Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory

  • Kalle M. Mikkola
  • Ilya M. Spitkovsky
Part of the Operator Theory: Advances and Applications book series (OT, volume 181)


The spectral or canonical factorization of matrix- or operator-valued function F defined on the imaginary axis is defined as F = Y *X, where Y ±1,X ±1 are H (bounded and holomorphic on Rez > 0), or, more generally, Y ±1,X ±1 belong to some weighted strong H2 space.

It is well known that the invertibility of the corresponding Toeplitz operator P + FP 0+ is necessary for this factorization to exist, where P + : L2 → H2 is the orthogonal projection. When F is positive, this condition is also sufficient for the factors to be H. In the general (indefinite) case, this is not so. However, if F is smooth enough, then the H canonical factorization does exist even in the indefinite case; we give a solution assuming that F is the Fourier transform of a measure with no singular continuous part.

If the (Popov function determined by the) transfer function of a control system has a canonical factorization, then a well-posed optimal state feedback exists for the corresponding control problem. Conversely, a well-posed optimal state feedback determines a canonical factorization of the transfer function. We generalize this to unstable systems, i.e., to transfer functions that are holomorphic and bounded on some right half-plane | Rez > r.

Then we show that if the generalized Popov Toeplitz operator is uniformly positive, then the canonical factorization exists (the stable case is well known). However, the results on the regularity of the factors and in the nonpositive case remain very few — we explain them and the remaining open problems.


Spectral factorization proper J-canonical factorization unstable canonical factorization regularity regular well-posed linear systems 


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  1. [BKS02]
    Albrecht Böttcher, Yuri I. Karlovich, and Ilya M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Operator Theory: Advances and Applications, vol. 131, Birkhäuser Verlag, Basel, 2002.zbMATHGoogle Scholar
  2. [BR86]
    R.G. Babadzhanyan and V.S. Rabinovich, Factorization of almost periodic operator functions, Differential and integral equations and complex analysis (Russian), Kalmytsk. Gos. Univ., Elista, 1986, pp. 13–22, i.Google Scholar
  3. [CD80]
    Frank M. Callier and Charles A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems, Ann. Soc. Sci. Bruxelles Sér. I 94 (1980), no. 1, 7–51.MathSciNetzbMATHGoogle Scholar
  4. [CG81]
    Kevin F. Clancey and Israel Gohberg, Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, vol. 3, Birkhäuser-Verlag, Basel Boston Berlin, 1981.zbMATHGoogle Scholar
  5. [CW99]
    Frank M. Callier and Joseph J. Winkin, The spectral factorization problem for multivariable distributed parameter systems, Integral Equations Operator Theory 34 (1999), no. 3, 270–292.CrossRefMathSciNetzbMATHGoogle Scholar
  6. [CW06]
    Ruth F. Curtain and George Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control Optim. (2006).Google Scholar
  7. [CZ95]
    Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Springer-Verlag, New York, 1995.zbMATHGoogle Scholar
  8. [FLT88]
    Franco Flandoli, Irena Lasiecka, and Roberto Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Ann. Mat. Pura Appl. 153 (1988), 307–382.CrossRefMathSciNetzbMATHGoogle Scholar
  9. [GKS03]
    Israel Gohberg, Marinus A. Kaashoek, and Ilya M. Spitkovsky, An overview of matrix factorization theory and operator applications, Factorization and integrable systems (Faro, 2000), Oper. Theory Adv. Appl., vol. 141, Birkhäuser, Basel, 2003, pp. 1–102.Google Scholar
  10. [GL73a]
    Israel C. Gohberg and Yuri Laiterer, A criterion for factorization of operator-functions with respect to a contour, Soviet Math. Dokl. 14 (1973), 425–429.zbMATHGoogle Scholar
  11. [GL73b]
    _____, The factorization of operator-functions relative to a contour. III. Factorization in algebras. (Russian), Math. Nachr. 55 (1973), 33–61.CrossRefMathSciNetGoogle Scholar
  12. [Gri93]
    Gustaf Gripenberg, Stability of Volterra equations with measure kernels in Banach spaces, J. Math. Anal. Appl. 178 (1993), 156–164.CrossRefMathSciNetzbMATHGoogle Scholar
  13. [IOW99]
    Vlad Ionescu, Cristian Oară, and Martin Weiss, Generalized Riccati theory and robust control. a Popov function approach, John Wiley, New York, London, 1999.zbMATHGoogle Scholar
  14. [LS87]
    Georgii S. Litvinchuk and Ilia M. Spitkovskii, Factorization of measurable matrix functions, Operator Theory: Advances and Applications, vol. 25, Birkhäuser Verlag, Basel, 1987, Translated from the Russian by Bernd Luderer, with a foreword by Bernd Silbermann.Google Scholar
  15. [LT00]
    Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. I abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000.Google Scholar
  16. [Mik02]
    Kalle M. Mikkola, Infinite-dimensional linear systems, optimal control and algebraic Riccati equations, Doctoral dissertation, technical report A452, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2002, Scholar
  17. [Mik06a]
    _____, Characterization of transfer functions of Pritchard-Salamon or other realizations with a bounded input or output operator, Integral Equations Operator Theory 54 (2006), no. 3, 427–440.Google Scholar
  18. [Mik06b]
    _____, State-feedback stabilization of well-posed linear systems, Integral Equations Operator Theory 55 (2006), no. 2, 249–271.Google Scholar
  19. [Mik07a]
    _____, Weakly coprime factorization and state-feedback stabilization of discrete-time systems, submitted, 2007.Google Scholar
  20. [Mik07b]
    _____, Indefinite optimal control and well-posed minimizing control, in preparation, 2007.Google Scholar
  21. [Mik08]
    _____, Fourier multipliers for L 2 functions with values in nonseparable Hilbert spaces and operator-valued Hp boundary functions, Ann. Acad. Sci. Fenn.Math. 33 (2008), no. 1, to appear.Google Scholar
  22. [RR85]
    Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford University Press, New York, Oxford, 1985.zbMATHGoogle Scholar
  23. [Sal89]
    Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory 21 (1989), 147–164.CrossRefMathSciNetzbMATHGoogle Scholar
  24. [Sta97]
    Olof J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc. 349 (1997), 3679–3715.CrossRefMathSciNetzbMATHGoogle Scholar
  25. [Sta98]
    _____, Feedback representations of critical controls for well-posed linear systems, Internat. J. Robust Nonlinear Control 8 (1998), 1189–1217.CrossRefMathSciNetzbMATHGoogle Scholar
  26. [Sta05]
    _____, Well-Posed Linear Systems, Encyclopedia Math. Appl., vol. 103, Cambridge University Press, Cambridge, 2005.Google Scholar
  27. [vK93]
    Bert van Keulen, H -control for distributed parameter systems: A state space approach, Birkhäuser Verlag, Basel Boston Berlin, 1993.Google Scholar
  28. [VSF82]
    Mathukumalli Vidyasagar, Hans Schneider, and Bruce A. Francis, Algebraic and topological aspects of feedback stabilization, IEEE Trans. Autom. Control 27 (1982), no. 5, 880–894.CrossRefMathSciNetzbMATHGoogle Scholar
  29. [Win89]
    Joseph Winkin, Spectral factorization and feedback control for infinite-dimensional control systems, Doctoral dissertation, Facultées Universitaires Notre-Dame de la Paix à Namur, 1989.Google Scholar
  30. [WW97]
    Martin Weiss and George Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems 10 (1997), 287–330.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Kalle M. Mikkola
    • 1
  • Ilya M. Spitkovsky
    • 2
  1. 1.Institute of MathematicsHelsinki University of TechnologyHUTFinland
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA

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