Victor S Kozyakin | Moscow Institute of Physics and Technology (original) (raw)

Papers by Victor S Kozyakin

arXiv (Cornell University), Oct 8, 2020

We consider the question of boundedness of matrix products AnBn • • • A1B1 with factors from two ... more We consider the question of boundedness of matrix products AnBn • • • A1B1 with factors from two sets of matrices, Ai ∈ A and Bi ∈ B, due to an appropriate choice of matrices {Bi}. It is assumed that for every sequence of matrices {Ai} there is a sequence of matrices {Bi} for which the sequence of matrix products {AnBn • • • A1B1} ∞ n=1 is norm bounded. Some situations are described where in this case the norms of the matrix products AnBn • • • A1B1 are uniformly bounded, that is, AnBn • • • A1B1 ≤ C for all natural numbers n, where C > 0 is a constant independent of the sequence {Ai} and the corresponding sequence {Bi}. For the general case, the question of the validity of the corresponding statement remains open.

Automatica, Nov 1, 2022

In the theory of linear switching systems with discrete time, as in other areas of mathematics, t... more In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products Aσ n • • • Aσ 0 with factors from a set of matrices A arises. So far, only for a relatively small number of classes of matrices A has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences {σn} of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of "good" properties of the sequences {Aσ n }, in particular the existence of a limiting frequency of occurrence of each matrix factor A i ∈ A in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two 2 × 2 matrices, similar to rotations in the plane, in which the sequence {Aσ n } maximizing the growth rate of the norms Aσ n • • • Aσ 0 is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.

Soviet physics. Doklady, Mar 1, 1981

Springer eBooks, Feb 6, 2006

ABSTRACT Desynchronized linear models are introduced as discrete linear models with the state coo... more ABSTRACT Desynchronized linear models are introduced as discrete linear models with the state coordinates changing at different times. Desynchronization is suggested as the easiest way to attain stability for some systems. Moreover, the limit hysteresis nonlinearities are introduced and the averaging principle is studied.

Nonlinear Analysis-theory Methods & Applications, Dec 1, 1997

Automation and Remote Control, Dec 1, 2007

IFAC Proceedings Volumes, Jul 1, 1993

Families of regimes for control systems are studied possessing the so called quasi-controllabilit... more Families of regimes for control systems are studied possessing the so called quasi-controllability property that is similar to the Kalman control lability property. A new approach is proposed to estimate the degree of transients overshooting in quasi-controllable systems. This approach is conceptually related with the principle of bounded regimes absence in the absolute stability problem. Its essence is in obtaining of constructive a priori bounds for degree of overshooting in terms of the so called quasi-control lability measure. It is shown that relations between stability, asymptotic stability and instability for quasi-controllable systems are similar to those for systems described by linear differential or difference equations in the case when the leading eigenvalue of the corresponding matrix is simple. The results are applicable for analysis of transients, classical absolute stability problem, stability problem for desynchronized systems and so on.

It is shown that the assumption of D-stability of the interconnection matrix, together with the s... more It is shown that the assumption of D-stability of the interconnection matrix, together with the standard assumptions on the activation functions, guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes. For the synchronous mode, these assumptions are also shown to imply local asymptotic stability of the equilibrium. For the asynchronous mode of operation, two results are derived. First, it is shown that symmetry and stability of the interconnection matrix guarantee local asymptotic stability of the equilibrium under a class of asynchronous modes-this is referred to as local absolute asymptotic stability. Second, it is shown that, under the standard assumptions, if the nonnegative matrix whose elements are the absolute values of the correspond- ing elements of the interconnection matrix is stable, then the equilibrium is globally absolutely asymptotically stable under a class of asynchronous modes. The results obtained are discussed from the points of view of their applications, robustness, and their relationship to earlier results.

arXiv (Cornell University), Oct 13, 2008

In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet ... more In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix products is the so-called joint or generalized spectral radius. In the work some explicit a priori estimates for the joint spectral radius with the help of the generalized Gelfand formula are obtained. These estimates are based on the notion of the measure of irreducibility (quasi-controllability) of matrix sets proposed previously by A. Pokrovskii and the author.

arXiv (Cornell University), Mar 8, 2019

An example of inconsistencies in information provided by popular bibliographic services is descri... more An example of inconsistencies in information provided by popular bibliographic services is described, and the reasons for such inconsistencies are discussed.

Linear & Multilinear Algebra, Dec 30, 2016

We prove the minimax equality for the spectral radius ρ(AB) of the product of matrices A ∈ A and ... more We prove the minimax equality for the spectral radius ρ(AB) of the product of matrices A ∈ A and B ∈ B, where A and B are compact sets of non-negative matrices of dimensions N × M and M × N , respectively, satisfying the so-called hourglass alternative.

IFAC Proceedings Volumes, 1991

Linear desynchronized sys~ems are considered ~ha~ are par~icular Cbu~ very impor~an~) case oC ~he... more Linear desynchronized sys~ems are considered ~ha~ are par~icular Cbu~ very impor~an~) case oC ~he so-called Discre~e Even~ Dynamic Sys~ems. The s~a bili~y problem oC linear desynchronized systems is studied Cor the most unCavorable si~ua~ion-when neither inCorma~ion concerning rules oC al~era~ion oC upda~ing momen~s for different subsys~ems nor inforn~~ion concerning ~he amoun~ oC subsys~ems upda~ing a~ any ~ime instanL is available. The Equivalen~ Norms Technique for sLabili~y analysis of desynchronized systems is presen~ed. Obs~a cles for applica~ion of Lhis technique are discussed and ~he way ~o overcome ~hem is shown. To demons~ra~e ~he po~enLial of ~he Equivalen~ Norms Technique some examples are presenled. Among ~hem lhe proof of robus~ness of linear desynchronized sys~ems wi~h regard ~o devia~ions of parame~ers inheren~ in a sysem. Wilh ~he help of ~he Equivalenl Norms Technique il is proved also ~ha~ asymplo~ically s~able linear desynchronized sys~ems are slable under constanlly ac~ing per~urba~ions. One new class of slable linear desynchronized sys~ems is described. Keywords. Desynchronized sys~ems; linear sys~ems; s~abili~y problem; equivalent norms ~echnique; robus~ness. I.NTRODUCTI ON =on~inuous Variable Dynamic S~~ems (CVDS) presen~ one of classic objec~s of s~udy in ~he con~rol ~heory.

Automation and Remote Control, 2019

We present a survey of results on models of consensus in asynchronous multiagent systems with dis... more We present a survey of results on models of consensus in asynchronous multiagent systems with discrete and continuous time. We consider mathematical methods developed over recent years, which are used in the analysis of stability, stabilization, and consensus problems for linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of a set of matrices to analyze the rate of convergence of matrix products with factors drawn from certain sets of matrices with special properties.

Journal of Communications Technology and Electronics, 2015

The investigation of the asymptotic behavior of trigonometric series near the origin is a promine... more The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G.H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to find the asymptotics of trigonomet ric series in several variables with the terms, having a form of "one minus the cosine" accurate to a decreasing power factor.

Theoretical Computer Science, Aug 1, 1995

A measure of the ability of a symbolic sequence to be coded by initial fragments of another symbo... more A measure of the ability of a symbolic sequence to be coded by initial fragments of another symbolic sequence-its self-similarity measure-is introduced and its basic properties are investigated. The selfsimilarity measure of symbolic sequences associated with tori shift mappings corresponding to a special partitioning of a torus are then considered.

Journal of Communications Technology and Electronics, 2019

An example of inconsistencies in information provided by popular bibliographic services is descri... more An example of inconsistencies in information provided by popular bibliographic services is described and the reasons for these inconsistencies are discussed.

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and m... more Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People's behaviors as small as possible, while Tribune wants to make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP\&coNP.

This paper investigates arbitrage chains involving four currencies and four foreign ex-change tra... more This paper investigates arbitrage chains involving four currencies and four foreign ex-change trader-arbitrageurs. In contrast with the three-currency case, we find that arbi-trage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a “balanced ” ensemble of exchange rates in which the law of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics.

One of fundamental results of the theory of joint/generalized spectral radius, the Berger-Wang th... more One of fundamental results of the theory of joint/generalized spectral radius, the Berger-Wang theorem, establishes equality between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law. We introduce the concepts of the joint and generalized spectral radii for products of matrices subjected to constraints on the sliding block relative frequencies of occurrences of different matrices, and prove an analog of the Berger-Wang theorem for this case.

We describe a new class of positive linear discrete-time switching systems for which the problems... more We describe a new class of positive linear discrete-time switching systems for which the problems of stability or stabilizability can be resolved constructively. This class generalizes the class of systems with independently switching state vector components. The distinctive feature of this class is that, for any system from this class, its components or blocks can be arbitrarily connected in parallel or in series without loss of the `constructive resolvability' property. It is shown also that, for such systems, it is possible to build constructively the individual positive trajectories with the greatest or the lowest rate of convergence to the zero.