Semi-implicit differential-algebraic equations constitute a normal form

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 (DAE's), this theorem does not apply immediately. The author generalizes known results to the latter case. More precisely, he gives a parameterized version of the Rank Theorem and results on functional dependence and presents a normal form for a class of nonlinear equations. Although these results are general in nature, the fundamental conclusion with respect to DAE's is that here the normal form exactly corresponds to semi-implicit DAE's. The author also generalizes results from the solution theory of DAE's in case differential geometric techniques fail to apply. Such DAE's occur, For example, in the analysis of certain circuits.<
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