Trudy Instituta Matematiki i Mekhaniki UrO RAN
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Volume 25, Number 4, Pages 31–43
DOI: https://doi.org/10.21538/0134-4889-2019-25-4-31-43
(Mi timm1667)
 

This article is cited in 1 scientific paper (total in 1 paper)

Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity

E. A. Barabanova, V. V. Bykovb

a Institute of Mathematics of the National Academy of Sciences of Belarus
b Lomonosov Moscow State University
Full-text PDF (255 kB) Citations (1)
References:
Abstract: Let ${\mathcal M}_n$ be the set of linear differential systems of order $n\geqslant 2$ whose coefficients are continuous and bounded on the time semiaxis $\mathbb{R}_+$. Denote by $\lambda_1(A)\leqslant\ldots\leqslant \lambda_n(A)$ the Lyapunov exponents of a system $A\in {\mathcal M}_n,$ by $\Lambda(A)=(\lambda_1(A),\ldots,\lambda_n(A))$ their spectrum, and by $\mathrm{es}(A)$ the exponential stability index of $A$ (the dimension of the linear subspace of solutions with negative characteristic exponents). For a system $A\in {\mathcal M}_n$ and a metric space $M,$ we consider the class ${\mathcal E}_n[A](M)$ of continuous $(n\times n)$ matrix-valued functions $Q\colon \mathbb{R}_+\times M\to \mathbb{R}^{n\times n}$ satisfying the bound $\|Q(t,\mu)\|\leqslant C_Q\exp(-\sigma_Qt)$ for all $(t,\mu)\in\mathbb{R}_+\times M,$ where $C_Q$ and $\sigma_Q$ are positive constants (possibly different for each function $Q$), and such that the Lyapunov exponents of the system $A+Q,$ which are functions of $\mu\in M$ and are denoted by $\lambda_1(\mu;A+Q)\leqslant\ldots\leqslant \lambda_n(\mu;A+Q),$ are not less than the corresponding Lyapunov exponents of the system $A$; i.e., $\lambda_k(\mu;A+Q)\geqslant \lambda_k(A),$ $k=\overline{1,n},$ for all $\mu\in M$. The problem is to obtain a complete description for each $n\in\mathbb{N}$ and each metric space $M$ of the class of pairs $\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)$ composed of the spectrum $\Lambda(A)\in\mathbb{R}^n$ of a system $A\in {\mathcal M}_n$ and the spectrum $\Lambda(\cdot\,;A+Q)\colon M\to \mathbb{R}^n$ of a family $A+Q,$ where $A$ ranges over ${\mathcal M}_n$ and the matrix-valued function $Q$ ranges over the class ${\mathcal E}_n[A](M)$ for each $A,$ i.e., of the class $\Pi {\mathcal E}_n(M)=\{\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\mathcal M}_{n},\,Q\in {\mathcal E}_n[A](M)\}$. The solution of this problem is provided by the following statement: for each integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(l,F(\cdot)\bigr),$ where $l=(l_1,\ldots,l_n)\in\mathbb{R}^n$ and $F(\cdot)=(f_1(\cdot),\ldots,f_n(\cdot))\colon M\to \mathbb{R}^n,$ belongs to the class $\Pi {\mathcal E}_n(M)$ if and only if the following conditions are met: (1) $l_1\leqslant \ldots \leqslant l_n,$ (2) $f_1(\mu)\leqslant \ldots \leqslant f_n(\mu)$ for all $\mu\in M,$ (3) $f_i(\mu)\geqslant l_i$ for all $i=\overline{1,n}$ and $\mu\in M,$ (4) for each $i=\overline{1,n}$, the function $f_i(\cdot)\colon M\to \mathbb{R}$ is bounded and, for any $r\in\mathbb{R}$, the preimage $f_i^{-1}([r,+\infty))$ of the half-interval $[r,+\infty)$ is a $G_{\delta}$-set. The solution of the similar problem of describing the pairs composed of the exponential stability index $\mathrm{es}(A)\in \{0,\ldots,n\}$ of a system $A$ and the exponential stability index $\mathrm{es}(\cdot\,;A+Q)\colon M\to \{0,\ldots,n\}$ of a family $A+Q,$ i.e., the class ${\mathcal I}{\mathcal E}_n(M)=\{\bigl(\mathrm{es}(A),\mathrm{es}(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\mathcal M}_{n},\,Q\in {\mathcal E}_n[A](M)\}$, is contained in the following statement: for any positive integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(d,f(\cdot)\bigr),$ where $d\in\{0,\ldots,n\}$ and $f\colon M\to\{0,\ldots,n\},$ belongs to the class ${\mathcal I}{\mathcal E}_n(M)$ if and only if $f(\mu)\leqslant d$ for all $\mu\in M$ and, for any $r\in\mathbb{R}$, the preimage $f^{-1}((-\infty,r])$ of the half-interval $(-\infty,r]$ is a $G_{\delta}$-set.
Keywords: linear differential system, Lyapunov exponents, perturbations vanishing at infinity, Baire classes.
Received: 30.09.2019
Revised: 08.11.2019
Accepted: 11.11.2019
Bibliographic databases:
Document Type: Article
UDC: 517.926.4
MSC: 34D08, 34D10
Language: Russian
Citation: E. A. Barabanov, V. V. Bykov, “Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 4, 2019, 31–43
Citation in format AMSBIB
\Bibitem{BarByk19}
\by E.~A.~Barabanov, V.~V.~Bykov
\paper Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 4
\pages 31--43
\mathnet{http://mi.mathnet.ru/timm1667}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-4-31-43}
\elib{https://elibrary.ru/item.asp?id=41455518}
Linking options:
  • https://www.mathnet.ru/eng/timm1667
  • https://www.mathnet.ru/eng/timm/v25/i4/p31
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Statistics & downloads:
    Abstract page:161
    Full-text PDF :55
    References:27
    First page:3
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024