## 3 A Generalized Parametrization Approach
\begin{array}{c}\widetilde{f}_{1}(P_{M};y(t),z(t),u(t),p, t)\\ \widetilde{f}_{2}(P_{M};y(t),z(t),u(t),p,t)\\ \vdots\\ \widetilde{f}_{n_{y}}(P_{M};y(t),z(t),u(t),p,t)\\ \widetilde{s}_{1}(P_{M};y(t),z(t),u(t),p,t)\\ \widetilde{s}_{2}(P_{M};y(t),z(t),u(t),p,t)\\ \vdots\\ \widetilde{s}_{n_{z}}(P_{M};y(t),z(t),u(t),p,t)\end{array} \right],\]  
where:  
\[P_{M}=\left[ \begin{array}{cccc}a_{1}^{1}&a_{1}^{2}&\cdots&a_{1}^{m}\\ a_{2}^{1}&a_{2}^{2}&\cdots&a_{2}^{m}\\ \vdots&\vdots&\cdots&\vdots\\ a_{n_{y}}^{1}&a_{n_{y}}^{2}&\cdots&a_{n_{y}}^{m}\\ a_{n_{y}+1}^{1}&a_{n_{y}+1}^{2}&\cdots&a_{n_{y}+1}^{m}\\ a_{n_{y}+2}^{1}&a_{n_{y}+2}^{2}&\cdots&a_{n_{y}+2}^{m}\\ \vdots&\vdots&\cdots&\vdots\\ a_{n_{y}+n_{z}}^{1}&a_{n_{y}+n_{z}}^{2}&\cdots&a_{n_{y}+n_{z}}^{m}\end{array}  \right]\in\mathcal{R}^{(n_{y}+n_{z})\times m}. \tag{18}\]  
Finally, in the considered DAE model (17), the parametrized control function \(\widetilde{u}(P_{u};t)\) can be introduced:  
\[\left[ \begin{array}{c}\dot{y}(t)\\ 0\end{array} \right]=\left[ \begin{array}{c}\widetilde{f}(P_{M};y(t),z(t), \mathcal{P}_{m_{u}}\{u(t)\},p,t)\\ \widetilde{g}(P_{M};y(t),z(t),\mathcal{P}_{m_{u}}\{u(t)\},p,t)\end{array}  \right]=\left[ \begin{array}{c}\widetilde{f}(P_{M};y(t),z(t),\widetilde{u}(P _{u};t),p,t)\\ \widetilde{g}(P_{M};y(t),z(t),\widetilde{u}(P_{u};t),p,t)\end{array} \right]. \tag{19}\]Let us consider the special case, when \(\mathcal{P}_{m}=\mathcal{P}_{ny+n_{z}+n_{u}}\). Then, the DAE model (1) takes a special parametrized form:  
\[\left[ \begin{array}{c}\dot{y}(t)\\ 0\end{array} \right]=\left[ \begin{array}{c}\mathcal{P}_{n_{y}+n_{z}+n_{u}}\{f_ {1}(y(t),z(t),u(t),p,t)\}\\ \mathcal{P}_{n_{y}+n_{z}+n_{u}}\{f_{2}(y(t),z(t),u(t),p,t)\}\\ \vdots\\ \mathcal{P}_{n_{y}+n_{z}+n_{u}}\{f_{n_{y}}(y(t),z(t),u(t),p,t)\}\\ \mathcal{P}_{n_{y}+n_{z}+n_{u}}\{g_{1}(y(t),z(t),u(t),p,t)\}\\ \mathcal{P}_{n_{y}+n_{z}+n_{u}}\{g_{2}(y(t),z(t),u(t),p,t)\}\\ \vdots\\ \mathcal{P}_{n_{y}+n_{z}+n_{u}}\{g_{n}(y(t),z(t),u(t),p,t)\}\end{array} \right] =\left[ \begin{array}{c}\widetilde{f}_{1}(P_{DAE};y(t),z(t),u(t),p,t)\\ \widetilde{f}_{2}(P_{DAE};y(t),z(t),u(t),p,t)\\ \vdots\\ \widetilde{f}_{n_{y}}(P_{DAE};y(t),z(t),u(t),p,t)\\ \widetilde{g}_{1}(P_{DAE};y(t),z(t),u(t),p,t)\\ \widetilde{g}_{2}(P_{DAE};y(t),z(t),u(t),p,t)\\ \vdots\\ \widetilde{g}_{n_{z}}(P_{DAE};y(t),z(t),u(t),p,t)\end{array} \right], \tag{20}\]  
where:  
\[P_{DAE}=\left[ \begin{array}{cccc}a_{1}^{1}&a_{1}^{2}&\cdots&a_{1}^{n_{y}+n_ {z}+n_{u}}\\ a_{2}^{1}&a_{2}^{2}&\cdots&a_{2}^{n_{y}+n_{z}+n_{u}}\\ \vdots&\vdots&\vdots\\ a_{n_{y}}^{1}&a_{n_{y}}^{2}&\cdots&a_{n_{y}}^{n_{y}+n_{z}+n_{u}}\\ a_{n_{y}+1}^{1}&a_{n_{y}+1}^{2}&\cdots&a_{n_{y}+1}^{n_{y}+n_{z}+n_{u}}\\ a_{n_{y}+2}^{1}&a_{n_{y}+2}^{2}&\cdots&a_{n_{y}+2}^{n_{y}+2}\\ \vdots&\vdots&\cdots&\vdots\\ a_{n_{y}+n_{z}}^{1}&a_{n_{y}+n_{z}}^{2}+n_{u}&\cdots&a_{n_{y}+n_{z}}^{n_{y}+n_{ z}+n_{u}}\end{array} \right]\in\mathcal{R}^{(n_{y}+n_{z})\times(n_{y}+n_{z}+n_{u})} \tag{21}\]  
with the condition:  
\[\min_{P_{DAE}}\int_{t_{0}}^{t_{f}}\|\left[ \begin{array}{c}f(y(t),z(t),u(t), p,t)\\ g(y(t),z(t),u(t),p,t)\end{array} \right]-\left[ \begin{array}{c}\widetilde{f}(P _{DAE};y(t),z(t),u(t),p,t)\\ \widetilde{g}(P_{DAE};y(t),z(t),u(t),p,t)\end{array} \right]\|. \tag{22}\]  
As was indicated, the parametrization \(\mathcal{P}_{ny+n_{z}+n_{u}}\) is applied to the model, not to the state trajectory. The appropriate size of the matrix \(P_{DAE}\) can be used to obtain a new, linear model of the DAE constraints:  
\[\left[ \begin{array}{c}\dot{y}(t)\\ 0\end{array} \right] = P_{DAE}\left[ \begin{array}{c}y(t)\\ z(t)\\ u(t)\end{array} \right]=\left[ \begin{array}{c|c}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\end{array} \right]\left[ \begin{array}{c}y(t)\\ z(t)\\ u(t)\end{array} \right]\] \[= \underbrace{\left[ \begin{array}{c|c}A_{11}&A_{12}\\ \hline A_{21}&A_{22}\end{array} \right]}_{A\in\mathcal{R}^{(ny+n_{z})\times(n_{y }+n_{u})}}\left[ \begin{array}{c}y(t)\\ z(t)\end{array} \right]+\underbrace{\left[ \begin{array}{c}A_{13}\\ A_{23}\end{array}