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Abstract
Time-domain models of dynamical systems are formulated in many applications in terms of differential-algebraic equations (DAEs). In the linear time-varying context, certain limitations of models of the form E(t)x′(t) + B(t)x(t) = q(t) have recently led to the properly stated formulation A(t)(D(t)x(t))′ + B(t)x(t) = q(t), which allows for explicit descriptions of problem solutions in regular DAEs with arbitrary index, and provides precise functional input-output characterizations of the system. In this context, the present paper addresses critical points of linear DAEs with properly stated leading term; such critical points describe different types of singularities in the system. Critical points are classified according to a taxonomy which reflects the phenomenon from which the singularity stems; this taxonomy is proved independent of projectors and also invariant under linear time-varying coordinate changes and refactorizations. Under certain working assumptions, the analysis of such critical problems can be carried out through a scalarly implicit decoupling, yielding a singular inherent ODE. Certain harmless problems for which this decoupling can be rewritten in explicit form are characterized. Some electrical circuit applications, including a linear time-varying analogue of Chua's circuit, are discussed for illustrative purposes.
Acknowledgements
Research supported by the DFG Forschungszentrum Mathematics for Key Technologies (MATHEON) in Berlin. The second author (corresponding author) acknowledges additional support from Vicerrectorado de Investigación, Universidad Politécnica de Madrid; his work is framed in Projects MTM2004-5316 and MTM2005-3894 of Ministerio de Educación y Ciencia, Spain.