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Solution Concepts for Linear DAEs: A Survey

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Surveys in Differential-Algebraic Equations I

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks.

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Notes

  1. 1.

    A polynomial matrix is called unimodular if it is invertible and its inverse is again a polynomial matrix.

  2. 2.

    The topology is such that a sequence \((\varphi_{k})_{k\in\mathbb{N}}\) of test functions converges to zero if, and only if, (1) the supports of all φ k are contained within one common compact set \(K\subseteq\mathbb{R}\) and (2) for all \(i\in\mathbb{N}\), \(\varphi_{k}^{(i)}\) converges uniformly to zero as k→∞.

  3. 3.

    Two locally integrable functions which only differ on a set of measure zero are identified with each other.

  4. 4.

    Some authors [30, 38] use a different definition for the matrix vector product which is due to the different viewpoint of a distributional vector x as a map from to \(\mathbb{R}\) instead of a map from to \(\mathbb{R}^{n}\). The latter seems the more natural approach in view of applying it to (1.1), but it seems that both approaches are equivalent at least with respect to the solution theory of DAEs.

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Trenn, S. (2013). Solution Concepts for Linear DAEs: A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34928-7_4

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