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
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.
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
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.
BibTex
@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 = {http://mfat.imath.kiev.ua/article/?id=840},
}
References
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.
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. MathSciNetCrossRef
Yury Arlinskii, The Kalman-Yakubovich-Popov inequality for passive discrete time-invariant systems, Oper. Matrices 2 (2008), no. 1, 15-51. MathSciNetCrossRef
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. MathSciNetCrossRef
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
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
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. MathSciNetCrossRef
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. MathSciNetCrossRef
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. MathSciNetCrossRef
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. MathSciNetCrossRef
Mihaly Bakonyi and Hugo J. Woerdeman, Matrix completions, moments, and sums of Hermitian squares, Princeton University Press, Princeton, NJ, 2011. MathSciNetCrossRef
J. A. Ball and V. Bolotnikov, De Branges-Rovnyak Spaces: Basics and Theory, Operator Theory, D. Alpay (ed.), Springer, Basel, 2015, pp. 631-680.
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
Allen Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458-495. MathSciNet
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. MathSciNetCrossRef
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
Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhauser Verlag, Basel, 1990. MathSciNetCrossRef
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. MathSciNetCrossRef
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MathSciNet
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
Bela Sz.-Nagy, Ciprian Foias, Hari Bercovici, and Laszlo Kerchy, Harmonic analysis of operators on Hilbert space, Universitext, Springer, New York, 2010. MathSciNetCrossRef
