## 4 The New Computational Procedure
### The DAE Constraints' Parametrization  
The presented \(MSM_{N}\)-based step of the new optimization task formulation does not indicate the exact way of the state and control trajectories' parametrization. These aspects will be considered in this section. The parametrization used should take into account the properties of the considered optimization task, in particular:  
* the physical (technological) properties of the system under consideration (e.g., oscillations, discontinuities, a nonlinear behavior like hysteresis, regions of instability),
* the properties of the DAE solver used (the ability to determine the consistent initial conditions, the model solution accuracy in the instability region),* the performance of the optimization algorithm (the available operating memory, the possibility of parallel computations, dedicated numerical algebra procedures for sparse matrices and large-scale tasks, the globalization strategy, the rate of convergence). The solutions presented in the literature represent two possible ways to parametrize the differential-algebraic constraints system (1). 1. Parametrize both the state and control trajectories, but without parametrization of the relations represented by the \(f\) and \(g\) functions. This approach results in a large-scale algebraic system of equations. Let us denote: \[ \begin{array}{ccc}\mathcal{P}_{m_{y}}\{y(t)\}&=&\widetilde{y}(P_{y};t)\\ \mathcal{P}_{m_{z}}\{z(t)\}&=&\widetilde{z}(P_{z};t)\\ \mathcal{P}_{m_{u}}\{u(t)\}&=&\widetilde{u}(P_{u};t)\end{array} \] (35) where \(m_{y},m_{z},m_{u}\in\mathcal{N}_{+}\) and \(P_{y},P_{z},P_{u}\) are the matrices of the approximating polynomial coefficients. Then, if: \[\dot{\widetilde{y}}(P_{y};t_{i})=\left.\frac{d\widetilde{y}(P_{y};t)}{dt} \right|_{t=t_{i}}\] (36) \(t\in\{t_{i}\}_{i=0}^{m}\), with the relation \(t_{0}<\cdots<t_{m}\), where \(m=\max\{m_{y},m_{z},m_{u}\}\), the following algebraic system of equations \(\mathcal{G}(P_{y},P_{z},P_{u})\) is obtained: \[\mathcal{G}(P_{y},P_{z},P_{u})=\left[ \begin{array}{c}\dot{\widetilde{y}}(P_{ y};t_{0})-f(\widetilde{y}(P_{y};t_{0}),\widetilde{z}(P_{z};t_{0}),\widetilde{u}(P _{u};t_{0}),p,t_{0})\\ \vdots\\ \dot{\widetilde{y}}(P_{y};t_{m})-f(\widetilde{y}(P_{y};t_{m}),\widetilde{z}(P _{z};t_{m}),\widetilde{u}(P_{u};t_{m}),p,t_{m})\\ g(\widetilde{y}(P_{y};t_{0}),\widetilde{z}(P_{z};t_{0}),\widetilde{u}(P_{u},t_ {0}),p,t_{0})\\ \vdots\\ g(\widetilde{y}(P_{y};t_{m}),\widetilde{z}(P_{z};t_{m}),\widetilde{u}(P_{u};t _{m}),p,t_{m})\end{array} \right]=\left[ \begin{array}{c}0\\ \vdots\\ 0\\ 0\\ \vdots\\ 0\end{array} \right]\] (37)
2. The parametrization of the constraints represented by the functions \(f\) and \(g\): Opposite the previous point, the state and control trajectories can be calculated by a numerical procedure implemented in a DAE solver. Recently, a special case of DAE model parametrization was used to introduce the variability constraints into the optimization task [22]. The choice of a particular parametrization method is dependent on the task under consideration, as well as the efficiency of the solution method used. The situation when the DAE constraints cannot be solved with the available implemented numerical procedure seems to be particularly interesting. Further considerations will be made to resolve this issue. Let us consider the nonlinear system of differential-algebraic constraints (1). In computer simulations of many real-life problems, the direct application of the numerical DAE solvers cannot result in an accurate and stable solution. These cases were reported, e.g., in the monograph [22]. Therefore, it is suggested to apply a simplified DAE model, which can reflect the dynamical features appropriately and does not exhibit numerical difficulties or instabilities. The commonly used methodologies in the context of a highly nonlinear DAE constraint solution are based on a polynomial representation of the state trajectories \(y(t)\) and \(z(t)\) ([23]) or a full state parametrization (direct transcription method [17]).