## 3 A Generalized Parametrization Approach  
In the presented work, the multiple shooting approach is used to parametrize both the control function, as well as the differential-algebraic model of the system. Usually, in the direct optimization algorithms, a partial state trajectory parametrization is often used. Therefore, to unify parametrization issues, the general method for the model, state, and control function parametrization is defined. The presented discussion is an extension for the understanding of the direct shooting approach. Definition: Let us denote by \(\mathcal{P}_{m}\left\{f(x)\right\}\) the parametrization procedure, where \(m\in\mathcal{N}_{+}\) and \(f:\mathcal{R}^{n_{x}}\rightarrow\mathcal{R}^{n_{f}}\). \(\mathcal{P}_{m}\left\{f(x)\right\}\) results in a new parametrized function \(\widetilde{f}(P;x)\), such that \(\widetilde{f}:\mathcal{R}^{n_{x}}\rightarrow\mathcal{R}^{n_{f}}\) and \(P\in\mathcal{R}^{n_{f}\times m}\) is a matrix of the function \(\widetilde{f}\) parameters, where:  
\[\min_{P}\int_{x_{0}}^{x_{f}}\left\|f(x)-\widetilde{f}(P,x)\right\| \tag{12}\]  
for \(x\in[x_{0}\quad x_{f}]\) and according to the chosen norm. In particular, the parametrization of a scalar function \(f(x)\) such that \(f:\mathcal{R}^{n_{x}}\rightarrow\mathcal{R}\) is:  
\[\mathcal{P}_{m}\left\{f(x)\right\}=\widetilde{f}\left(\left[ \begin{array}{ cc}p^{1}\quad p^{2}\quad\cdots\quad p^{m}\end{array} \right];x\right)= \widetilde{f}(P;x). \tag{13}\]  
Then, according to Definition 2, the control function \(u(t)\in\mathcal{R}^{n_{u}}\) can take a new form:  
\[\mathcal{P}_{m_{u}}\{u(t)\} = \mathcal{P}_{m_{u}}\left\{\left[ \begin{array}{c}u_{1}(t)\\ u_{2}(t)\\ \vdots\\ u_{n_{u}}(t)\end{array} \right]\right\}=\left[ \begin{array}{c}\mathcal{P}_{m_ {u}}\{u_{1}(t)\\ \mathcal{P}_{m_{u}}\{u_{2}(t)\}\\ \vdots\\ \mathcal{P}_{m_{u}}\{u_{n}(t)\}\end{array} \right]=\left[ \begin{array}{c} \tilde{u}_{1}([u_{1}^{1}\quad u_{1}^{2}\quad\cdots\quad u_{1}^{n_{u}}];t)\\ \tilde{u}_{2}([u_{2}^{1}\quad u_{2}^{2}\quad\cdots\quad u_{2}^{n_{u}}];t)\\ \vdots\\ \tilde{u}_{n_{u}}([u_{n_{u}}^{1}\quad u_{n_{u}}^{2}\quad\cdots\quad u_{n_{u}}^{ n_{u}}];t)\end{array} \right]\] \[= \widetilde{u}\left(\underbrace{\left[ \begin{array}{cccc}u_{1}^{ 1}\quad u_{1}^{2}\quad\cdots\quad u_{1}^{n_{u}}\\ u_{2}^{1}\quad u_{2}^{2}\quad\cdots\quad u_{2}^{n_{u}}\\ \vdots\quad\vdots\\ u_{n_{u}}\quad u_{n_{u}}^{2}\quad\cdots\quad u_{n_{u}}^{n_{u}}\end{array}  \right]}_{P_{k}\in\mathcal{R}^{n_{u}\times m_{u}}};t\right)=\widetilde{u}(P_{u };t)\]  
and briefly:  
\[\mathcal{P}_{m_{u}}\{u(t)\}=\widetilde{u}(P_{u};t). \tag{15}\]The parameter matrix \(P_{u}\) should be found with the condition:  
\[\min_{P_{u}}\int_{t_{0}}^{tf}\|u(t)-\widetilde{u}(P_{u};t)\| \tag{16}\]  
for \(t\in[t_{0}\quad t_{f}]\) and according to the chosen norm. Moreover, the parametrization procedure \(\mathcal{P}\) can be applied to the DAE constraints (1). In other words, it can be used with regard to the model relations, not the state trajectories. The parametrization procedure \(\mathcal{P}\) can result in a new form of the model dependencies:  
\[\mathcal{P}_{m}\left\{\left[ \begin{array}{c}\dot{y}(t)\\ 0\end{array} \right]\right\} = \mathcal{P}_{m}\left\{\left[ \begin{array}{c}f_{1}(y(t),z(t),u( t),p,t)\\ f_{2}(y(t),z(t),u(t),p,t)\\ \vdots\\ f_{n_{y}}(y(t),z(t),u(t),p,t)\\ g_{1}(y(t),z(t),u(t),p,t)\\ g_{2}(y(t),z(t),u(t),p,t)\\ \vdots\\ g_{n_{z}}(y(t),z(t),u(t),p,t)\end{array} \right]\right\}=\left[ \begin{array} []{c}\mathcal{P}_{m}\{f_{1}(y(t),z(t),u(t),p,t)\\ \mathcal{P}_{m}\{f_{2}(y(t),z(t),u(t),p,t)\}\\ \vdots\\ \mathcal{P}_{m}\{f_{n_{y}}(y(t),z(t),u(t),p,t)\}\\ \mathcal{P}_{m}\{g_{1}(y(t),z(t),u(t),p,t)\}\\ \mathcal{P}_{m}\{g_{2}(y(t),z(t),u(t),p,t)\}\\ \vdots\\ \mathcal{P}_{m}\{g_{n_{z}}(y(t),z(t),u(t),p,t)\}\end{array} \right]\] \[= \left[