## 4 Admissibility of 2D Singular Continuous Systems
\]  
and right-multiply by \(S_{1}^{\top}\) as follows:  
\[\text{Sym}\Big{\{}S_{1}\tilde{A}\tilde{P}\tilde{E}S_{1}^{\top} \Big{\}}+S_{1}\tilde{A}\Big{(}I-\tilde{E}\Big{)}\Big{(}\tilde{Y}\Big{)}\Big{(} I-\tilde{E}\Big{)}^{\top}\tilde{A}^{\top}S_{1}^{\top}\] \[= \text{Sym}\Big{\{}S_{1} \begin{bmatrix}\tilde{A}_{11}&\tilde{A}_ {12}\\ \tilde{A}_{21}&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}\tilde{P}_{11}&0\\ \tilde{P}_{21}&0\end{bmatrix} S_{1}^{\top}\Big{\}}\] \[+ S_{1} \begin{bmatrix}\tilde{A}_{11}&\tilde{A}_{12}\\ \tilde{A}_{21}&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&\tilde{Y}_{22}\end{bmatrix}\begin{bmatrix}\tilde{A}_{11}^{\top}&\tilde{A}_ {21}^{\top}\\ \tilde{A}_{12}^{\top}&\tilde{A}_{22}^{\top}\end{bmatrix} S_{1}^{\top}\]  
In order to transform \(\tilde{A}\) into a block diagonal matrix, we introduce matrix  
\[S_{2}= \begin{bmatrix}I&0\\ -\tilde{A}_{22}^{-1}\tilde{A}_{21}&I\end{bmatrix} \]  
into (19) to obtain  
\[\text{Sym}\Big{\{}S_{1}\tilde{A}S_{2}S_{2}^{-1}\tilde{P}\tilde{E} S_{1}^{\top}\Big{\}}+S_{1}\tilde{A}S_{2}S_{2}^{-1}\Big{(}I-\tilde{E}\Big{)}\Big{(} \tilde{Y}\Big{)}\Big{(}I-\tilde{E}\Big{)}^{\top}S_{2}^{-\top}S_{2}^{\top} \tilde{A}^{\top}S_{1}^{\top} \tag{20}\] \[= \text{Sym}\Bigg{\{} \begin{bmatrix}\mathcal{A}&0\\ 0&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}\tilde{P}_{11}&0\\ \tilde{A}_{22}^{-1}\tilde{A}_{21}\tilde{P}_{11}+\tilde{P}_{21}&0\end{bmatrix}  \Bigg{\}}+ \begin{bmatrix}\mathcal{A}&0\\ 0&\tilde{A}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&\tilde{Y}_{22}\end{bmatrix}\begin{bmatrix}\mathcal{A}&0\\ 0&\tilde{A}_{22}\end{bmatrix} ^{\top}\]  
from which we obtain  
\[\text{Sym}\Big{\{}\mathcal{A}\tilde{P}_{11}\Big{\}}<0 \tag{21}\]  
a condition that states the stability of system (12) if matrix \(\tilde{P}_{11}\) is positive definite. A condition that imposes the positivity on \(\tilde{P}_{11}\) could be given as follows:  
\[-\text{Sym}\Bigg{\{} \begin{bmatrix}\tilde{P}_{11}&0\\ \tilde{P}_{21}&0\end{bmatrix} \Bigg{\}}+ \begin{bmatrix}0&0\\ 0&\tilde{X}_{22}\end{bmatrix} <0\]Notice that  
\[-\text{Sym}\bigg{\{} \begin{bmatrix}\tilde{P}_{11}&0\\ \tilde{P}_{21}&0\end{bmatrix} \bigg{\}}+ \begin{bmatrix}0&0\\ 0&\tilde{X}_{22}\end{bmatrix} \] \[= -\text{Sym}\bigg{\{} \begin{bmatrix}\tilde{P}_{11}&\tilde{P}_{12}\\ \tilde{P}_{21}&\tilde{P}_{22}\end{bmatrix}\begin{bmatrix}I&0\\ 0&0\end{bmatrix} \bigg{\}}+ \begin{bmatrix}0&0\\ 0&I\end{bmatrix}\begin{bmatrix}\tilde{X}_{11}&\tilde{X}_{12}\\ \tilde{X}_{21}&\tilde{X}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&I\end{bmatrix} \] \[= -\text{Sym}\bigg{\{} \begin{aligned} \top&V^{-1}& PV^{-\top}&I&0\\ 0&0\end{aligned} \bigg{\}}+ \begin{bmatrix}0&0\\ 0&I\end{bmatrix}\begin{bmatrix}\tilde{X}_{11}&\tilde{X}_{12}\\ \tilde{X}_{21}&\tilde{X}_{22}\end{bmatrix}\begin{bmatrix}0&0\\ 0&I\end{bmatrix} \]  
The previous condition is equivalent to  
\[-\text{Sym}\bigg{\{}PV^{-\top} \begin{bmatrix}I&0\\ 0&0\end{bmatrix} \bigg{J}^{\top}V^{\top}\bigg{\}}+V \begin{bmatrix}0&0\\ 0&I\end{bmatrix} \tilde{X} \begin{bmatrix}0&0\\ 0&I\end{bmatrix} J^{\top}V^{\top}<0\] \[-\text{Sym}\Big{\{}PV^{-\top}\tilde{E}^{\top}V^{\top}\Big{\}}+V \big{(}I-\tilde{E}\big{)}\tilde{X} \begin{pmatrix}I-\tilde{E}\end{pmatrix} ^{ \top}J^{\top}V^{\top}<0\] \[-\text{Sym}\Big{\{}PV^{-\top}\tilde{E}^{\top}V^{\top}\Big{\}}+V \big{(}I-J^{\top}\tilde{E}\big{)}\tilde{X}\begin{pmatrix}I-J^{\top}\tilde{E} \big{)}\Big{\}}^{\top}J^{\top}V^{\top}<0\] \[-\text{Sym}\Big{\{}PV^{-\top}\tilde{E}^{\top}V^{\top}\Big{\}}+V \big{(}I-\tilde{E}\big{)}J\tilde{X}J^{\top}(I-\tilde{E})U\Big{\}}^{\top}>0\] \[\text{Sym}\Big{\{}PE^{\ddagger}\Big{\}}+E^{\ddagger}XE^{\ddagger \top}<0\]  
with  
\[E^{\ddagger} = \big{(}VIE\big{)}^{\top}\] \[\tilde{X} = UXU^{\top}\]  
Finally, if there exists a symmetric matrix \(X\) such that the condition  
\[-\text{Sym}\Big{\{}PE^{\ddagger}\Big{\}}+E^{\ddagger}XE^{\ddagger \top}<0 \tag{22}\]  
holds, then \(\tilde{P}_{11}\) in (21) is positive definite.