Gunther Reißig.

Semi-Implicit Differential-Algebraic Equations Constitute a Normal
Form.

IEEE Trans. CAS, Part I, vol. 42, no. 7, July 1995, pp. 399-402.

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**Abstract:**

Continuously differentiable functions, the total derivative or a
partial derivative of which is of constant rank, play a part in many
engineering problems. One usually exploits this property of constancy
of rank by applying the Rank Theorem. However, in case only a partial
derivative is of constant rank, which is the natural situation for
functions involved in Differential-Algebraic Equations (DAEs), this
theorem does not apply immediately. In this note, we generalize known
results to the latter case. More precisely, we give a parametrized
version of the Rank Theorem and results on functional dependence and
present a normal form for a class of nonlinear equations. Although
these results are general in nature, the fundamental conclusion with
respect to DAEs is that here the normal form exactly corresponds to
semi-implicit DAEs. We also generalize results from the solution
theory of DAEs in case differential geometric techniques fail to apply.
Such DAEs occur, for example, in the analysis of certain circuits.

**BibTeX entry:**

@article{Reissig95b,
AUTHOR = {Rei{\ss}ig, Gunther},
TITLE = {Semi-implicit differential-algebraic equations constitute a normal form},
JOURNAL = {IEEE Trans. Circuits Systems I Fund. Theory Appl.},
VOLUME = {42},
YEAR = {1995},
NUMBER = {7},
PAGES = {399-402},
doi = {10.1109/81.401157}
}

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