Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the standard canonical form of time-varying linear DAEs


Authors: Thomas Berger and Achim Ilchmann
Journal: Quart. Appl. Math. 71 (2013), 69-87
MSC (2010): Primary 34A09, 65L80
DOI: https://doi.org/10.1090/S0033-569X-2012-01285-1
Published electronically: August 27, 2012
MathSciNet review: 3075536
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Abstract: We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) $ E(t)\dot x=A(t)x$ which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE $ \dot z_1 = J(t)z_1$ and a pure DAE $ N(t) \dot z_2 = z_2$, where $ N$ is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF.


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Additional Information

Thomas Berger
Affiliation: Institute for Mathematics, Ilmenau University of Technology, Weimarer Straße 25, 98693 Ilmenau, Germany
Email: thomas.berger@tu-ilmenau.de

Achim Ilchmann
Affiliation: Institute for Mathematics, Ilmenau University of Technology, Weimarer Straße 25, 98693 Ilmenau, Germany
Email: achim.ilchmann@tu-ilmenau.de

DOI: https://doi.org/10.1090/S0033-569X-2012-01285-1
Keywords: Time-varying linear differential algebraic equations, standard canonical form, analytically solvable, generalized transition matrix
Received by editor(s): March 2, 2011
Received by editor(s) in revised form: May 26, 2011
Published electronically: August 27, 2012
Additional Notes: Supported by DFG grant IL 25/9.
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.


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