Self-adjoint differential-algebraic equations
- 282 Downloads
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.
KeywordsSelf-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.
- 1.Abou-Khandil H, Freiling G, Ionescu V, Jank G (2000) Matrix Riccati equations in control and systems theory. Birkhäuser, BaselGoogle Scholar
- 2.Backes A (2006) Optimale Steuerung der linearen DAE im Fall Index 2. Dissertation, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universität zu Berlin, BerlinGoogle Scholar
- 3.Balla K, Kurina G, März R (2003) Index criteria for differential algebraic equations arising from linear-quadratic optimal control problems. Preprint 2003–14, Institut für Mathematik, Humboldt-Universität zu Berlin, BerlinGoogle Scholar
- 8.Boltyanskii V, Gamkrelidze R, Mishenko E, Pontryagin LS (1962) The mathematical theory of optimal processes. Interscience, New YorkGoogle Scholar
- 15.Friswell MI, Penny JET, Garvey SD, Lees AW (2010) Dynamics of rotating machines. Cambridge University Press, CambridgeGoogle Scholar
- 19.Hesteness MR (1966) Calculus of variations and optimal control theory. John Wiley and Sons, New YorkGoogle Scholar
- 26.Kunkel P, Mehrmann V (2008) Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index. Math Control Signals Sys 20:227–269Google Scholar
- 29.Kunkel P, Mehrmann V, Scholz L (2011) Self-adjoint differential-algebraic equations. Technical Report 13/2011, Institut für Mathematik, TU Berlin, Berlin. http://www.math.tu-berlin.de/preprints/
- 36.M do R de Pinho, Vinter RB (1997) Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J Math Anal Appl 212:493–516Google Scholar