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% !TeX encoding = UTF-8
% !Mode:: "BibTeX:UTF-8"
% Для правильной обработки URL в основном документе должны быть
% загружены пакеты ulr или hyperref:
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% или
%
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% \usepackage{breakurl}
%
% последний пакет нужен для переносов длинных гиперссылок
% и допустимости в них символов подчеркивания и т.п.
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% Этот пакет надо добавлять, если компиляция идет не в режиме
% PDFLaTeX, например, при обработке файла в ArXiv
%
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\providecommand{\bbldec}[0]{December}"}
@ARTICLE{Abder:05,
author = "Abderrahmane, Ould M.",
title = "On the {L}ojasiewicz exponent and {N}ewton polyhedron",
journal = "Kodai Math. J.",
fjournal = "Kodai Mathematical Journal",
year = "2005",
volume = "28",
number = "1",
pages = "106--110",
issn = "0386-5991",
mrclass = "32S05 (14B05)",
mrnumber = "2122194 (2006c:32031)",
mrreviewer = "Toshizumi Fukui",
zblnumber = "1082.14006",
doi = "10.2996/kmj/1111588040",
url = "https://dx.doi.org/10.2996/kmj/1111588040",
language = "english",
}
@INPROCEEDINGS{AhmJun:CDC13,
author = "Ahmadi, Amir Ali and Jungers, Rapha{\"e}l M.",
title = "Switched stability of nonlinear systems via {SOS}-convex
{L}yapunov functions and semidefinite programming",
booktitle = "Proceedings of the 52nd IEEE Annual Conference on Decision
and Control (CDC)",
year = "2013",
pages = "727--732",
doi = "10.1109/CDC.2013.6759968",
url = "https://ieeexplore.ieee.org/document/6759968",
language = "english",
annote = "We introduce the concept of sos-convex Lyapunov functions
for stability analysis of discrete time switched systems.
These are polynomial Lyapunov functions that have an
algebraic certificate of convexity, and can be efficiently
found by semidefinite programming. We show that convex
polynomial Lyapunov functions are universal (i.e.,
necessary and sufficient) for stability analysis of
switched linear systems. On the other hand, we show via an
explicit example that the minimum degree of an sos-convex
Lyapunov function can be arbitrarily higher than a
(non-convex) polynomial Lyapunov function. (The proof is
omitted.) In the second part, we show that if the switched
system is defined as the convex hull of a finite number of
nonlinear functions, then existence of a non-convex common
Lyapunov function is not a sufficient condition for
switched stability, but existence of a convex common
Lyapunov function is. This shows the usefulness of the
computational machinery of sos-convex Lyapunov functions
which can be applied either directly to the switched
nonlinear system, or to its linearization, to provide proof
of local switched stability for the nonlinear system. An
example is given where no polynomial of degree less than 14
can provide an estimate to the region of attraction under
arbitrary switching.",
}
@INPROCEEDINGS{AhmPar:CDC12,
author = "Ahmadi, Amir Ali and Parrilo, Pablo A.",
title = "Joint Spectral Radius of Rank One Matrices and the Maximum
Cycle Mean Problem",
booktitle = "Proceedings of the 51st Annual Conference on Decision and
Control (CDC), 10-13 Dec.",
organization = "IEEE",
year = "2012",
pages = "731--733",
doi = "10.1109/CDC.2012.6425992",
url = "https://ieeexplore.ieee.org/document/6425992",
language = "english",
annote = "We prove several exact results on approximability of joint
spectral radius by matrix norms induced by Euclidean norms.
We point out, perhaps for the first time in this context, a
difference between complex and real cases. New connections
of joint spectral radius to convex geometry and
combinatorics are established. Several open problems are
posed.",
}
@BOOK{AizGant:e,
author = "Aizerman, M. A. and Gantmacher, F. R.",
title = "Absolute stability of regulator systems",
publisher = "Holden-Day Inc.",
address = "San Francisco, Calif.",
year = "1964",
pages = "viii+172",
mrclass = "93.20",
mrnumber = "0183556 (32 \#1036)",
language = "english",
note = "Translated by E. Polak",
}
@BOOK{AKPRS:e,
author = "Akhmerov, R. R. and Kamenski{\u\i}, M. I. and Potapov, A. S.
and Rodkina, A. E. and Sadovski{\u\i}, B. N.",
title = "Measures of noncompactness and condensing operators",
series = "Operator Theory: Advances and Applications",
publisher = "Birkh{\"a}user Verlag",
address = "Basel",
year = "1992",
volume = "55",
pages = "viii+249",
isbn = "3-7643-2716-2",
mrclass = "47H09 (34G20 34K05 47H15 58C30)",
mrnumber = "1153247 (92k:47104)",
zblnumber = "0748.47045",
language = "english",
note = "Translated from the 1986 Russian original by A. Iacob",
}
@INPROCEEDINGS{AGGG:LIPIcs17,
author = "Akian, Marianne and Gaubert, St{\'e}phane and
Grand-Cl{\'e}ment, Julien and Guillaud, J{\'e}r{\'e}mie",
editor = "Vollmer, Heribert and Vall{\'e}e, Brigitte",
title = "The Operator Approach to Entropy Games",
booktitle = "34th Symposium on Theoretical Aspects of Computer Science
(STACS 2017)",
series = "Leibniz International Proceedings in Informatics (LIPIcs)",
publisher = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
address = "Dagstuhl, Germany",
year = "2017",
volume = "66",
pages = "6:1--6:14",
keywords = "stochastic games; shapley operators; policy iteration;
{P}erron eigenvalues; risk sensitive control",
isbn = "978-3-95977-028-6",
issn = "1868-8969",
mrnumber = "3655333",
zblnumber = "1402.91017",
doi = "10.4230/LIPIcs.STACS.2017.6",
url = "https://drops.dagstuhl.de/opus/volltexte/2017/7026",
language = "english",
annote = "Entropy games and matrix multiplication games have been
recently introduced by Asarin et al. They model the
situation in which one player (Despot) wishes to minimize
the growth rate of a matrix product, whereas the other
player (Tribune) wishes to maximize it. We develop an
operator approach to entropy games. This allows us to show
that entropy games can be cast as stochastic mean payoff
games in which some action spaces are simplices and
payments are given by a relative entropy (Kullback-Leibler
divergence). In this way, we show that entropy games with a
fixed number of states belonging to Despot can be solved in
polynomial time. This approach also allows us to solve
these games by a policy iteration algorithm, which we
compare with the spectral simplex algorithm developed by
Protasov.",
}
@MISC{AGN11,
author = "Akian, Marianne and Gaubert, Stephane and Nussbaum, Roger",
title = "A {C}ollatz-{W}ielandt characterization of the spectral
radius of order-preserving homogeneous maps on cones",
howpublished = "ArXiv.org e-Print archive",
year = "2011",
month = dec,
eprinttype = "arXiv",
eprint = "1112.5968",
doi = "10.48550/arXiv.1112.5968",
url = "https://arxiv.org/abs/1112.5968",
language = "english",
annote = "Several notions of spectral radius arise in the study of
nonlinear order-preserving positively homogeneous self-maps
of cones in Banach spaces. We give conditions that
guarantee that all these notions lead to the same value. In
particular, we give a Collatz-Wielandt type formula, which
characterizes the growth rate of the orbits in terms of
eigenvectors in the closed cone or super-eigenvectors in
the interior of the cone. This characterization holds when
the cone is normal and when a quasi-compactness condition,
involving an essential spectral radius defined in terms of
$k$-set-contractions, is satisfied. Some fixed point
theorems for non-linear maps on cones are derived as
intermediate results. We finally apply these results to
show that non-linear spectral radii commute with respect to
suprema and infima of families of order preserving maps
satisfying selection properties.",
}
@ARTICLE{ARS08,
author = "Akram, Q. Farooq and Rime, Dagfinn and Sarno, Lucio",
title = "Arbitrage in the Foreign Exchange Market: Turning on the
Microscope",
journal = "J. Int. Econ.",
fjournal = "Journal of International Economics",
year = "2008",
volume = "76",
number = "2",
pages = "237--253",
month = dec,
doi = "10.1016/j.jinteco.2008.07.004",
url = "https://www.sciencedirect.com/science/article/pii/S0022199608000706",
language = "english",
}
@ARTICLE{NM:JAMSA06,
author = "Al-Badarneh, Anwar A. and Maaitah, Rabaa K.",
title = "Robustness of solutions of linear differential equations
with infinite delay",
journal = "J. Appl. Math. Stoch. Anal.",
fjournal = "Journal of Applied Mathematics and Stochastic Analysis.
JAMSA",
year = "2006",
pages = "Art. ID 38769, 13",
issn = "1048-9533",
mrclass = "34K06 (34E10)",
mrnumber = "2220987 (2007k:34207)",
zblnumber = "1115.93024",
doi = "10.1155/JAMSA/2006/38769",
url = "https://www.hindawi.com/journals/ijsa/2006/038769/",
language = "english",
}
@ARTICLE{NKP:JMAA97,
author = "Al-Nayef, A. A. and Kloeden, P. E. and Pokrovskii, A. V.",
title = "Expansivity of nonsmooth functional-differential equations",
journal = "J. Math. Anal. Appl.",
fjournal = "Journal of Mathematical Analysis and Applications",
year = "1997",
volume = "208",
number = "2",
pages = "453--461",
coden = "JMANAK",
issn = "0022-247X",
mrclass = "34K30 (34K15 47H99 47N20)",
mrnumber = "1441447 (97m:34156)",
mrreviewer = "Bernhard Lani-Wayda",
zblnumber = "0899.58050",
doi = "10.1006/jmaa.1997.5337",
url = "https://www.sciencedirect.com/science/article/pii/S0022247X97953373",
language = "english",
}
@ARTICLE{ABT:PIMA56,
author = "Aljan{\v{c}}i{\'c}, S. and Bojani{\'c}, R. and Tomi{\'c},
M.",
title = "Sur le comportement asymtotique au voisinage de z{\'e}ro des
s{\'e}ries trigonom{\'e}triques de sinus {\`a} coefficients
monotones",
journal = "Acad. Serbe Sci., Publ. Inst. Math.",
fjournal = "Acad{\'e}mie Serbe des Sciences, Publications de l'Institut
Math{\'e}matique",
year = "1956",
volume = "10",
pages = "101--120",
mrnumber = "0082579",
zblnumber = "0071.06003",
language = "english",
}
@BOOK{AllenL:10,
author = "Allen, Linda J. S.",
title = "An Introduction to Stochastic Processes with Applications to
Biology",
publisher = "CRC Press",
address = "Boca Raton, FL",
year = "2011",
pages = "xxiv+466",
edition = "Second",
isbn = "978-1-4398-1882-4",
mrclass = "60-01 (60H30 60J10 60J20 60J28 92-01 92B05)",
mrnumber = "2560499",
zblnumber = "1263.92001",
language = "english",
}
@ARTICLE{AM:AMUC00,
author = "Alsed{\`a}, L. and Moreno, J. M.",
title = "On the rotation sets for non-continuous circle maps",
journal = "Acta Math. Univ. Comenian. (N.S.)",
fjournal = "Acta Mathematica Universitatis Comenianae. New Series",
year = "2000",
volume = "69",
number = "1",
pages = "115--125",
issn = "0862-9544",
mrclass = "37E45 (37E10 54H20)",
mrnumber = "1796792",
mrreviewer = "Peter Raith",
zblnumber = "0967.37025",
zblreviewer = "J.Sm{\'\i}tal (Opava)",
language = "english",
annote = "We give some examples of non-continuous circle maps whose
rotation sets lack the good properties they have in the
case of continuous maps. It is well-known that the rotation
sets of continuous maps of the circle have certain nice
properties. E.g., let $f$ be such a map, and let
$F:\mathbb{R}\to\mathbb{R}$ be a degree one lifting of $f$.
Then the rotation set of $F$ is connected and depends
continuously on $F$. Moreover, if $p$ and $q$ are
relatively prime positive integers and $p/q$ is an interior
point of the rotation set then $F$ has a lifted cycle of
period $q$ and rotation number $p/q$. The authors show that
in the class of piecewise continuous maps of the circle
these and other properties are not preserved.",
}
@ARTICLE{AM:AMUC96,
author = "Alsed{\`a}, Ll. and Ma{\~n}osas, F.",
title = "Kneading theory for a family of circle maps with one
discontinuity",
journal = "Acta Math. Univ. Comenian. (N.S.)",
fjournal = "Acta Mathematica Universitatis Comenianae. New Series",
year = "1996",
volume = "65",
number = "1",
pages = "11--22",
issn = "0862-9544",
mrclass = "58F03 (34C35 54C70 54H20)",
mrnumber = "1422291",
mrreviewer = "Jos{\'e} Miguel Moreno",
zblnumber = "0863.34046",
zblreviewer = "K.H.Kim (Montgomery)",
language = "english",
annote = "The authors extend kneading theory to maps of the circle
having a single discontinuity, bounded and non-decreasing
(in terms of the reals) away from it.",
}
@ARTICLE{ALM:N90,
author = "Alsed{\`a}, Llu{\'\i}s and Ma{\~n}osas, Francesc",
title = "Kneading theory and rotation intervals for a class of circle
maps of degree one",
journal = "Nonlinearity",
fjournal = "Nonlinearity",
year = "1990",
volume = "3",
number = "2",
pages = "413--452",
issn = "0951-7715",
mrclass = "58F11 (58F20)",
mrnumber = "1054582",
mrreviewer = "L{\'u}cia Helena Vilas B{\^o}as Mendes",
zblnumber = "0735.54026",
zblreviewer = "Andrzej Pelczar (Krak{\'o}w)",
url = "https://iopscience.iop.org/article/10.1088/0951-7715/3/2/008",
language = "english",
annote = "The authors give a kneading theory for the class of
continuous maps of the circle of degree with a single
maximum and a single minimum. For a map of this class they
characterise the set of itineraries depending on the
rotation interval. From this result they obtain lower and
upper bounds of the topological entropy and of the number
of periodic orbits of each period. These lower bounds
appear to be valid for a general continuous map of the
circle of degree one.",
}
@ARTICLE{AndoShih:SIAM:98,
author = "Ando, Tsuyoshi and Shih, Mau-Hsiang",
title = "Simultaneous contractibility",
journal = "SIAM J. Matrix Anal. Appl.",
fjournal = "SIAM Journal on Matrix Analysis and Applications",
year = "1998",
volume = "19",
number = "2",
pages = "487--498 (electronic)",
issn = "0895-4798",
mrclass = "15A60 (47A30)",
mrnumber = "1614074 (99c:15046)",
mrreviewer = "Sivaram K. Narayan",
zblnumber = "0912.15033",
zblreviewer = "Alexey Alimov (Moskva)",
doi = "10.1137/S0895479897318812",
url = "https://epubs.siam.org/doi/10.1137/S0895479897318812",
language = "english",
annote = "Let $C$ be a set of $n\times n$ complex matrices. For $m =
1,2,\ldots$, $C^{m}$ is the set of all products of matrices
in $C$ of length $m$. Denote by $\hat{r}(\mathcal{C})$ the
joint spectral radius of $C$, that is,
\[\hat{r}(\mathcal{C})\stackrel{\rm
def}{=}\limsup_{m\to\infty}[\sup_{A\in \mathcal{C}^m}\| A\|
]^{\frac{1}{m}}.\] We call $C$ \emph{simultaneously
contractible} if there is an invertible matrix $S$ such
that \[\sup\{\| S^{-1}AS\|;\ A\in \mathcal{C}\}<1,\] where
$\| \cdot\|$ is the spectral norm. This paper is primarily
devoted to determining the optimal joint spectral radius
range for simultaneous contractibility of bounded sets of
$n\times n$ complex matrices, that is, the maximum subset
$J$ of $[0,1)$ such that if $C$ is a bounded set of
$n\times n$ complex matrices and $\hat{r}(\mathcal{C})\in
J$, then $C$ is simultaneously contractible. The central
result proved in this paper is that this maximum subset is
$[0,\frac{1}{\sqrt{n}})$. Our method of proof is based on a
matrix-theoretic version of complex John's ellipsoid
theorem and the generalized Gelfand spectral radius
formula.",
}
@BOOK{AndrVitt:e,
author = "Andronov, A. A. and Vitt, A. A. and Kha{\u\i}kin, S.
{\`E}.",
title = "Theory of oscillators",
publisher = "Dover Publications Inc.",
address = "New York",
year = "1987",
pages = "xxxiv+815",
isbn = "0-486-65508-3",
mrclass = "34C15 (34Cxx 70-01)",
mrnumber = "925417 (88j:34069)",
language = "english",
note = "Translated from the Russian by F. Immirzi, Reprint of the
1966 translation",
}
@ARTICLE{Anosov:PSIM67:e,
author = "Anosov, D. V.",
title = "Geodesic flows on closed {R}iemann manifolds with negative
curvature",
journal = "Proc. Steklov Inst. Math.",
fjournal = "Proceedings of the Steklov Institute of Mathematics",
year = "1967",
volume = "90",
pages = "235",
mrclass = "57.36 (28.00)",
mrnumber = "0224110 (36 \#7157)",
mrreviewer = "M. Eisenberg",
zblnumber = "0176.19101",
language = "english",
}
@BOOK{ArnVGZ:e,
author = "Arnol'd, V. I. and Guse{\u\i}n-Zade, S. M. and Varchenko, A.
N.",
title = "Singularities of differentiable maps. {V}ol. {I}",
series = "Monographs in Mathematics",
publisher = "Birkh{\"a}user Boston Inc.",
address = "Boston, MA",
year = "1985",
volume = "82",
pages = "xi+382",
isbn = "0-8176-3187-9",
mrclass = "58C27",
mrnumber = "777682 (86f:58018)",
language = "english",
note = "The classification of critical points, caustics and wave
fronts, translated from the Russian by Ian Porteous and
Mark Reynolds",
}
@BOOK{ArnoldL:98,
author = "Arnold, Ludwig",
title = "Random dynamical systems",
series = "Springer Monographs in Mathematics",
publisher = "Springer-Verlag",
address = "Berlin",
year = "1998",
pages = "xvi+586",
isbn = "3-540-63758-3",
mrclass = "37Hxx (37-02 60H10)",
mrnumber = "1723992 (2000m:37087)",
mrreviewer = "Yuri Kifer",
zblnumber = "0906.34001",
language = "english",
}
@MISC{AFHS:ArXiv21,
author = "Artigiani, Mauro and Fougeron, Charles and Hubert, Pascal
and Skripchenko, Alexandra",
title = "A note on double rotations of infinite type",
howpublished = "ArXiv.org e-Print archive",
year = "2021",
month = feb,
eprinttype = "arXiv",
eprint = "2102.11803",
doi = "10.48550/arXiv.2102.11803",
url = "https://arxiv.org/abs/2102.11803",
language = "english",
annote = "We introduce a new renormalization procedure on double
rotations, which is reminiscent of the classical Rauzy
induction. Using this renormalization we prove that the set
of parameters which induce infinite type double rotations
has Hausdorff dimension strictly smaller than $3$.
Moreover, we construct a natural invariant measure
supported on these parameters and show that, with respect
to this measure, almost all double rotations are uniquely
ergodic.",
}
@ARTICLE{ADFP:93,
author = "Asarin, E. and Diamond, Phil and Fomenko, I. and Pokrovskii,
A.",
title = "Chaotic phenomena in desynchronized systems and stability
analysis",
journal = "Comput. Math. Appl.",
fjournal = "Computers {\&} Mathematics with Applications. An
International Journal",
year = "1993",
volume = "25",
number = "1",
pages = "81--87",
coden = "CMAPDK",
issn = "0898-1221",
mrclass = "58F13 (58F03 58F10 93D05)",
mrnumber = "1192675 (93k:58144)",
mrreviewer = "Peter E. Kloeden",
zblnumber = "0774.93057",
zblreviewer = "T. Duncan (Lawrence)",
doi = "10.1016/0898-1221(93)90214-G",
url = "https://www.sciencedirect.com/science/article/pii/089812219390214G",
language = "english",
annote = "Complex systems tend to be desynchronized, as part and
parcel of their internal organisation and internal
connections. One way that this may arise is from quite
small mismatching of operating times of system components.
On the other hand, lack of synchronization can be built
into the system. For example, ``chaotic'' iterations and
asynchronous algorithms are exploited in parallel
processing. Too, a system may be susceptible of change by
numerous factors at different, asynchronous times. In all
cases, it is important to understand the effect that
desynchronization can have on the stability of the system
and the convergence properties of processes.\par We
consider the symbolic dynamics of desynchronized switching
times and extract a numerical quantity whose values
determine the stability characteristics of the system. An
important role in the proof is played by Hilbert's
projective metric.",
}
@BOOK{AuFrank:09,
author = "Aubin, Jean-Pierre and Frankowska, H{\'e}l{\`e}ne",
title = "Set-valued analysis",
series = "Modern Birkh{\"a}user Classics",
publisher = "Birkh{\"a}user Boston Inc.",
address = "Boston, MA",
year = "2009",
pages = "xx+461",
isbn = "978-0-8176-4847-3",
mrclass = "49-02 (28-02 46-02 47H04 47J05 49J52 49J53 58C06)",
mrnumber = "2458436",
language = "english",
note = "Reprint of the 1990 edition [MR1048347]",
}
@ARTICLE{AulSieg:01,
author = "Aulbach, Bernd and Siegmund, Stefan",
title = "The dichotomy spectrum for noninvertible systems of linear
difference equations",
journal = "J. Differ. Equations Appl.",
fjournal = "Journal of Difference Equations and Applications",
year = "2001",
volume = "7",
number = "6",
pages = "895--913",
coden = "JDEAEA",
issn = "1023-6198",
mrclass = "39A05 (37B55)",
mrnumber = "1870729 (2003f:39001)",
zblnumber = "1001.39003",
doi = "10.1080/10236190108808310",
url = "https://www.tandfonline.com/doi/abs/10.1080/10236190108808310",
language = "english",
note = "On the occasion of the 60th birthday of Calvin Ahlbrandt",
}
@INPROCEEDINGS{AulSieg:02,
author = "Aulbach, Bernd and Siegmund, Stefan",
title = "A spectral theory for nonautonomous difference equations",
booktitle = "New trends in difference equations ({T}emuco, 2000)",
publisher = "Taylor {\&} Francis",
address = "London",
year = "2002",
pages = "45--55",
mrclass = "39A05",
mrnumber = "2016054 (2005e:39001)",
zblnumber = "1062.39014",
language = "english",
}
@ARTICLE{BGM:SPMJ12,
author = "Badea, C. and Grivaux, S. and M{\"u}ller, V.",
title = "The rate of convergence in the method of alternating
projections",
journal = "St. Petersbg. Math. J.",
fjournal = "St. Petersburg Mathematical Journal",
year = "2012",
volume = "23",
number = "3",
pages = "413--434",
eprinttype = "arXiv",
eprint = "1006.2047",
issn = "1061-0022",
mrclass = "46C05 (41A65)",
mrnumber = "2896163 (2012j:46032)",
zblnumber = "1294.47026",
doi = "10.1090/S1061-0022-2012-01202-1",
url = "https://www.ams.org/journals/spmj/2012-23-03/S1061-0022-2012-01202-1/home.html",
language = "english",
annote = "The cosine of the Friedrichs angle between two subspaces is
generalized to a parameter associated with several closed
subspaces of a Hilbert space. This parameter is employed to
analyze the rate of convergence in the von Neumann-Halperin
method of cyclic alternating projections. General dichotomy
theorems are proved, in the Hilbert or Banach space
situation, providing conditions under which the alternative
QUC/ASC (quick uniform convergence versus arbitrarily slow
convergence) holds. Several meanings for ASC are
proposed.",
}
@ARTICLE{BXMS:IEEETSP13,
author = "Bajovic, Dragana and Xavier, Joao and Moura, Jose M. F. and
Sinopoli, Bruno",
title = "Consensus and Products of Random Stochastic Matrices:
{E}xact Rate for Convergence in Probability",
journal = "IEEE Trans. Signal Process.",
fjournal = "IEEE Transactions on Signal Processing",
year = "2013",
volume = "61",
number = "10",
pages = "2557--2571",
eprinttype = "arXiv",
eprint = "1202.6389",
mrnumber = "3053826",
zblnumber = "1393.90025",
doi = "10.1109/TSP.2013.2248003",
url = "https://ieeexplore.ieee.org/document/6466430",
language = "english",
annote = "Distributed consensus and other linear systems with system
stochastic matrices $W_k$ emerge in various settings, like
opinion formation in social networks, rendezvous of robots,
and distributed inference in sensor networks. The matrices
$W_k$ are often random, due to, e.g., random packet
dropouts in wireless sensor networks. Key in analyzing the
performance of such systems is studying convergence of
matrix products $W_kW_{k-1} \cdots W_1$. In this paper, we
find the exact exponential rate $I$ for the convergence in
probability of the product of such matrices when time $k$
grows large, under the assumption that the $W_k$'s are
symmetric and independent identically distributed in time.
Further, for commonly used random models like with gossip
and link failure, we show that the rate $I$ is found by
solving a min-cut problem and, hence, easily computable.
Finally, we apply our results to optimally allocate the
sensors' transmission power in consensus+innovations
distributed detection.\par Our analysis reveals that the
exponential rate of convergence in probability depends only
on the statistics of the support graphs of the random
matrices. Further, we show how to compute this rate for
commonly used random models: gossip and link failure. With
these models, the rate is found by solving a min-cut
problem, and hence it is easily computable. Finally, as an
illustration, we apply our results to solving power
allocation among networked sensors in a
consensus+innovations distributed detection problem.",
}
@ARTICLE{BJ:SIAMJCO11,
author = "Balde, Moussa and Jouan, Philippe",
title = "Geometry of the limit sets of linear switched systems",
journal = "SIAM J. Control Optim.",
fjournal = "SIAM Journal on Control and Optimization",
year = "2011",
volume = "49",
number = "3",
pages = "1048--1063",
eprinttype = "arXiv",
eprint = "1004.5302",
coden = "SJCODC",
issn = "0363-0129",
mrclass = "93D20 (37N35 93C30)",
mrnumber = "2806574 (2012e:93136)",
mrreviewer = "Peng Nian Chen",
zblnumber = "1228.93104",
doi = "10.1137/100793153",
url = "https://epubs.siam.org/doi/10.1137/100793153",
language = "english",
annote = "This paper is concerned with asymptotic stability properties
of linear switched systems. Under the hypothesis that all
the subsystems share a nonstrict quadratic Lyapunov
function, we provide a large class of switching signals for
which a large class of switched systems is asymptotically
stable. For this purpose we define what we call nonchaotic
inputs, which generalize the different notions of inputs
with dwell time. Next, we turn our attention to the
behavior for possibly chaotic inputs. Finally, we give a
sufficient condition for a system composed of a pair of
Hurwitz matrices to be asymptotically stable for all
inputs.",
}
@INCOLLECTION{Ball:97,
author = "Ball, Keith",
title = "An elementary introduction to modern convex geometry",
booktitle = "Flavors of geometry",
series = "Math. Sci. Res. Inst. Publ.",
publisher = "Cambridge Univ. Press",
address = "Cambridge",
year = "1997",
volume = "31",
pages = "1--58",
mrclass = "52-02 (46B07 52A21 52A40)",
mrnumber = "1491097 (99f:52002)",
mrreviewer = "Peter M. Gruber",
zblnumber = "0901.52002",
language = "english",
}
@ARTICLE{Bar:AIT88-2:e,
author = "Barabanov, N. E.",
title = "{L}yapunov indicator of discrete inclusions. {I}",
journal = "Autom. Remote Control",
fjournal = "Automation and Remote Control",
year = "1988",
volume = "49",
number = "2",
pages = "152--157",
issn = "0005-1179",
mrclass = "34A60 (34D05 39A12)",
mrnumber = "940263 (89e:34025)",
mrreviewer = "J. W. Macki",
zblnumber = "0665.93043",
zblreviewer = "O. Pastravanu",
language = "english",
annote = "An equivalence is proved between the negativity of the,
Lyapunov indicator corresponding to a discrete inclusion
defined by $F(y)=\{Ay: A\in \mathcal{U}\}$, a polyhedral
set: \[ \mathcal{U}=\{A+\sum\sp{m}\sb{i=1}b\sb i\nu\sb
ic\sp T\sb i,\quad 0\le \nu\sb i\le \mu\sb i,\quad b\sb
i,c\sb i\in {\mathbb{R}}\sp n\},\] and the absolute
stability of the discrete control system: \[
x\sb{k+1}=Ax\sb k+\sum\sp{m}\sb{i=1}b\sb i\xi\sp i\sb
k,\quad \sigma\sp i\sb k=c\sp T\sb ix\sb k,\quad \xi\sp
i\sb k=\phi\sb i(\sigma\sp i\sb k,k) \] in a class of
nonlinearities that satisfy the conditions $0\le \phi\sb
i(\sigma,k)\sigma \le \mu\sb i\sigma\sp 2$ for any
$\sigma\in R$, $i=1,\ldots,m$, $k=0,1,2,\ldots$. It is also
shown that the existence of an unbounded solution of a
nonsingular inclusion is equivalent to a positive Lyapunov
indicator.",
}
@ARTICLE{Bar:ARC88,
author = "Barabanov, N. E.",
title = "The {L}yapunov exponent of discrete inclusions. {I-III}",
journal = "Autom. Remote Control",
fjournal = "Automation and Remote Control",
year = "1988",
volume = "49",
pages = "152--157, 283--287, 558--565",
issn = "0005-1179",
language = "english",
}
@ARTICLE{Bar:AIT88-3:e,
author = "Barabanov, N. E.",
title = "The {L}yapunov indicator of discrete inclusions. {II}",
journal = "Autom. Remote Control",
fjournal = "Automation and Remote Control",
year = "1988",
volume = "49",
number = "3",
pages = "283--287",
issn = "0005-1179",
mrclass = "34A60 (34D05 39A12)",
mrnumber = "943889 (89e:34026)",
mrreviewer = "J. W. Macki",
zblnumber = "0665.93044",
zblreviewer = "O. Pastravanu",
language = "english",
annote = "The concept of conjugate inclusion is formulated and its
properties are studied. The problem of determination of the
Lyapunov indicator for an original inclusion is reduced to
the analysis of a conjugate autonomous system of discrete
equations.",
}
@ARTICLE{Bar:AIT88-5:e,
author = "Barabanov, N. E.",
title = "The {L}yapunov indicator of discrete inclusions. {III}",
journal = "Autom. Remote Control",
fjournal = "Automation and Remote Control",
year = "1988",
volume = "49",
number = "5",
pages = "558--565",
issn = "0005-1179",
mrclass = "34D05 (34A60 39A12 93C55)",
mrnumber = "952665 (89m:34060)",
mrreviewer = "J. W. Macki",
zblnumber = "0665.93045",
zblreviewer = "O. Pastravanu",
language = "english",
annote = "Certain algebraic formulas characterizing the Lyapunov
indicator $\rho$ of a discrete inclusion are presented. For
$\rho <0$ it is shown that the inclusion state space can be
embedded in a space of larger dimension in which the cubic
norm decreases along any solution of the inclusion that
corresponds to the original one. A similar necessary and
sufficient condition is derived for the case $\rho\ge 0$
too. An algorithm which allows to determine the sign of
$\rho$ in a finite number of steps is also formulated.",
}
@ARTICLE{Bar:SMZh88:e,
author = "Barabanov, N. E.",
title = "Absolute characteristic exponent of a class of linear
nonstatinoary systems of differential equations",
journal = "Sib. Math. J.",
publisher = "Springer Science and Business Media LLC",
year = "1989",
volume = "29",
number = "4",
pages = "521--530",
zblnumber = "0688.93051",
doi = "10.1007/BF00969859",
url = "https://link.springer.com/article/10.1007/BF00969859",
language = "english",
annote = "For a nondegenerate class of systems of differential
equations, the existence of an extreme norm is proved. The
problem of absolute stability of the original class of
nonstationary systems of differential equations is reduced
to the analysis of an autonomous Hamiltonian system. All
theorems are formulated using the absolute characteristic
exponent (Lyapunov exponent) of the class of original
systems. Some algebraic properties of the Lyapunov exponent
are established. Some results known in the theory of
automatic control are extended to broader classes of
systems of differential equations.",
}
@INPROCEEDINGS{Bar:ACC95,
author = "Barabanov, N. E.",
authauthor = "Barabanov, N.",
title = "Stability of inclusions of linear type",
booktitle = "American Control Conference, Proceedings of the 1995",
year = "1995",
volume = "5",
pages = "3366--3370",
month = jun,
doi = "10.1109/ACC.1995.532231",
url = "https://ieeexplore.ieee.org/document/532231/",
language = "english",
annote = "Introduced several new concepts for the differential
inclusions of linear type. The main result concerns the
existence of an extremal norm for nonsingular inclusions.
It allows the author to present the numerical algorithm for
calculating the Lyapunov index up to arbitrary accuracy.
For 3-ordered automatic control systems the author
describes all extremal solutions and in such a way the
author solves the problem of absolute stability in terms of
coefficients of transfer functions",
}
@ARTICLE{Bar:LAA08,
author = "Barabanov, N. E.",
title = "Asymptotic behavior of extremal solutions and structure of
extremal norms of linear differential inclusions of order
three",
journal = "Linear Algebra Appl.",
fjournal = "Linear Algebra and its Applications",
year = "2008",
volume = "428",
number = "10",
pages = "2357--2367",
coden = "LAAPAW",
issn = "0024-3795",
mrclass = "34D08 (15A18 93D09)",
mrnumber = "2408032 (2009a:34089)",
mrreviewer = "Jialin Hong",
zblnumber = "1142.34009",
zblreviewer = "Ludv{\'\i}k Jano{\v{s}} (Claremont)",
doi = "10.1016/j.laa.2007.10.039",
url = "https://www.sciencedirect.com/science/article/pii/S0024379507005010",
language = "english",
annote = "Asymptotic properties of extremal solutions of linear
inclusions of order three with zero Lyapunov exponent are
investigated. Under certain conditions it is shown that all
extremal solutions of such inclusions tend to the same (up
to a multiplicative factor) solution, which is central
symmetric. The structure of the convex set of extremal norm
is studied. A number of extremal points of this set are
described.\par The author considers a linear differential
inclusion \[\frac{dx}{dt}\in \{Ax: A\in M\},\] where $M$ is
a bounded closed set of $3\times 3$-matrices. It is assumed
that the Lyapunov exponent of this inclusion is zero and
that no proper subspace in $\mathbb{R}^3$ is invariant with
respect to all matrices $A\in M$. He proves that if all
matrices $A\in\text{conv}(M)$ are nonsingular, then there
exists a single periodic solution $x(t)$ on the surface $S=
\{x: v(x)= 1\}$ and all the extremal solutions of this
inclusion with initial data on $S$ tend to the closed curve
$\{x(t): t\ge 0\}$. Here $v$ is a certain norm on
$\mathbb{R}^3$ which the author calls extremal.",
}
@INPROCEEDINGS{Bar:CDC05,
author = "Barabanov, Nikita",
title = "Lyapunov Exponent and Joint Spectral Radius: Some Known and
New Results",
booktitle = "Proceedings of the 44th {IEEE} Conference on Decision and
Control and European Control Conference 2005, Seville,
Spain, December 12--15",
year = "2005",
pages = "2332--2337",
doi = "10.1109/CDC.2005.1582510",
language = "english",
annote = "The logarithm of joint spectral radius of a set of matrices
coincides with Lyapunov exponent of corresponding linear
inclusions. Main results about Lyapunov exponents of
discrete time and continuous time linear inclusions are
presented. They include the existence of extremal norm;
relations between Lyapunov indices of dual inclusions;
maximum principle for linear inclusions; algebraic criteria
for stability of linear inclusions; algorithm to find out
the sign of Lyapunov exponents. The main result is extended
to linear inclusions with delays. The Aizerman problem for
three-ordered timevarying continuous time systems with one
nonlinearity is solved. The Perron-Frobenius theorem is
extended for three-ordered continuous time linear
inclusions.",
}
@ARTICLE{BV:JGA12,
author = "Barnsley, Michael F. and Vince, Andrew",
title = "Real projective iterated function systems",
journal = "J. Geom. Anal.",
fjournal = "Journal of Geometric Analysis",
year = "2012",
volume = "22",
number = "4",
pages = "1137--1172",
eprinttype = "arXiv",
eprint = "1003.3473",
issn = "1050-6926",
mrclass = "37B25",
mrnumber = "2965365",
mrreviewer = "Jacek R. Jachymski",
zblnumber = "1256.28002",
doi = "10.1007/s12220-011-9232-x",
url = "https://link.springer.com/article/10.1007/s12220-011-9232-x",
language = "english",
annote = "This paper contains four main results associated with an
attractor of a projective iterated function system (IFS).
The first theorem characterizes when a projective IFS has
an attractor which avoids a hyperplane. The second theorem
establishes that a projective IFS has at most one
attractor. In the third theorem the classical duality
between points and hyperplanes in projective space leads to
connections between attractors that avoid hyperplanes and
repellers that avoid points, as well as hyperplane
attractors that avoid points and repellers that avoid
hyperplanes. Finally, an index is defined for attractors
which avoid a hyperplane. This index is shown to be a
nontrivial projective invariant.",
}
@BOOK{Bary:e,
author = "Bary, N. K.",
title = "A treatise on trigonometric series. {V}ols. {I}, {II}",
series = "Authorized translation by Margaret F. Mullins. A Pergamon
Press Book",
publisher = "The Macmillan Co.",
address = "New York",
year = "1964",
pages = "Vol. I: xxiii+553, Vol. II: xix+508",
mrclass = "42.00",
mrnumber = "0171116 (30 \#1347)",
mrreviewer = "R. P. Boas, Jr.",
language = "english",
}
@ARTICLE{Baudet:JASSCM78,
author = "Baudet, G{\'e}rard M.",
title = "Asynchronous iterative methods for multiprocessors",
journal = "J. Assoc. Comput. Mach.",
fjournal = "Journal of the Association for Computing Machinery",
year = "1978",