Remarks on DAEs Clause Samples

Remarks on DAEs. DAEs are distinct from ODEs in that they are not completely solvable for the derivatives of all components of the functionx(t) because these may not all appear (i.e., some equations are algebraic). Consider following DAE given in the general form f(x,˙x, y, t) = 0 (4) wherex=x(t),˙x=˙x(t)∈R n, y=y(t)∈R m, f:G⊆R n ×R n ×R m ×R→R n+m. For a set of initial conditions (x0,˙x0, y0, t0) = 0 to be consistent, it must satisfy system (4) at an initial timet 0: f(x 0,˙x0, y0, t0) = 0. (5) The consistent initialization of such systems is the problem that ▇▇▇▇▇▇▇▇▇▇ considered in his famous paper [Pan88]. Note that “initial conditions” here refers to the vector (x0,˙x0, y0) rather than simply (x0, y0), hence elements of˙x0 may appear in (5). There is another complication as [Pan88] describes: Differentiating some or even all of the original equations produces new equa- tions which must also be satisfied by the initial conditions. This need not necessarily constrain the vector (x0,˙x0, y0) further: differentiation can also introduce new variables (time-derivatives of˙▇▇▇▇▇) and it may well be the case that the new equations can be satisfied for all possible values of the initial conditions and appropriate choices of values for the new variables. Thus, in this case no useful information is generated by differentiation. ▇▇▇▇▇▇▇▇▇▇ then proposes an algorithm to analyze the structure of the system equations and determine the minimal subset for which differentiation may yield useful information in the sense that it imposes further constraints on the vector of initial conditions. This algorithm is not only useful for the initialization of a DAE system, but also for transform- ing the DAE system to a corresponding ODE system. The algorithm is further detailed below. An important property for DAEs is the notion of theDAE index. There exist different definitions for DAE index in the literature. The following informal definition is from [Fri14, Section 18.2.2]. A more formal definition can be found in [AP98, Definition 9.1, p. 236]. Definition 2.1.DAE index or differential index. Thedifferential indexof a general DAE system is the minimum number of times that certain equations in the system need to be differentiated to reduce the system to a set of ODEs, which can then be solved by the usual ODE solvers. The DAE index can therefore be seen as measurement for the distance between a DAE and a corresponding ODE. An ODE system inexplicit state-space formis a DAE system ofindex 0: ˙x...