Linear differential-algebraic equations with properly stated leading term: B-critical points

Abstract

We examine in this article so-called B-critical points of linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x(t))′ + B(t)x(t) = q(t). These critical or singular points, which cannot be handled by classical projector methods, require adapting a recently introduced framework based on Π-projectors. Via a continuation of certain invariant spaces through the singularity, we arrive at a scenario which accommodates both A- and B-critical DAEs. The working hypotheses apply in particular to standard-form analytic systems although, in contrast to other approaches to critical problems, the scope of our approach extends beyond the analytic setting. Some examples illustrate the results.

Keywords:

• differential-algebraic equation
• index
• projector
• critical point
• singularity

AMS Subject Classifications:

• 34A09
• 34A30

Acknowledgements

The authors wish to thank several suggestions and remarks from the reviewers of a former version of this manuscript. This research was supported by the DFG Forschungszentrum Mathematics for Key Technologies (MATHEON) in Berlin. R. Riaza was supported by Research Project MTM2007-62064 of Ministerio de Educación y Ciencia, Spain.