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Self-adjoint differential-algebraic equations

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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References

  1. 1.

    Abou-Khandil H, Freiling G, Ionescu V, Jank G (2000) Matrix Riccati equations in control and systems theory. Birkhäuser, Basel

  2. 2.

    Backes A (2006) Optimale Steuerung der linearen DAE im Fall Index 2. Dissertation, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universität zu Berlin, Berlin

  3. 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, Berlin

  4. 4.

    Balla K, Kurina G, März R (2006) Index criteria for differential algebraic equations arising from linear-quadratic optimal control problems. J Dyn Control Syst 12:289–311

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Balla K, Linh VH (2005) Adjoint pairs of differential-algebraic equations and Hamiltonian systems. Appl Numer Math 53:131–148

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Balla K, März R (2002) A unified approach to linear differential algebraic equations and their adjoints. Z Anal Anwendungen 21:783–802

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Barkwell L, Lancaster P (1992) Overdamped and gyroscopic vibrating systems. Trans AME: J Appl Mech 59:176–181

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Boltyanskii V, Gamkrelidze R, Mishenko E, Pontryagin LS (1962) The mathematical theory of optimal processes. Interscience, New York

    Google Scholar 

  9. 9.

    Brenan KE, Campbell SL, Petzold LR (1996) Numerical solution of initial-value problems in differential algebraic equations, 2nd edn. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  10. 10.

    Bunch JR (1982) A note on the stable decomposition of skew-symmetric matrices. Math Comp 38:475–479

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Byers R, Mehrmann V, Xu H (2007) A structured staircase algorithm for skew-symmetric/symmetric pencils. Electr Trans Num Anal 26:1–33

    MATH  MathSciNet  Google Scholar 

  12. 12.

    Campbell SL (1987) A general form for solvable linear time varying singular systems of differential equations. SIAM J Math Anal 18:1101–1115

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Devdariani EN, Ledyaev YuS (1999) Maximum principle for implicit control systems. Appl Math Optim 40:79–103

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Doležal V (1964) The existence of a continuous basis of a certain subspace of \(E_r\) which depends on a parameter. Cas Pro Pest Mat 89:466–468

    MATH  Google Scholar 

  15. 15.

    Friswell MI, Penny JET, Garvey SD, Lees AW (2010) Dynamics of rotating machines. Cambridge University Press, Cambridge

  16. 16.

    Griepentrog E, März R (1986) Differential-algebraic equations and their numerical treatment. Teubner, Leipzig

    MATH  Google Scholar 

  17. 17.

    Grubb G (2009) Distributions and operators. Springer, New York

    MATH  Google Scholar 

  18. 18.

    Hairer E, Lubich C, Wanner G (2002) Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Springer, Berlin

    MATH  Google Scholar 

  19. 19.

    Hesteness MR (1966) Calculus of variations and optimal control theory. John Wiley and Sons, New York

    Google Scholar 

  20. 20.

    Heuser H (1992) Funktionalanalysis, 3rd edn. B. G. Teubner, Stuttgart

    MATH  Google Scholar 

  21. 21.

    Kantorovitz S (2003) Introduction to modern analysis. Oxford University Press, Oxford

    MATH  Google Scholar 

  22. 22.

    Knobloch HW, Kwakernaak H (1985) Lineare Kontrolltheorie. Springer, Berlin

    Book  MATH  Google Scholar 

  23. 23.

    Kunkel P, Mehrmann V (1991) Smooth factorizations of matrix valued functions and their derivatives. Numer Math 60:115–132

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Kunkel P, Mehrmann V (1994) Canonical forms for linear differential-algebraic equations with variable coefficients. J Comput Appl Math 56:225–259

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Kunkel P, Mehrmann V (2006) Differential-algebraic equations. Analysis and numerical solution. EMS Publishing House, Zürich

    Book  MATH  Google Scholar 

  26. 26.

    Kunkel P, Mehrmann V (2008) Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index. Math Control Signals Sys 20:227–269

    Google Scholar 

  27. 27.

    Kunkel P, Mehrmann V (2011) Formal adjoints of linear DAE operators and their role in optimal control. Electr J Lin Alg 22:672–693

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Kunkel P, Mehrmann V, Rath W, Weickert J (1997) A new software package for linear differential-algebraic equations. SIAM J Sci Comput 18:115–138

    Article  MATH  MathSciNet  Google Scholar 

  29. 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/

  30. 30.

    Kunkel P, Mehrmann V, Stöver R (2004) Multiple shooting for unstructured nonlinear differential-algebraic equations of arbitrary index. SIAM J Numer Anal 42:2277–2297

    Article  Google Scholar 

  31. 31.

    Kunkel P, Mehrmann V, Stöver R (2004) Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index. Numer Math 98:277–304

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Kurina GA, März R (2004) On linear-quadratic optimal control problems for time-varying descriptor systems. SIAM J Cont Optim 42:2062–2077

    Article  MATH  Google Scholar 

  33. 33.

    Mehrmann V (1991) The autonomous linear quadratic control problem. Springer, Berlin

    Book  MATH  Google Scholar 

  34. 34.

    O’Malley RE (1991) Singular perturbation methods for ordinary differential equations. Springer, New York

    Book  MATH  Google Scholar 

  35. 35.

    Paige CC, Van Loan CF (1981) Schur decomposition for Hamiltonian matrices. Lin Alg Appl 14:11–32

    Article  Google Scholar 

  36. 36.

    M do R de Pinho, Vinter RB (1997) Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J Math Anal Appl 212:493–516

    Google Scholar 

  37. 37.

    Roubicek T, Valasek M (2002) Optimal control of causal differential-algebraic systems. J Math Anal Appl 269:616–641

    Article  MATH  MathSciNet  Google Scholar 

  38. 38.

    Thompson RC (1976) The characteristic polynomial of a principal submatrix of a Hermitian pencil. Lin Alg Appl 14:135–177

    Article  MATH  Google Scholar 

  39. 39.

    Thompson RC (1991) Pencils of complex and real symmetric and skew matrices. Lin Alg Appl 147:323–371

    Article  MATH  Google Scholar 

  40. 40.

    Van Dooren P (1979) The computation of Kronecker’s canonical form of a singular pencil. Lin Alg Appl 27:103–141

    Article  MATH  Google Scholar 

  41. 41.

    Van Loan CF (1984) A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Lin Alg Appl 16:233–251

    Article  Google Scholar 

  42. 42.

    Vinter R (2000) Optimal control. Birkhäuser, Boston

    MATH  Google Scholar 

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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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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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Keywords

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