## 2 Optimization with Differential-Algebraic Constraints  
The starting point for further considerations is the statement of a dynamic optimization task with the differential-algebraic constraints. In some practical applications, the DAE system can be presented in a semi-explicit form:  
\[ \begin{array}{rcl}\dot{y}(t)&=&f(y(t),z(t),u(t),p,t)\\ 0&=&g(y(t),z(t),u(t),p,t)\end{array}  \tag{1}\]  
where the specified functions and variables have appropriate properties and perform the assumed roles in the model; in particular, \(y(t)\in\mathcal{R}^{n_{y}}\) is a vector of differential state trajectories, and \(\dot{y}(t)=\frac{dy}{dt}\). Moreover, \(z(t)\in\mathcal{R}^{n_{z}}\) is a vector of algebraic state trajectories. \(u(t)\in\mathcal{R}^{n_{u}}\) denotes the vector of control functions. \(p\in\mathcal{R}^{n_{p}}\) is a vector of parameters constant in time. \(t\in\mathcal{R}\) is an independent variable with \(t\in[t_{0}\quad t_{f}]\). The important assumptions are connected with the functions \(f\) and \(g\) and the vector of consistent initial conditions \([y(t_{0})\quad z(t_{0})]^{T}=[y_{0}\quad z_{0}]^{T}\). **Assumption 1**. : _The initial conditions \([y(t_{0})\quad z(t_{0})]^{T}=[y_{0}\quad z_{0}]^{T}\) for the system (1) are consistent, iff for a given initial value of the control function \(u(t_{0})\), the equality:_  
\[g(y_{0},z_{0},u(t_{0}),p,t_{0})=0 \tag{2}\]  
_is fulfilled._  
The consistent initialization of the differential-algebraic systems is an important and classical task considered in the literature. The vector \([y(t_{0})\quad z(t_{0})]^{T}\) can be obtained as a result of analytical considerations [12], characterized by the real-life process initial conditions [13], computed by the deterministic optimization procedure [14] or obtained by heuristic optimization algorithms [15]. The relation between the initial conditions' consistency and the index of the considered system can be obtained. **Assumption 2**. : _Let us assume that the following \(C^{2}\) class functions \(f\) and \(g\) are considered:_  
\[ \begin{array}{rl}f:&\mathcal{R}^{n_{y}}\times\mathcal{R}^{n_{z}}\times \mathcal{R}^{n_{u}}\times\mathcal{R}^{n_{\tau}}\times\mathcal{R}\to\mathcal{R} ^{n_{y}}\\ g:&\mathcal{R}^{n_{y}}\times\mathcal{R}^{n_{z}}\times\mathcal{R}^{n_{u}}\times \mathcal{R}^{n_{\tau}}\times\mathcal{R}\to\mathcal{R}^{n_{z}}\end{array}  \tag{3}\]  
**Theorem 1**. : _If a matrix \(\frac{\partial g}{\partial z}\) is nonsingular, then the system (1) is an implicit ODE of the following form:_  
\[ \begin{array}{rcl}\dot{y}(t)&=&f(y(t),z(t),u(t),p,t)\\ \dot{z}(t)&=&\Big{(}\frac{\partial g}{\partial z}\Big{)}^{-1}\Big{(}-\frac{ \partial g}{\partial\dot{y}}\dot{y}-\frac{\partial g}{\partial u}\dot{u}- \frac{\partial g}{\partial t}\Big{)}\end{array}  \tag{4}\]  
_and has index-\(1\)._  
Proof of Theorem 1.: Let us differentiate the algebraic part of Equation (1) with respect to \(t\):  
\[0=\frac{d}{dt}\Big{(}g(y(t),z(t),u(t),t)\Big{)} \tag{5}\]  
which results in:  
\[0=\frac{\partial g}{\partial\dot{y}}\dot{y}+\frac{\partial g}{\partial z}\dot{ z}+\frac{\partial g}{\partial u}\dot{u}+\frac{\partial g}{\partial t} \tag{6}\]and:  
\[\dot{z}(t)=\Big{(}\frac{\partial g}{\partial z}\Big{)}^{-1}\Big{(}-\frac{ \partial g}{\partial y}\dot{y}-\frac{\partial g}{\partial u}\dot{u}-\frac{ \partial g}{\partial t}\Big{)} \tag{7}\]  
This is true, iff a matrix \(\frac{\partial g}{\partial z}\) is nonsingular. **Definition 1**. : _The minimum number of times that all or part of the system (1) must be differentiated with respect to \(t\) in order to determine \(\dot{z}(t)\) as a continuous function of \(z\) and \(t\) is the index of the DAE (1) [16]._  
From the computational methods' point of view, the consistency of the proposed initial conditions and the index of a system are the main properties of the differential-algebraic equations. The index-1 assumption is common for many real-life applications, although some index-2, as well as higher index formulations are the subject of research.