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 | References | Similar Articles | Additional Information

Abstract: We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE and a pure DAE , where 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.