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Generalized solutions of Riccati equalities and inequalities


The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinskiĭ in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.

Key words: Discrete time-invariant systems, scattering supply rate, passive systems, Riccati equality, Riccati inequality, Kalman-Yakubovich-Popov inequality.

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TitleGeneralized solutions of Riccati equalities and inequalities
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 95-116
MathSciNet MR3522854
zbMATH 06665382
MilestonesReceived 07/02/2016; Revised 03/03/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

D. Z. Arov
Division of Applied Mathematics and Informatics, Institute of Physics and Mathematics, South-Ukrainian Pedagogical University, Odessa, 65020, Ukraine

M. A. Kaashoek
Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands

D. R. Pik
Faculty of Social and Behavioural Sciences, University of Amsterdam, Amsterdam, The Netherlands

Citation Example

D. Z. Arov, M. A. Kaashoek, and D. R. Pik, Generalized solutions of Riccati equalities and inequalities, Methods Funct. Anal. Topology 22 (2016), no. 2, 95-116.


@article {MFAT840,
    AUTHOR = {Arov, D. Z. and Kaashoek, M. A. and Pik, D. R.},
     TITLE = {Generalized solutions of Riccati equalities and inequalities},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {2},
     PAGES = {95-116},
      ISSN = {1029-3531},
  MRNUMBER = {MR3522854},
 ZBLNUMBER = {06665382},
       URL = {},

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  1. T. Ando, De Branges Spaces and Analytic Operator Functions, Lecture notes of the division of Applied Mathematics Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan, 1990.
  2. Yury M. Arlinskii, Seppo Hassi, and Henk S. V. de Snoo, Parametrization of contractive block operator matrices and passive discrete-time systems, Complex Anal. Oper. Theory 1 (2007), no. 2, 211-233.  MathSciNet CrossRef
  3. Yury Arlinskii, The Kalman-Yakubovich-Popov inequality for passive discrete time-invariant systems, Oper. Matrices 2 (2008), no. 1, 15-51.  MathSciNet CrossRef
  4. D. Z. Arov, M. A. Kaashoek, and D. R. Pik, Minimal and optimal linear discrete time-invariant dissipative scattering systems, Integral Equations Operator Theory 29 (1997), no. 2, 127-154.  MathSciNet CrossRef
  5. D. Z. Arov, M. A. Kaashoek, and D. R. Pik, Minimal representations of a contractive operator as a product of two bounded operators, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 313-336.  MathSciNet
  6. D. Z. Arov, M. A. Kaashoek, and D. R. Pik, The Kalman-Yakubovich-Popov inequality for discrete time systems of infinite dimension, J. Operator Theory 55 (2006), no. 2, 393-438.  MathSciNet
  7. D. Z. Arov and M. A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), no. 1, 1-45.  MathSciNet CrossRef
  8. D. Z. Arov and M. A. Nudel′man, Criterion of unitary similarity of minimal passive scattering systems with a given transfer function, Ukrainian Math. J. 52 (2000), no. 2, 161-172.  MathSciNet CrossRef
  9. D. Z. Arov and M. A. Nudel′man, Conditions for the similarity of all minimal passive realizations of a given transfer function (scattering and resistance matrices), Mat. Sb. 193 (2002), no. 6, 3-24.  MathSciNet CrossRef
  10. Damir Z. Arov and Olof J. Staffans, The infinite-dimensional continuous time Kalman-Yakubovich-Popov inequality, The extended field of operator theory, Oper. Theory Adv. Appl., vol. 171, Birkhauser, Basel, 2007, pp. 37-72.  MathSciNet CrossRef
  11. Mihaly Bakonyi and Hugo J. Woerdeman, Matrix completions, moments, and sums of Hermitian squares, Princeton University Press, Princeton, NJ, 2011.  MathSciNet CrossRef
  12. J. A. Ball and V. Bolotnikov, De Branges-Rovnyak Spaces: Basics and Theory, Operator Theory, D. Alpay (ed.), Springer, Basel, 2015, pp. 631-680.
  13. S. S. Boiko and V. K. Dubovoi, On some extremal problem connected with the suboperator of the scattering through inner channels of the system, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (1997), no. 4, 7-11.  MathSciNet
  14. Allen Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458-495.  MathSciNet
  15. Michael A. Dritschel and James Rovnyak, The operator Fejer-Riesz theorem, A glimpse at Hilbert space operators, Oper. Theory Adv. Appl., vol. 207, Birkhauser Verlag, Basel, 2010, pp. 223-254.  MathSciNet CrossRef
  16. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons\ New York-London, 1963.  MathSciNet
  17. Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhauser Verlag, Basel, 1990.  MathSciNet CrossRef
  18. C. Foias, A. E. Frazho, I. Gohberg, and M. A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Operator Theory: Advances and Applications, vol. 100, Birkhauser Verlag, Basel, 1998.  MathSciNet CrossRef
  19. Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  MathSciNet
  20. M. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 431-495.  MathSciNet
  21. Bela Sz.-Nagy, Ciprian Foias, Hari Bercovici, and Laszlo Kerchy, Harmonic analysis of operators on Hilbert space, Universitext, Springer, New York, 2010.  MathSciNet CrossRef

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