@article{kal:koe58,
title = "Optimal synthesis of linear sampling control systems using generalized performance indexes",
journal = "Transactions of the American Society of Mechanical Engineers",
volume = "80",
pages = "1820 - 1826",
year = "1958",
author ={R. E. Kalman and R. W. Koepcke}
}
%
@book{tou:63,
  title={Optimum Design of Digital Control Systems},
  author={L. T. Tou},
series={Mathematics in science and engineering},
volume={10},
  year={1963},
  publisher={Academic Press}
}
%
@book{fridman,
  title={Stochastic Differential Equations and Applications},
  author={A. Freedman},
volume={I},
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  publisher={Academic, New York}
}
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@book{freedman,
  title={Markov Chains},
  author={D. Freedman},
volume={},
  year={1983},
  publisher={Springer-Verlag New York}
}
%
@book{oksendal,
  title={Stochastic Differential Equations},
  author={B. \O{}ksendal},
  volume={},
  year={2003},
  publisher={Springer-Verlag Berlin Heidelberg}
}
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@Article{hal:63,
title = {An Optimization Problem for Discrete-Time Systems (Romanian)},
author = {A. Halanay},
doi = {10.1137/140951758},
journal = {Probleme de Automatizare},
volume = V,
year = 1963,
pages = {103 - 109},
}
%
%
@Article{kal:60,
title = {A New Approach to Linear Filtering and Prediction Problems},
author = {R. E. Kalman},
doi = {10.1115/1.3662552},
journal = {J. Basic Eng. Mar},
volume = 82,
number = {1},
year = 1960,
pages = {35 - 45},
}
%
%
@Article{aliev,
title = {Optimization problems for periodic systems},
author = {F. A. Aliev and V. B. Larin},
doi = {10.1007/s10778-010-0257-9},
journal = {International Applied Mechanics},
volume = 45,
year = 2009,
pages = {1162 - 1188},
}
@book{abou,
  title={Matrix Riccati Equations in Control and Systems Theory},
  author={H. Abou-Kandil and G. Freiling and V. Ionescu and G. Jank},
series={Systems \& Control: Foundations \& Applications},
  year={2003},
  publisher={Birkh\"{a}user Basel}
}
%
%
@book{costacarte,
  title={Discrete-Time Markov Jump Linear Systems},
  author={O. L. V. Costa and R. P. Marques and M. D. Fragoso},
series={Probability and Its Applications},
  year={2005},
  publisher={Springer-Verlag London}
}
%
%
@book{ghersh,
  title={$H_\infty-$Control and Estimation of State-multiplicative Linear Systems},
  author={E. Gershon and U. Shaked and I. Yaesh},
series={Lecture Notes in Control and Information Sciences},
volume={318},
  year={2005},
  publisher={Springer-Verlag London}
}
%
%
@book{green,
  title={Linear Robust Control},
  author={M. Green and D. J. N. Limebeer},
  year={2013},
  publisher={Dover Publications}
}
%
%
@book{mariton,
  title={Jump Linear Systems in Automatic Control},
  author={M. Mariton},
  year={1990},
  publisher={Marcel Dekker, New York}
}
%
%
@book{goodwin,
  title={Digital Control and Estimation: A Unified Approach},
  author={R. H. Middleton and  G. C. Goodwin},
  year={1990},
  series={Information and System Sciences},
  publisher={Prentice Hall}
}
%
%
@incollection{Bittanti,
  author      = "S. Bittanti and P. Colaneri and G.  Nicolao",
  title       = "The Periodic {Riccati} Equation",
  editor      = "S. Bittanti and A. J. Laub and J. C. Willems",
  booktitle   = "The Riccati Equation",
  publisher   = "Springer, Berlin, Heidelberg",
  series={Communications and Control Engineering},
  year        = 1991,
  pages       = "127 - 162",
  chapter     = 6,
}
%
%
@book{bit-colaneri,
  title={Periodic Systems: Filtering and Control},
  author={S. Bittanti and P. Colaneri},
series={Communications and Control Engineering},
  year={2009},
  publisher={Springer-Verlag London}
}
%
%
@book{halion,
  title={Time-Varying Discrete Linear Systems: Input-Output Operators. {Riccati} Equations. Disturbance Attenuation},
  author={A. Halanay and V. Ionescu},
series={Operator Theory: Advances and Applications},
volume={68},
  year={1994},
  publisher={Birkh\"{a}user Basel}
}
%
%
@article{Damhin,
title = "{Newton’s} method for concave operators with resolvent positive derivatives in ordered {Banach} spaces",
journal = "Linear Algebra and its Applications",
volume = "363",
pages = "43 - 64",
year = "2003",
note = "Special Issue on Nonnegative matrices, M-matrices and their generalizations",
issn = "0024-3795",
doi = "https://doi.org/10.1016/S0024-3795(02)00328-2",
url = "http://www.sciencedirect.com/science/article/pii/S0024379502003282",
author = "T. Damm and D. Hinrichsen",
keywords = "Newton’s method, Positive operators, Concave operators, Riccati equation",
abstract = "We prove a non-local convergence result for Newton’s method applied to a class of nonlinear equations in ordered real Banach spaces. The key tools in our approach are special notions of concavity and the spectral theory of resolvent positive operators."
}
%
%
@book{carte2010,
  title={Mathematical Methods in Robust Control of Discrete-Time Linear Stochastic Systems},
  author={V. Dr\u{a}gan and T. Morozan and A.-M. Stoica},
  year={2010},
  publisher={Springer-Verlag New York}
}
%
%
@ARTICLE{hench,
  author={J. J. {Hench} and A. J. {Laub}},
  journal={IEEE Transactions on Automatic Control},
  title={Numerical solution of the discrete-time periodic {Riccati} equation},
  year={1994},
  volume={39},
  number={6},
  pages={1197 - 1210},
  }
%
%
@article{oara,
title = "Stabilizing solution to the reverse discrete-time {Riccati} equation: A matrix-pencil-based approach",
journal = "Linear Algebra and its Applications",
volume = "246",
pages = "113 - 130",
year = "1996",
issn = "0024-3795",
doi = "https://doi.org/10.1016/0024-3795(94)00341-6",
url = "http://www.sciencedirect.com/science/article/pii/0024379594003416",
author = "C. Oar\u{a}",
abstract = "We derive necessary and sufficient conditions for the existence of the stabilizing solution to the reverse (discrete-time) Riccati equation, a particular type of algebraic {Riccati} equation which is related to the solution of the discrete-time version of the extended (two-block) Nehari problem. The conditions are expressed in terms of the left-stable deflating subspace of an associated symplectic matrix pencil. In particular, a maximum-phase spectral factorization of the Popov function is obtained under very relaxed conditions imposed on the initial data. It is also proved that under a restrictive additional assumption, the stabilizing solution to the reverse Riccati equation reduces to the antistabilizing solution to the usual (discrete-time) Riccati equation. A reliable numerical algorithm for computing the stabilizing solution to the reverse Riccati equation is also presented, together with formulae for the solution to the extended Nehari problem."
}
%
%
@article{ungureanu2017,
title = {Stabilizing Solution for a Discrete-Time Modified Algebraic {Riccati} Equation in Infinite Dimensions},
journal = {Discrete Dynamics in Nature and Society, Article ID 293930, 11 pages},
year = {2015},
author = {V. M. Ungureanu},
}
%
@article{UDM-2012,
title = {Global Solutions of a Class of Discrete-Time Backward Nonlinear Equations on Ordered Banach Spaces with Applications to {Riccati} Equations of Stochastic Control},
journal = {Optimal Control Applications and Methods},
volume = {34},
number = {2},
pages = {164 - 190},
year = {2013},
author = {V. M. Ungureanu and V. Dr\u{a}gan and T. Morozan},
}
%
%
@article{rantzel1996,
title = "On the {Kalman}$—${Yakubovich}$—${Popov} Lemma",
journal = "Systems \& Control Letters",
volume = "28",
number = "1",
pages = "7 - 10",
year = "1996",
issn = "0167-6911",
doi = "https://doi.org/10.1016/0167-6911(95)00063-1",
url = "http://www.sciencedirect.com/science/article/pii/0167691195000631",
author = "A. Rantzer",
abstract = "The purpose of this note is to present a new elementary proof for the multivariable K-Y-P lemma. A minimum of linear algebra and finite dimensional convexity theory is used."
}
%
%
@article{Varga,
author = {A. Varga},
title = {On Solving Periodic {Riccati} Equations},
journal = {Numerical Linear Algebra with Applications},
volume = {15},
number = {9},
pages = {809 - 835},
keywords = {periodic systems, periodic Riccati equation, periodic Lyapunov equation, periodic deadbeat control, numerical methods},
doi = {10.1002/nla.604},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.604},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nla.604},
abstract = {Abstract Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, the multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represent extensions of the periodic QZ algorithm to non-square periodic pairs, whereas the third method represents an extension of a quotient-product swapping and collapsing ‘fast’ algorithm. All proposed approaches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting matrices. Copyright © 2008 John Wiley \& Sons, Ltd.},
year = {2008}
}
%
@article{chinezii-SIAM,
title = {Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic {Riccati} Equation},
journal = {SIAM Journal on Control and Optimization},
volume = {56},
number = {1},
pages = {367 - 387},
year = {2018},
author = {J. Zheng and L. Qiu},
}
%
%
@conference{chinezii,
  author       = {J. Zheng and L. Qiu},
  title        = {On the Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic {Riccati} Equation},
  booktitle    = {Proceedings of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014},
   pages        = 6988 - 6993
}
%
%
@conference{mtns2014,
  author       = {J. Zheng and L. Qiu},
  title        = {On the Existence of a Mean-square Stabilizing Solution to a Continuous-time Modified Algebraic {Riccati} Equation},
  booktitle    = {21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands },
  year         = 2014,
   pages        = {694 - 700}
}
%
@article{wonham68,
title = {On a Matrix {Riccati} Equation of Stochastic Control},
journal = {SIAM Journal on Control},
volume = {6},
number = {4},
pages = {681 - 697},
year = {1968},
author = {W. M. Wonham},
}
%
@book{reid:72,
  title={{Riccati} Differential Equations},
  author={W. T. Reid},
  year={1972},
  series={Mathematics in science and engineering},
volume={86},
  publisher={Academic Press}
}
%
%
@book{coppel:71,
  title={Disconjugacy},
  author={W. A. Coppel},
  year={1971},
  series={Lecture Notes in Mathematics},
volume={220},
  publisher={Springer-Verlag Berlin Heidelberg}
}
%
%
@book{dale,
  title={Stability of Solutions of Differential Equations in Banach Space},
  author={Ju. L. {Dalecki\u\i} and M. G. {Kre\u{\i}n}},
  volume={43},
  year={1974},
  publisher={Translations of Mathematical Monographs, American Mathematical Society}
}
%
%
@book{halanay,
  title={Differential Equations: Stability, Oscillations, Time Lags},
  author={A. Halanay},
  volume={23},
  year={1966},
  publisher={Academic Press}
}
%
%
@book{ciucutudor,
  title={Teoria probabilitatilor si aplicatii},
  author={G. {Ciucu} and C. {Tudor}},
  volume={},
  year={1983},
  publisher={Editura Stiintifica si Enciclopedica}
}
%
%
@book{chung,
  title={Markov Chains with Stationary Transition Probabilities},
  author={K. L. Chung},
  volume={},
  year={1967},
  publisher={Springer, Berlin, Heidelberg}
}
%
%
@book{doob,
  title={Stochastic Processes},
  author={J. L. Doob},
  volume={},
  year={1967},
  publisher={New York, Wiley}
}
%
%
@INPROCEEDINGS{ma-hou-zhang,
  author={H. {Ma} and T. {Hou} and W. {Zhang}},
  booktitle={2016 American Control Conference (ACC)},
  title={Stability and structural properties of stochastic periodic systems: An operator-spectral approach},
  year={2016},
  volume={},
  number={},
  pages={3880 - 3885},}
%
%
@book{book2013,
  title={Mathematical Methods in Robust Control of Linear Stochastic Systems},
  author={V. Dr\u{a}gan and T. Morozan and A.-M. Stoica},
  volume={},
  year={2013},
  publisher={Springer-Verlag New York}
}
%
%
@book{delong,
  title={Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications},
  author={L. Delong},
  year={2013},
  publisher={Springer-Verlag London}
}
%
%
%
@INPROCEEDINGS{ma-hou-zhang,
  author={H. {Ma} and T. {Hou} and W. {Zhang}},
  booktitle={2016 American Control Conference (ACC)},
  title={Stability and structural properties of stochastic periodic systems: An operator-spectral approach},
  year={2016},
  volume={},
  number={},
  pages={3880 - 3885},}
%
%
@book{j.zhang,
  title={Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory},
  author={J. Zhang},
  year={2017},
  series={Probability Theory and Stochastic Modelling},
volume={86},
  publisher={Springer-Verlag New York}
}
%
@article{kalman:70,
title = {Contributions on the theory of optimal control},
journal = {Bult. Soc. Math. Mexicana Segunda Ser.},
volume = {5},
number = {1},
pages = {102 - 119},
year = {1960},
author = {R. Kalman},
}
%
%
@book{dr-hal1999,
  title={ Stabilization of Linear Systems},
  author={V. Dr\u{a}gan and A. {Halanay}},
  year={1999},
  publisher={Birkhä\"{a}user Basel}
  }
%
@article{qiu-1999,
title = "On the generalized eigenspace approach for solving {Riccati} equations",
journal = "IFAC Proceedings Volumes",
volume = "32",
number = "2",
pages = "2945 - 2950",
year = "1999",
note = "14th IFAC World Congress 1999, Beijing, China, 5-9 July",
issn = "1474-6670",
doi = "https://doi.org/10.1016/S1474-6670(17)56502-7",
url = "http://www.sciencedirect.com/science/article/pii/S1474667017565027",
author = "L. Qiu",
keywords = "Riccati equations, Numerical methods, Optimal control, Matrix equations",
abstract = "In this paper, we show that a general discrete time Riccati equation can be solved by finding a deflating subspace of a 2n × 2n matrix pencil. In many applications, the data in the Riccati equation is formed from the squares of the raw physical data. In this case, we show that continuous-time and discrete-time Riccati equations can be solved without squaring the raw data by finding deflating subspaces of larger matrix pencils. Examples show that the solutions without squaring the data have numerical advantages over the existing solutions of Riccati equations using the generalized eigenspace approach."
}
%
%
@book{lancaster,
  title={ Algebraic Riccati Equations},
  author={ P. {Lancaster} and L. {Rodman}},
  year={1995},
  series={},
volume={},
  publisher={Clarendon Press, Oxford}
}
%
@article{fragoso-98,
title = {A New Approach to Linearly Perturbed {Riccati} Equations Arising in Stochastic Control},
journal = {Applied Mathematics and Optimization},
volume = {37},
pages = {99 - 126},
year = {1998},
author = {M. D. {Fragoso} and O. L. V. {Costa} and C. E. {de Souza}},
}
%
%
@article{tang,
title = {General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic {Riccati} Equations},
journal = {SIAM Journal on Control and Optimization},
volume = {42},
number = {1},
pages = {53 - 75},
year = {2003},
author = {S. Tang},
}
%
%
@book{zhang-carte,
  title={Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory},
  author={J. Zhang},
  year={2017},
  series={Probability Theory and Stochastic Modelling},
volume={86},
  publisher={Springer-Verlag New York}
}
%
%
%
@article{yakubovich,
title = {A linear-quadratic optimization problem and the frequency theorem for nonperiodic systems. {I}},
journal = {Siberian Mathematical Journal},
volume = {27},
number = {},
pages = {614 - 630},
year = {1986},
author = {V. A. Yakubovich },
}
%
%
%
%
@article{pastor,
title = {Differential periodic {Riccati} equations: Existence and uniqueness of nonnegative definite solutions},
journal = {Mathematics of Control, Signals and Systems},
volume = {6},
number = {},
pages = {341 - 362},
year = {1993},
author = {A. Pastor and V. Hern\'{a}ndez},
}
%
%
@ARTICLE{kawano,
  author={Y. {Kawano} and T. {Ohtsuka}},
  journal={IEEE Transactions on Automatic Control},
  title={Nonlinear Eigenvalue Approach to Differential {Riccati} Equations for Contraction Analysis},
  year={2017},
  volume={62},
  number={12},
  pages={6497-6504},
  doi={10.1109/TAC.2017.2655443}}