Nonlinear DAEs

Abstract

We discuss general nonlinear DAEs with properly involved derivative and characterize the class of regular nonlinear DAEs by means of admissible matrix function sequences and associated projector functions which are pointwise generated from the first partial Jacobians of the given DAE data, and which directly generalize those given for linear DAEs. We do not use higher derivatives and derivative arrays. We show that, in particular, all Hessenberg form DAEs and large classes of DAEs resulting from the modified nodal analysis in circuit simulation are regular in this sense. We provide new local solvability assertions and perturbation results. We further introduce so-called regularity regions of a DAE and prove a practically useful theorem concerning linearizations. Several characteristic values including the tractability index are attributed to each regularity region, which generalizes the Kronecker index and the Kronecker structural data of constant matrix pencils. It is pointed out that a DAE may have several regularity regions, also those with different characteristics. Solutions may cross the borders of these regions featuring a critical behavior.