The index of general nonlinear DAEs

In thelast few years there has been considerable researchon differential algebraic equations (DAEs)(Formula presented.)where (Formula presented.) is identically singular. Much of themathematical effort has focused on computing a solutionthat is assumed to exist. More recently there has beensome discussion of solvability of DAEs. There hashistorically been some imprecision in the use of the twokey concepts of solvability and index for DAEs. Theindex is also important in control and systems theorybut with different terminology. The consideration ofincreasingly complex nonlinear DAEs makes aclear and correct development necessary. This paper willtry to clarify several points concerning the index. Afterestablishing some new and more precise terminology thatwe need, some inaccuracies in the literature will becorrected. The two types of indices most frequently used,the differentiation index and the perturbation index, aredefined with respect to solutions of unperturbedproblems. Examples are given to show that these indicescan be very different for the same problem. We definenew "maximum indices," which are the maxima of earlierindices in a neighborhood of the solution over a set ofperturbations and show that these indices are simplyrelated to each other. These indices are also related to anindex defined in terms of Jacobians.