## 2 The Problem Formulation
\tag{8}\]  
where \(y_{0}^{i}\in\mathcal{R}^{n_{y^{i}}}\) and \(z_{0}^{i}\in\mathcal{R}^{n_{x^{i}}}\) denote the vectors of the initial conditions for the differential and algebraic state trajectories, respectively. It was assumed that the input function \(u^{i}(t^{i})\) is constant in each subinterval. The presented procedure, known as the multiple-shooting method, is a basis for computational optimization algorithms for problems with a highly nonlinear dynamics. This is because the dynamical relations can be solved more accurately on shorter subintervals than over one long range. Moreover, the initial conditions vectors \(y_{0}^{i}\) and \(z_{0}^{i}\), with the state trajectories' continuity requirements:  
\[y^{i}(t_{f}^{i})-y^{i+1}(t_{0}^{i+1}) = 0\] \[z^{i}(t_{f}^{i})-z^{i+1}(t_{0}^{i+1}) = 0 \tag{9}\] \[i = 1,\cdots,N-1\]  
can be used to introduce additional discrete process constraints. Therefore, inequality constraints on the differential state trajectory:  
\[y^{L} \leq y(t) \leq y^{U} \tag{10}\] \[t \in [t_{0}\quad t_{f}]\]with \(y^{L},y^{U}\in\mathcal{R}\), can be effectively considered in the following pointwise form:  
\[y^{L} \leq y^{i}(t_{0}^{i}) \leq y^{U} \tag{11}\] \[i = 1,\cdots,N.\]  
The presented multiple-shooting approach enables us to approximate an infinite-dimensional optimization task by a finite-dimensional formulation. The further considerations concern a new approach for optimization according to the sequential approach rules, where the differential-algebraic Equation (8) is fulfilled in each step of a computational procedure. Finally, the application of the multiple-shooting approach can be used to transform the classical task (1) and (2) to an optimization problem with pointwise-continuous constraints:  
\[\min_{\mathbf{X}}F(\mathbf{X}) \tag{12}\]  
where \(\mathbf{X}\) is a matrix of decision variables:  
\[\mathbf{X} = \left[ \begin{array}{cccc}(\mathbf{x}_{y_{0}}^{1})^{T}&(\mathbf{ x}_{z_{0}}^{1})^{T}&(\mathbf{x}_{u}^{1})^{T}\\ (\mathbf{x}_{y_{0}}^{2})^{T}&(\mathbf{x}_{z_{0}}^{2})^{T}&(\mathbf{x}_{u}^{2} )^{T}\\ \vdots&\vdots&\vdots\\ (\mathbf{x}_{y_{0}}^{N})^{T}&(\mathbf{x}_{z_{0}}^{N})^{T}&(\mathbf{x}_{u}^{N })^{T}\end{array} \right]\in\mathcal{R}^{N\times(n_{y}+n_{z}+n_{u})} \tag{13}\]  
and the objective function:  
\[F:\mathcal{R}^{N\times(n_{y}+n_{z}+n_{u})}\rightarrow\mathcal{R} \tag{14}\]  
should be optimized subject to the parametrized DAEs:  
\[(DAE^{i}(\mathbf{X})) \tag{15}\]  
the continuity constraints of the state trajectories:  
\[ \begin{array}{rcl}y^{i}(t_{f}^{i})-\mathbf{x}_{y_{0}}^{i+1}&=&0\\ z^{i}(t_{f}^{i})-\mathbf{x}_{z_{0}}^{i+1}&=&0\\ i&=&1,\cdots,N-1,\end{array}  \tag{16}\]  
as well as the initial conditions consistency constraints (6) in the new parametrized form:  
\[g(\mathbf{x}_{y_{0}}^{i},\mathbf{x}_{z_{0}}^{i},\mathbf{x}_{u}^{i},p,t_{0}^{ i}) = 0 \tag{17}\] \[i = 1,\cdots,N.\]The presented reformulation is a consequence of the multiple-shooting approach. Moreover, the continuous model equations can be solved, independently in each subinterval, by an outer numerical procedure. It is worth noticing that the application of the outer DAE solver is easy to implement and more flexible, because various numerical procedures can be used, such as ode15s in MATLAB or NDSolve in Mathematica. On the other hand, the full parametrization of the state trajectories leads us to a problem formulation that is independent of the outer DAE system solution procedure. Unfortunately, it needs, in many cases, much more computational effort and experience. Then, to obtain a feasible solution, an efficient large-scale nonlinear programming procedure is necessary. In the full-parametrization approach, computational experience seems to be crucial for an efficient solution algorithm design and implementation. Therefore, in the present work, the following assumptions were made. The system of differential-algebraic equations (15) should be solved by an outer numerical procedure.