## 4 The New Computational Procedure
### The Multiple Shooting Approach  
The multiple shooting approach \(MSM_{N}\) enables us to divide the independent variable domain into an assumed number of \(N\) subintervals. Then, the relation between obtained subintervals takes the form of Equations (10) and (11). Moreover, in a sense, the differential-system of Equation (1) is divided into subintervals also. In the other words, the DAE systems can be considered independently in subintervals. This situation is presented with details in Equation (25):  
\[ \begin{array}{llllll}\text{EQs:}&\dot{y}(t)&=&f(y(t),z(t),u(t),p,t)\\ &0&=&g(y(t),z(t),u(t),p,t)\\ \text{TI:}&t&\in&[t_{0}\quad t_{f}]\\ \text{CIC:}&0&=&g(y(t_{0}),z(t_{0}),u(t_{0}),p,t_{0})\\ \text{IC:}&y(t_{0})&=&y_{0}\\ \end{array} \] \[ \begin{array}{llllll}\text{LQs:}&\dot{y}^{1}(t^{1})&=&f^{1}(y^{1}(t ^{1}),z^{1}(t^{1}),u^{1}(t^{1}),p,t^{1})\\ &0&=&g^{1}(y^{1}(t^{1}),z^{1}(t^{1}),u^{1}(t^{1}),p,t^{1})\\ \text{TI:}&t^{1}&\in&[t_{0}^{1}\quad t_{f}^{1}]\\ \text{CIC:}&0&=&g^{1}(x_{y^{1}},z^{1}(t_{0}^{1}),u^{1}(t_{0}^{1}),p,t_{0}^{1} )\\ \text{IC:}&y^{1}(t_{0}^{1})&=&x_{y^{1}}\\ &\vdots\\ \text{LQs:}&\dot{y}^{N}(t^{N})&=&f^{N}(y^{N}(t^{N}),z^{N}(t^{N}),u^{N}(t^{N}), p,t^{N})\\ &0&=&g^{N}(y^{N}(t^{N}),z^{N}(t^{N}),u^{N}(t^{N}),p,t^{N})\\ \text{TI:}&t^{N}&\in&[t_{0}^{N}\quad t_{f}^{N}]\\ \text{CIC:}&0&=&g^{N}(x_{y^{N}},z^{N}(t_{0}^{N}),u^{N}(t_{0}^{N}),p,t_{0}^{N} )\\ \text{IC:}&y^{N}(t_{0}^{N})&=&x_{y^{N}}\\ \end{array}  \tag{25}\]  
The new system obtained by the \(MSM_{N}\) approach consists of the following components:  
* \(i=1\),..., \(N\) is the number of the current subintervals. * The current \(i\)-th subinterval is considered in a time interval (TI) \(t^{i}\in[t_{0}^{i}\quad t_{f}^{i}]\). * The initial conditions (ICs) of the differential state trajectories are treated as decision variables and take a structured concatenated form: \[\left[ \begin{array}{c}y^{1}(t_{0}^{1})\\ \vdots\\ y^{N}(t_{0}^{N})\end{array} \right]=\left[ \begin{array}{c}x_{y^{i}}\\ \vdots\\ x_{y^{N}}\end{array} \right]\in\mathcal{R}^{(n_{y^{1}}+\cdots+n_{y^{N}})}\] (26)* EQsdenotes the system of the DAE constraints connected to the \(i\)-th subinterval. The considered system consists of the vector-valued functions \(f^{i}\) and \(g^{i}\) such that: \[ \begin{array}{ll}f^{i}:&\mathcal{R}^{n_{i}}\times\mathcal{R}^{n_{i}}\times \mathcal{R}^{n_{u}}\times\mathcal{R}^{n_{\mu}}\times\mathcal{R}\rightarrow \mathcal{R}^{n_{\mu}}\\ g^{i}:&\mathcal{R}^{n_{\mu}}\times\mathcal{R}^{n_{\mu}}\times\mathcal{R}^{n_{ \mu}}\times\mathcal{R}^{n_{\mu}}\times\mathcal{R}\rightarrow\mathcal{R}^{n_{ \mu}}\end{array} \] (27) and \(y^{i}(t^{i})\in\mathcal{R}^{n_{\mu}}\) is the vector of the differential state trajectories in the \(i\)-th subinterval. \(z^{i}(t^{i})\in\mathcal{R}^{n_{\mu}}\) is the vector of the algebraic state trajectories in the \(i\)-th subinterval. \(u^{i}(t^{i})\in\mathcal{R}^{n_{\mu}}\) denotes the vector of the control functions in the \(i\)-th subinterval. \(p\in\mathcal{R}^{n_{\mu}}\) is the vector of the parameters constant in time. \(t^{i}\in\mathcal{R}\) is the independent variable, and \(t^{i}\in[t^{i}_{0}\quad t^{i}_{f}]\). * It is assumed that: \[f^{1}\
eq f^{2}\
eq\cdots\
eq f^{N}\] (28) \[g^{1}\
eq g^{2}\
eq\cdots\
eq g^{N}\] (29) \[y^{1}(t^{1})\
eq y^{2}(t^{2})\
eq\cdots\
eq y^{N}(t^{N})\] (30) \[z^{1}(t^{1})\
eq z^{2}(t^{2})\
eq\cdots\
eq z^{N}(t^{N})\] (31) \[u^{1}(t^{1})\
eq u^{2}(t^{2})\
eq\cdots\
eq u^{N}(t^{N})\] (32) and this is true, e.g., for DAE models in multi-stage systems [22]. * To solve the DAE system on the appropriate subintervals (25), the consistent initial conditions (CICs) are needed. In particular, the consistent initial conditions can be obtained with the following equations: \[\left[ \begin{array}{c}g^{1}(x_{y^{i}},z^{1}(t^{1}_{0}),u^{1}(t^{1}_{0}),p,t^ {1}_{0})\\ \vdots\\ g^{N}(x_{y^{i}},z^{N}(t^{N}_{0}),u^{N}(t^{N}_{0}),p,t^{N}_{0})\end{array}  \right]=\left[ \begin{array}{c}0\\ \vdots\\ 0\end{array}