New answers to an old question in the theory of differential–algebraic equations: Essential underlying ODE versus inherent ODE

Abstract

In the context of linear differential–algebraic equations (DAEs) one finds different associated explicit ordinary differential equations (ODEs), among them essential underlying and inherent explicit regular ones, abbreviated: EUODEs and IERODEs. EUODEs have been introduced in 1991 for index-2 DAEs in Hessenberg form by means of special transformations. IERODEs result within the framework of the projector based decoupling. Each such explicit ODE is occasionally considered to rule the flow of the DAE.The question to which extend EUODEs and IERODEs are related to each other has been asked promptly after 1991. For index-2 Hessenberg-form DAEs, answers have been given in 2005, saying that EUODEs represent somehow condensed IERODEs. Recently, EUODEs have been indicated for general arbitrary-index DAEs and it has been proved that they are condensed IERODEs. The understanding of the relation between the IERODE and the EUODEs enables to uncover the stability behavior of the DAE flow.In the present paper we show that both, the IERODEs and EUODEs of a DAE with arbitrary high index do not at all depend on derivatives of the right-hand side. We consider adjoint pairs of DAEs and provide generalizations of the classical Lagrange identity. Furthermore, we address Lyapunov spectra and Lyapunov regularity.