## 2 The Problem Formulation  
Let us consider a classical optimization task subject to the differential-algebraic model equations:  
\[\min_{u(t)}\mathcal{F}(y(t),z(t),u(t),p,t) \tag{1}\]  
s.t  
\[(DAE)\qquad\left\{ \begin{array}{rcl}\dot{y}(t)&=&f(y(t),z(t),u(t),p,t)\\ 0&=&g(y(t),z(t),u(t),p,t)\\ t&\in&[t_{0}\quad t_{f}]\\ y(t_{0})&=&y_{0}\\ z(t_{0})&=&z_{0}\end{array} \right. \tag{2}\]  
where:  
\[\mathcal{F}:\mathcal{R}^{n_{y}}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_ {u}}\times\mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow\mathcal{R} \tag{3}\]  
is an optimized performance index, \(\mathcal{R}\) denotes a set of real numbers, \(y(t)\in\mathcal{R}^{n_{y}}\) is a vector of state variables modeled by differential equations with \(\dot{y}(t)=\frac{dy(t)}{dt}\), and \(z(t)\in\mathcal{R}^{n_{z}}\) represents a vector of state variables described by algebraic constraints. Moreover, \(u(t)\in\mathcal{R}^{n_{u}}\) denotes a vector of input functions; \(p\in\mathcal{R}^{n_{p}}\) is a vector of global constant parameters; \(t\in\mathcal{R}\) is an independent variable with a known a priori range \(t\in[t_{0}\quad t_{f}]\). The considered relations take the form of the differential-algebraic equations in a semi-explicit form with:  
\[ \begin{array}{rl}f:&\mathcal{R}^{n_{y}}\times\mathcal{R}^{n_{z}}\times \mathcal{R}^{n_{u}}\times\mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow \mathcal{R}^{n_{y}}\\ g:&\mathcal{R}^{n_{y}}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_{u}} \times\mathcal{R}^{n_{p}}\times\mathcal{R}\rightarrow\mathcal{R}^{n_{z}}\end{array}  \tag{4}\]  
To solve the model equations (2), consistent initial conditions:  
\[[y(t_{0})\quad z(t_{0})]^{It is worth explaining what the main difference between the differential-algebraic equations and differential-algebraic constraints is. The DAEs in their classical understanding can be solved by specialized single- or multi-step numerical schemes [31]. For the given consistent initial conditions (6), the DAEs (2) can be solved by [PERSON]'s method or modified Runge-Kutta procedures [32]. Finally, starting from the consistent initial conditions, a set of feasible pointwise solutions can be found. This set can be further used to construct the solution trajectories \(y(t)\) and \(z(t)\). On the other hand, there is a group of methods that treat the DAE relations as the constraint system. While searching for a solution, the imposed restrictions do not have to be met. Even the starting solution may be infeasible. Finally, only the last result of the searching procedure should meet all of the model equations in the form of the differential-algebraic constraints. Therefore, it is possible to find a feasible solution without the knowledge about an explicit formulation of the constraint functions. The effectiveness of this approach in the context of simultaneous optimization has been described in the literature [33; 34].  
Although some solution procedures for the differential-algebraic equations have been just listed, it is worth presenting the main features of the direct shooting method--an approach designed for solving the highly nonlinear DAE systems. To solve the system of DAEs (2) characterized by a highly nonlinear dynamics, according to the multiple-shooting approach, the range of the independent variable is divided into an assumed number \(N\) of subintervals. Then, in each subinterval \(t^{i}\in[t_{0}^{i}\quad t_{f}^{i}]\), for \(i=1,\ldots,N\) and:  
\[t_{0}=t_{0}^{1}<t_{f}^{1}=t_{0}^{2}<\cdots<t_{f}^{N-1}=t_{0}^{N}<t_{f}^{N}=t_ {f}, \tag{7}\]  
the DAE system can be considered independently in each subinterval:  
\[(DAE^{i})\quad\left\{ \begin{array}{rcl}\dot{y}^{i}(t^{i})&=&f(y^{i}(t^{i}), z^{i}(t^{i}),u^{i}(t^{i}),p,t^{i})\\ 0&=&g(y^{i}(t^{i}),z^{i}(t^{i}),u^{i}(t^{i}),p,t^{i})\\ y^{i}(t_{0}^{i})&=&y_{0}^{i}\\ z^{i}(t_{0}^{i})&=&z_{0}^{i}\\ u^{i}(t^{i})&=&u^{i}=\text{const}\\ t^{i}&\in&[t_{0}^{i}\quad t_{f}^{i}]\\ i&=&1,\cdots,N\end{array} \right.