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Regularity of distributional differential algebraic equations

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Abstract

Time-varying differential algebraic equations (DAEs) of the form \({E\dot{x}=Ax+f}\) are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.

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  1. Institute for Mathematics, Ilmenau University of Technology, Ilmenau, Germany

    Stephan Trenn

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Trenn, S. Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21, 229–264 (2009). https://doi.org/10.1007/s00498-009-0045-4

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