## 3 Regularity and Impulse-Freeness of 2D Singular Continuous Systems  
For a singular system to have a smooth solution, it must be both regular and impulse-free. This means the system can be transformed into a standard system, subject to an algebraic constraint on the system state vector. Let matrices \(A\) and \(B\) be defined as follows:  
\[A= \begin{bmatrix}A^{hh}&A^{hv}\\ A^{vh}&A^{vv}\end{bmatrix} ,\qquad B= \begin{bmatrix}B^{h}\\ B^{v}\end{bmatrix} u(t_{1},t_{2}).\]  
Let \(U=\text{diag}\left\{U^{h},\ U^{v}\right\}\), \(V=\text{diag}\left\{V^{h},\ V^{v}\right\}\), where  
\(U^{i}\) and \(V^{i}\), \(i=h,v\) are some nonsingular matrices such that  
\[E^{i}=U^{i}E^{i}V^{i}= \begin{bmatrix}I_{i^{\prime}}&0\\ 0&0\end{bmatrix} \]  
for \(i=h,v\). The same transformation applied to matrix \(A\) yields  
\[\bar{A}=UAV= \begin{bmatrix}U^{h}A^{hh}V^{h}&U^{h}A^{hv}V^{v}\\ U^{v}A^{vh}V^{h}&U^{v}A^{vv}V^{v}\end{bmatrix} = \begin{bmatrix}\bar{A}^{hh}& \bar{A}^{hv}\\ \bar{A}^{vh}&\bar{A}^{vv}\end{bmatrix} .\]  
Let also \(x^{i}=V^{i} \begin{bmatrix}\bar{x}^{i}_{1}\\ \bar{x}^{i}_{2}\end{bmatrix} \) with \(\bar{x}^{i}_{1}\in\mathbb{R}^{x^{i}}\), for \(i=h,v\). System (1) with \(u(t_{1},t_{2})=0\) can be transformed as follows  
\[ \begin{bmatrix}\bar{E}^{h}\frac{\partial\bar{x}^{h}(t_{1},t_{2})}{\partial t _{1}}\\ \bar{E}^{v}\frac{\partial\bar{x}^{v}(t_{1},t_{2})}{\partial t_{2}}\end{bmatrix}  = \begin{bmatrix}\bar{A}^{hh}&\bar{A}^{hv}\\ \bar{A}^{vh}&\bar{A}^{vv}\end{bmatrix}\begin{bmatrix}\bar{x}^{h}(t_{1},t_{2}) \\ \bar{x}^{v}(t_{1},t_{2})\end{bmatrix}  \tag{6}\]  
The differential equation in (6) concerning \(\bar{x}^{h}(t_{1},t_{2})\) is then as follows:  
\[\bar{E}^{h}\frac{\partial\bar{x}^{h}(t_{1},t_{2})}{\partial t_{1}} = \bar{A}^{hh}\bar{x}^{h}(t_{1},t_{2})+\bar{A}^{hv}\bar{x}^{v}(t_{1},t _{2}) \tag{7}\]and taking into account the partitioning implied by the structure of \(\bar{E}^{h}\), we obtain  
\[ \begin{bmatrix}\frac{\partial\bar{x}_{1}^{h}(t_{1},t_{2})}{\partial t_{1}}\\ 0\end{bmatrix}  = \begin{bmatrix}\bar{A}_{11}^{hh}&\bar{A}_{12}^{hh}\\ \bar{A}_{21}^{hh}&\bar{A}_{22}^{hh}\end{bmatrix}\begin{bmatrix}\bar{x}_{1}^{h}( t_{1},t_{2})\\ \bar{x}_{2}^{h}(t_{1},t_{2})\end{bmatrix} + \begin{bmatrix}\bar{A}_{11}^{hv}&\bar{A}_ {12}^{hv}\\ \bar{A}_{21}^{hv}&\bar{A}_{22}^{hv}\end{bmatrix}\begin{bmatrix}\bar{x}_{1}^{v}( t_{1},t_{2})\\ \bar{x}_{2}^{v}(t_{1},t_{2})\end{bmatrix}  \tag{8}\]  
The same can be performed for \(\bar{x}_{1}^{v}(t_{1},t_{2})\), obtaining  
\[ \begin{bmatrix}\frac{\partial\bar{x}_{1}^{v}(t_{1},t_{2})}{\partial t_{2}}\\ 0\end{bmatrix}  = \begin{bmatrix}\bar{A}_{11}^{hh}&\bar{A}_{12}^{hh}\\ \bar{A}_{21}^{hh}&\bar{A}_{22}^{hh}\end{bmatrix}\begin{bmatrix}\bar{x}_{1}^{h} (t_{1},t_{2})\\ \bar{x}_{2}^{h}(t_{1},t_{2})\end{bmatrix} + \begin{bmatrix}\bar{A}_{11}^{vv}& \bar{A}_{12}^{vv}\\ \bar{A}_{21}^{vv}&\bar{A}_{22}^{vv}\end{bmatrix}\begin{bmatrix}\bar{x}_{1}^{v} (t_{1},t_{2})\\ \bar{x}_{2}^{v}(t_{1},t_{2})\end{bmatrix}  \tag{9}\]  
Stacking up the different Equations (8) and (9), we obtain  
\[\frac{\partial\bar{x}_{1}(t_{1},t_{2})}{\partial(t_{1},t_{2})} = \bar{A}_{11}\bar{x}_{1}(t_{1},t_{2})+\bar{A}_{12}\bar{x}_{2}(t_{1 },t_{2}) \tag{10}\] \[0 = \bar{A}_{21}\bar{x}_{1}(t_{1},t_{2})+\bar{A}_{22}\bar{x}_{2}(t_{ 1},t_{2}) \tag{11}\]  
with  
\[\bar{x}_{i}(t_{1},t_{2}) = \begin{bmatrix}\bar{x}_{i}^{h}(t_{1},t_{2})\\ \bar{x}_{i}^{v}(t_{1},t_{2})\end{bmatrix} ,\quad i=1,\,2\] \[\frac{\partial\bar{x}_{1}(t_{1},t_{2})}{\partial(t_{1},t_{2})} = \begin{bmatrix}\frac{\partial\bar{x}_{1}^{h}(t_{1},t_{2})}{ \partial t_{1}}\\ \frac{\partial\bar{x}_{1}^{v}(t_{1},t_{2})}{\partial t_{2}}\end{bmatrix} \] \[\bar{A}_{ij} = \begin{bmatrix}\bar{A}_{ij}^{hh}&\bar{A}_{ij}^{hv}\\ \bar{A}_{ij}^{vh}&\bar{A}_{ij}^{vv}\end{bmatrix} ,\quad\text{for $i=1,\,2$ and $j=1,\,2$}\]  
Equation (10) is a differential equation for a standard system. Its solution is unique and depends on the fast component \(\bar{x}_{2}(t_{1},t_{2})\).