## 4 The New Computational Procedure
### The Similarity between the Original and Parametrized DAE Systems  
An important issue is related to the method of measuring the similarity between the original DAE system and the modified model with the estimated coefficient matrices \(P_{f}\) and \(P_{g}\). As was mentioned in the previous section, the difference between the systems can have an integral form adapted to the multiple shooting method, according to the following definition.  
**Definition 3**: _The difference between the original and modified DAE systems in a semi-explicit form (1) can be expressed as:_  
\[\Delta_{DAE}=\sum_{i=1}^{N}\sum_{j=1}^{n_{y}}\int_{t_{0}^{i}}^{t_{i}^{i}}\|f_{ j}^{i}(\cdot)-\widetilde{f}_{j}^{i}(\cdot)\|+\sum_{i=1}^{N}\sum_{j=1}^{n_{z}} \int_{t_{0}^{i}}^{t_{f}^{i}}\|g_{j}^{i}(\cdot)-\widetilde{g}_{j}^{i}(\cdot)\|, \tag{41}\]  
_where \(\|\cdot\|\) denotes a chosen norm, \(i\) is the index of a subinterval, and \(j\) denotes the index of the function vector component._  
Further considerations require the following assumption.  
**Assumption 3**: _Functions \(f_{j}^{i}(\cdot,t^{i})\), \(\widetilde{f}_{j}^{i}(\cdot,t^{i})\), \(g_{k}^{i}(\cdot,t^{i})\) and \(\widetilde{g}_{k}^{i}(\cdot,t^{i})\) for \(i=1,\ldots,N\), \(j=1,\ldots n_{y}\), \(k=1,\ldots,n_{z}\) are continuous, real-valued, and defined on the appropriate subintervals \(t^{i}\in[t_{0}^{i}\quad t_{f}^{i}]\) for all considered consistent initial conditions and control functions \(u^{i}(t^{i})\) and \(\widetilde{u}^{i}(\cdot,t^{i})\)._  
Based on Assumption 3, let us consider the two following DAE constraint systems:  
\[\begin{array}{rcl}\dot{y}(t)&=&f(y(t),z(t),\widetilde{u}(P_{u};t),p,t)\\ 0&=&g(y(t),z(t),\widetilde{u}(P_{u};t),p,t)\end{array} \tag{42}\]  
and:  
\[\begin{array}{rcl}\dot{\tilde{y}}(t)&=&\widetilde{f}(P_{f};\tilde{y}(t),z(t ),\widetilde{u}(P_{u};t),p,t)\\ 0&=&\widetilde{g}(P_{g};\tilde{y}(t),z(t),\widetilde{u}(P_{u};t),p,t)\end{array} \tag{43}\]  
To estimate the difference between the differential state trajectories:  
\[\Delta_{y}=y-\tilde{y} \tag{44}\]  
the implications of the mean value theorem can be applied.  
**Theorem 3**: _If \(\dot{y}(t)=\dot{\tilde{y}}(t)\) for all \(t\) in an interval \(t\in(t_{0}\quad t_{f})\) of the domain of these functions, then \(y(t)-\tilde{y}(t)\) is constant or \(y(t)=\dot{y}(t)+c\), where \(c\in\mathcal{R}\) is constant on \((t_{0}\quad t_{f})\)._  
Let \(\Delta_{y}=y-\tilde{y}\), then \(\Delta_{\tilde{y}}=\dot{y}-\dot{\tilde{y}}=f(\cdot)-\widetilde{f}(\cdot)=0\) on the interval \((t_{0}\quad t_{f})\). Therefore, \(\Delta_{y}=y-\dot{\tilde{y}}=f-\widetilde{f}=const\) or \(y=\dot{y}+c\), constant \(c\in\mathcal{R}\).  
As a result of Theorem 3, to supervise the fulfillment of the condition:  
\[\left[\begin{array}{rcl}\dot{y}^{1}(t^{1})&=&\dot{\tilde{y}}^{1}(t^{1})\\ &\vdots&\\ \dot{y}^{N}(t^{N})&=&\dot{\tilde{y}}^{N}(t^{N})\end{array}\right] \tag{45}\]  
the vector of equality constraints is considered as a part of the new optimization task:\[\left[\begin{array}{ccccc}f^{1}(y^{1}(t^{1}_{0}),z^{1}(t^{1}_{0}),\tilde{u}^{1}(P _{u};t^{1}_{0}),p,t^{1}_{0})&-&\tilde{f}^{1}(P_{f};g^{1}(t^{1}_{0}),\tilde{z}^{1 }(t^{1}_{0}),\tilde{u}^{1}(P_{u};t^{1}_{0}),p,t^{1}_{0})\\ f^{1}(y^{1}(t^{1}_{f}),z^{1}(t^{1}_{f}),\tilde{u}^{1}(P_{u};t^{1}_{f}),p,t^{1}_ {f})&-&\tilde{f}^{1}(P_{f};g^{1}(t^{1}_{f}),\tilde{z}^{1}(t^{1}_{f}),\tilde{u}^{ 1}(P_{u};t^{1}_{f}),p,t^{1}_{f})\\ &&\vdots&\\ f^{N}(y^{N}(t^{N}_{0}),z^{N}(t^{N}_{0}),\tilde{u}^{N}(P_{u};t^{N}_{0}),p,t^{N}_ {0})&-&\tilde{f}^{N}(P_{f};\tilde{y}^{N}(t^{N}_{0}),\tilde{z}^{N}(t^{N}_{0}), \tilde{u}^{N}(P_{u};t^{N}_{0}),p,t^{N}_{0})\\ f^{N}(y^{N}(t^{N}_{f}),z^{N}(t^{N}_{f}),\tilde{u}^{N}(P_{u};t^{N}_{f}),p,t^{N}_ {f})&-&\tilde{f}^{N}(P_{f};\tilde{y}^{N}(t^{N}_{f}),\tilde{z}^{N}(t^{N}_{f}), \tilde{u}^{N}(P_{u};t^{N}_{f}),p,t^{N}_{f})\\ \end{array}\right] \tag{46}\]  
**Remark 1**.: _The values of \(z^{i}(t^{i}_{0})\), \(i=1,\ldots,N\) are a consequence of the consistent initial conditions._