On linear differential-algebaic equations with properly stated leading term. I: Regular points

Roswitha März , Ricardo Riaza

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

34A09 Implicit equations, differential-algebraic equations

34A30 Linear equations and systems, general

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.