## 2 Optimization with Differential-Algebraic Constraints
Throughout this work, it is assumed that the considered model constraints can be described by the system of index-1 highly nonlinear DAEs in a semi-explicit form (1). In the context of the differential-algebraic model constraints, a scalar-valued performance index in a general form can be considered:  
\[\min_{u(t)}\mathcal{J}(y(t),z(t),u(t),t) \tag{8}\]  
and in particular:  
\[\mathcal{J}(y(t),z(t),u(t),t)=\phi(y(t_{f}),z(t_{f}),t_{f})+\int_{t_{0}}^{t_{f }}\psi(y(t),z(t),u(t),t). \tag{9}\]  
To optimize the performance index (8) subject to the DAE system (1), both indirect and direct methods can be applied. The indirect methods are based on the solution of the necessary optimality conditions. Thus, for the indirect methods, it is necessary to explicitly derive a system of the Euler-Lagrange equations consisting of adjoint equations, control equations, and transversality conditions, separately for each considered task [17]. Although the indirect methods have some known disadvantages, the recent research results indicate efficient indirect schemes [18], as well as combinations of direct-indirect approaches [19]. The optimization of systems described by the nonlinear differential-algebraic constraints requires a reliable approach to obtain the state trajectories for the given consistent initial conditions. Therefore, the further considerations related to the computational optimization with the nonlinear DAE model are based on the multiple shooting approach. The application of the multiple shooting step needs to divide the independent variable domain into an assumed number \(N\) of subintervals:  
\[t^{i}\in[t_{0}^{i}\quad t_{f}^{i}],\qquad i=1,\dots,N, \tag{10}\]  
according to the following rule:  
\[t_{0}=t_{0}^{1}<t_{f}^{1}=t_{0}^{2}<t_{f}^{2}\quad\dots\quad t_{0}^{N}<t_{f}^ {N}=t_{f} \tag{11}\]  
and can be used to parametrize the dynamic optimization task. An appropriate optimization problem parametrization enables us to apply efficient nonlinear optimization procedures and obtain a suboptimal solution even for highly nonlinear and possibly unstable systems. There are three main dynamic optimization task approaches  
1. Control vector parametrization (CVP): The parametrization is related only to the control function and results in a medium-scale nonlinear optimization task [20]. 2. Direct shooting method: The parametrized control function, as well as the initial conditions of the state trajectories are treated as additional decision variables. The differential-algebraic equationsare solved for each subinterval independently. Therefore, an efficient DAE solver is required. Additionally, the equality constraints providing the state trajectory continuity need to be taken into account [21]. 3. Direct transcription formulation: The values of the state trajectories are parametrized and treated as additional decision variables. To parametrize the state trajectories, the Runge-Kutta, Hermite-Simpson, or any other numerical scheme can be used. The direct transcription formulation results in a large-scale and sparse nonlinear optimization task. Moreover, only the final result can be considered as an applicable solution [17]. Recently, a new family of methods was constituted by a combination of the direct shooting method with a DAE model approximation. The multiple shooting approach enables us to consider other systems in each subinterval. Moreover, the appropriate approximation makes the highly nonlinear DAE models possible to solve. Then, the new model properties can be analyzed from a computational optimization point of view.