{n}\left([\zeta_{n,0},\zeta_{n,1},...,\zeta_{n,N-1}]\right)\right)$

          $=\sum_{(k_{1},k_{2},\cdots,k_{n})\in\Omega_{0,N-1}^{n}}\left|\mathrm{det}\left(\left[\beta_{n,k_{1}},\beta_{n,k_{2}},\cdots,\beta_{n,k_{n}}\right]\right)\right|$

where $\det(P)=1$ , $\Omega_{0,N-1}^{n}$ is constituted by the all possible multi-tuple $$ which elements are picked from the set $\{0,1,2,\cdots,N-1\}$ and sorted by the values..

      Because that above $\beta_{n,k}$ has sth. to do with $b_{1}$ but not $b_{i}(i>1)$ , the reachable abundance is only related to the first row, but not other rows, of the input matrix $B$ according to the upper Jordan matrix $A$ . The above conclusion is consistency with the following reachability conclusion in the control theory.

“The reachability is only related to the first/last row of the input matrix $B$ according to the upper/lower Jordan matrix $A$ .”