On linear differential-algebraic equations with properly stated leading term

Abstract

We consider in this work linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x(t))'+B(t)x(t)=q(t) through a projector approach. Our analysis applies in particular to linear DAEs in standard form E(t)x'(t)+F(t)x(t)=q(t). Under mild smoothness assumptions, we introduce local regularity and index notions, showing that they hold uniformly in intervals and are independent of projectors. Several algebraic and geometric properties supporting these notions are addressed. This framework is aimed at supporting a complementarity analysis of so-called critical points, where the assumptions for regularity fail. Our results are applied here to the analysis of a linear time-varying analogue of Chua's circuit with current-controlled resistors, displaying a rich variety of indices depending on the characteristics of resistive and reactive devices.