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Self-adjoint differential-algebraic equations

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Abstract

Motivated by the structure which arises e.g., in the necessary optimality boundary value problem of DAE constrained linear-quadratic optimal control, a special class of structured DAEs, so-called self-adjoint DAEs, is studied in detail. It is analyzed when and how this structure is actually associated with a self-conjugate operator. Local structure preserving condensed forms under constant rank assumptions are developed that allow to study the existence and uniqueness of solutions. A structured global condensed form and structured reduced models based on derivative arrays are developed as well. Furthermore, the relationship between DAEs with self-conjugate operator and Hamiltonian systems are analyzed and it is characterized when there is an underlying symplectic flow.

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Acknowledgments

We thank three anonymous referees for their helpful comments which significantly improved the content and readability of the paper. This work was partially supported by the Research In Pairs program of Mathematisches Forschungsinstitut Oberwolfach, whose hospitality is gratefully acknowledged. Peter Kunkel was partially supported by the Deutsche Forschungsgemeinschaft through Project KU964/7-1. Volker Mehrmann and Lena Scholz were partially supported by the Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon Mathematics for key technologies in Berlin.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 , Leipzig, Germany

    Peter Kunkel

  2. Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 , Berlin, Germany

    Volker Mehrmann & Lena Scholz

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  1. Peter Kunkel
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  2. Volker Mehrmann
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  3. Lena Scholz
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Correspondence to Volker Mehrmann.

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Kunkel, P., Mehrmann, V. & Scholz, L. Self-adjoint differential-algebraic equations. Math. Control Signals Syst. 26, 47–76 (2014). https://doi.org/10.1007/s00498-013-0109-3

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