Linear Constant Coefficient DAEs

Differential-algebraic equations (DAEs) differ in main aspects from explicit or regularly implicit ordinary differential equations. This concerns theory, e.g. solvability and representation of the solution, as well as numerical aspects, e.g. convergence behavior under discretization. Both aspects depend essentially on the index of the DAE. Thus, we first define the index. 

The solution of these equations can be obtained explicitly by modal transformation, i.e. computing eigenvalues and eigenvectors. Prom the computation of the solution the main differences between explicit ODEs and DAEs are figured out. 

Definition 2.6.1 The smallest number k with =0 is called the index of nilpo-tency of N and is identical to the size of the largest Jordan block of N, 

Substituting the definition of J into Eq. (2.6.3), using the coordinate transformation X = T x and premultiplying by T yields 

Because of the nonsingularity of R, Eq. (2.6.5a) is an explicit ODE, the state space form of the system. 

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In Sec. 2.6 we discussed generalized eigenvalues of linear constant coefficient DAEs and found three groups of eigenvalues... 

In these approaches, control variables are approximated by piecewise constant, piecewise linear, or, in general, polynomial functions, over a specified number of control intervals. As shown in Figure 14.1, an NLP is formulated at the outer level, with coefficients of these polynomials as optimization variables. With fixed profiles for control variables, there are no degrees of freedom, and the DAE system can be solved at an inner level with any commercial DAE solver. As the number of control variables is usually small compared to the total number of variables, the resulting NLP problem has a relatively small number of optimization variables. 

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