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Surveys in Differential-Algebraic Equations II pp 163–285Cite as

Differential-Algebraic Equations from a Functional-Analytic Viewpoint: A Survey

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Abstract

The purpose of this paper is to provide an overview on the state of the art concerning functional-analytic properties associated with differential-algebraic equations (DAEs). We summarize the relevant literature and develop a basic theory of linear and nonlinear differential-algebraic operators. In particular, we consider Fredholm properties, normal solvability, generalized inverses, least-squares solutions, splittings of regular linear differential-algebraic operators, bounded outer inverses, local solvability of equations with regular nonlinear differential-algebraic operators, Newton–Kantorovich iterations, and regularizations of the ill-posed problems arising from higher-index operators.

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Notes

  1. 1.

    In contrast to Sect. 3, here we do not fix these projectors to be orthogonal.

  2. 2.

    ker Π μ−1 and im  Π μ−1 are used in \(\mathbb{R}^{m}\) and in \(\mathcal{C}_{D}^{1}\) , but no confusion should arise.

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  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Roswitha März

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  1. Technische Universität Ilmenau Institut für Mathematik, Ilmenau, Germany

    Achim Ilchmann

  2. Universität Hamburg Fachbereich Mathematik, Hamburg, Germany

    Timo Reis

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März, R. (2015). Differential-Algebraic Equations from a Functional-Analytic Viewpoint: A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations II. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-11050-9_4

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