## 4 Admissibility of 2D Singular Continuous Systems  
To check the stability of the system, consider the Lyapunov inequality  
\[\text{Sym}\Big{\{}APE^{\top}\Big{\}}+Z<0 \tag{13}\]with \(P=\text{diag}\Big{\{}P^{h},\ P^{v}\Big{\}}\). Matrix \(Z\) has to be chosen in order to satisfy (13). Let  
\[J = \begin{bmatrix}I_{r^{h}}&0&0&0\\ 0&0&I_{n^{h}-r^{h}}&0\\ 0&I_{r^{n}}&0&0\\ 0&0&0&I_{n^{\pi}-r^{n}}\end{bmatrix} \] \[\text{satisfying }JJ^{\top}=J^{\top}J=I\] \[\text{Sym}\Big{\{}J^{\top}UAPE^{\top}U^{\top}J\Big{\}} = \text{Sym}\Big{\{}\Big{(}J^{\top}UAVJ\Big{)}\left(J^{\top}V^{-1} PV^{-\top}J\right)\left(J^{\top}V^{\top}E^{\top}U^{\top}J\right)\Big{\}}\] \[= \text{Sym}\Big{\{}\Big{(}J^{\top}\bar{A}J\Big{)}\left(J^{\top} \bar{P}J\right)\left(J^{\top}\bar{E}^{\top}J\right)\Big{\}}\] \[= \text{Sym}\Big{\{}\bar{A}\bar{B}\bar{E}^{\top}\Big{\}}\]  
Notice that  
\[\bar{A} = J^{\top}\bar{A}J= \begin{bmatrix}\bar{A}_{11}&\bar{A}_{12}\\ \bar{A}_{21}&\bar{A}_{22}\end{bmatrix} \] \[\bar{E} = J^{\top}\bar{E}J= \begin{bmatrix}I_{r^{h}}&0&0&0\\ 0&I_{r^{n}}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} \] \[\bar{P} = J^{\top}\bar{P}J= \begin{bmatrix}\bar{P}_{11}^{hh}&0&\vdots\bar {P}_{12}^{hh}&0\\ 0&\bar{P}_{11}^{vv}&\vdots&0&\bar{P}_{12}^{vv}\\ \bar{P}_{21}^{vv}&0&\vdots\bar{P}_{22}^{vv}&0\\ 0&\bar{P}_{21}^{vv}&\vdots&0&\bar{P}_{22}^{vv}\end{bmatrix} = \begin{bmatrix} \bar{P}_{11}&\bar{P}_{12}\\ \bar{P}_{21}&\bar{P}_{22}\end{bmatrix} \]  
It comes then that  
\[\text{Sym}\Big{\{}\bar{A}\bar{P}\bar{E}^{\top}\Big{\}} = \text{Sym}\Big{\{} \begin{bmatrix}\bar{A}_{11}&\bar{A}_{12}\\ \bar{A}_{21}&\bar{A}_{22}\end{bmatrix}\begin{bmatrix}\bar{P}_{11}&0\\ \bar{P}_{21}&0\end{bmatrix} \Big{\}} \tag{14}\]  
Notice that in (14), matrix \(\bar{A}_{22}\) is eliminated, and as a consequence, matrix \(Z\) in (13) should include matrix \(\bar{A}_{22}\) in order to constraint its invertibility, which can be insured if there exists a symmetric matrix \(\bar{Y}_{22}\) such as \(\bar{A}_{22}\bar{Y}_{22}\bar{A}_{22}^{\top}<0\). Notice that  
\[ \begin{bmatrix}*&*\\ *&\bar{A}_{22}\bar{Y}_{22}\bar{A}_{22}^{\top}\end{bmatrix}  = \begin{bmatrix}\bar{A}_{11}&\bar{A}_{12}\\ \bar{A}_{21}&\bar{A}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&\bar{Y}_{22}\end{bmatrix}\begin{bmatrix}\bar{A}_{11}^{\top}&\bar{A}_{21}^{ \top}\\ \bar{A}_{12}^{\top}&\bar{A}_{22}^{\top}\end{bmatrix} \] \[=\bar{A}\Big{(}I-\bar{E}\Big{)}\Big{(}\bar{Y}\Big{)}\left(I-\bar{ E}\right)^{\top}\bar{A}^{\top} \tag{15}\] \[=J^{\top}UZU^{\top}J\]  
with  
\[Z = AE^{\dagger}YE^{\dagger\top}A^{\top} \tag{16}\]  
and \(E^{\dagger}=V(I-UEV)U\). As a consequence, condition (13) becomes  
\[\text{Sym}\Big{\{}APE^{\top}\Big{\}}+AE^{\dagger}YE^{\dagger\top} A^{\top} \tag{17}\]If we complete (14) with (15), we obtain  
\[\text{Sym}\Big{\{}\tilde{A}\tilde{P}\tilde{E}\Big{\}}+\tilde{A} \Big{(}I-\tilde{E}\Big{)}\Big{(}\tilde{Y}\Big{)}\Big{(}I-\tilde{E}\Big{)}^{\top} \tilde{A}^{\top} \tag{18}\] \[= \text{Sym}\Big{\{} \begin{bmatrix}\tilde{A}_{11}&\tilde{A}_{12}\\ \tilde{A}_{21}&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}\tilde{P}_{11}&0\\ \tilde{P}_{21}&0\end{bmatrix} \Big{\}}+ \begin{bmatrix}\tilde{A}_{11}&\tilde{A}_ {12}\\ \tilde{A}_{21}&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&\tilde{Y}_{22}\end{bmatrix}\begin{bmatrix}\tilde{A}_{11}^{\top}&\tilde{A}_{2 1}^{\top}\\ \tilde{A}_{12}^{\top}&\tilde{A}_{22}^{\top}\end{bmatrix} \]  
which implies that, if (12) holds, matrix \(\tilde{A}_{22}\) satisfies \(\tilde{A}_{22}\tilde{Y}_{22}\tilde{A}_{22}^{\top}<0\) which, in fact, yields the invertibility of matrix \(\tilde{A}_{22}\). Notice that the invertibility of matrix \(\tilde{A}_{22}\) implies that the system is regular and impulse-free. At this step, we have to check the stability of system (12). For this aim, let us left-multiply condition (18) by  
\[S_{1}= \begin{bmatrix}I&-\tilde{A}_{12}\tilde{A}_{22}^{-1}\\ 0&I\end{bmatrix}