The index of general nonlinear DAEs

Abstract

In the last few years there has been considerable research on differential algebraic equations (DAEs) F (t, y, y ) = 0 where F y is identically singular. Much of the mathematical effort has focused on computing a solution that is assumed to exist. More recently there has been some discussion of solvability of DAEs. There has historically been some imprecision in the use of the two key concepts of solvability and index for DAEs. The index is also important in control and systems theory but with different terminology. The consideration of increasingly complex nonlinear DAEs makes a clear and correct development necessary. This paper will try to clarify several points concerning the index. After establishing some new and more precise terminology that we need, some inaccuracies in the literature will be corrected. The two types of indices most frequently used, the differentiation index and the perturbation index, are defined with respect to solutions of unperturbed problems. Examples are given to show that these indices can be very different for the same problem. We define new "maximum indices, " which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other. These indices are also related to an index defined in terms of Jacobians. (1991): 65L20 Mathematics Subject Classification

Conversion methods for improving structural analysis of differential-algebraic equation systems

“…The equations in block 8 are f 3 , f 4 , f 5 , f 6 . In these equations, we perform the following substitutions.…”

“…Various definitions of index exist in the literature: differentiation index [6,16,17], geometric index [51,53], structural index [12,43,46], perturbation index [20], tractability index [18], and strangeness index [25]. Among these definitions, the differentiation index is the most popular one; see its definition below.…”

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

“…Most of the results that we present below carry over directly to this system. Due to the use of orthogonal transformations, it is also clear that the two transformed systems (13) and (14) are globally kinematically equivalent. It is important to note in addition that the form (13) generalizes the semi-explicit form which appears frequently in applications, see [9].…”

“…Consider the DAE system (13) and suppose that the boundedness condition (17) holds. Then, the DAE system (13) is Lyapunov-regular if and only if the underlying ODE system (15) is Lyapunov-regular.…”

Difference Indices of Quasi-Regular Difference Algebraic Systems

“…Actually, there are several definitions of differential indices of a differential algebraic system in the literature (see for instance [4,11,13,15]), which are not completely equivalent. However, the corresponding notion of difference indices for difference algebraic systems has been rarely studied yet.…”