Self-adjoint differential-algebraic equations

  • Peter Kunkel
  • Volker MehrmannEmail author
  • Lena Scholz
Original Article


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.


Self-adjoint pair of matrix functions Self-conjugate operator  Optimal control with DAEs Condensed form Congruence transformation Hamiltonian system 



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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Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany
  2. 2.Institut für MathematikTU BerlinBerlinGermany

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