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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 3, Pages 318–337 (Mi vuu487)  

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

MATHEMATICS

Uniform complete controllability and global control over asymptotic invariants of linear systems in the Hessenberg form

V. A. Zaitsev

Department of Differential Equations, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: We prove that a linear control system
\begin{equation} \dot x=A(t)x+B(t)u,\qquad t\in\mathbb R,\quad x\in\mathbb R^n,\quad u\in\mathbb R^m, \end{equation}
with matrix coefficients of the Hessenberg form is uniformly completely controllable in the sense of Kalman under rather weak conditions imposed on coefficients. It is shown that some obtained sufficient conditions are essential. Corollaries are derived for quasi-differential equations. We construct feedback control $u=Ux$ for the system (1) and study the problem of global control over asymptotic invariants of the closed-loop system
\begin{equation} \dot x=(A(t)+B(t)U)x,\qquad t\in\mathbb R,\quad x\in\mathbb R^n, \end{equation}
The conditions on coefficients are weakened in the known results of S. N. Popova. For the system (2) with matrix coefficients of the Hessenberg form, the obtained results and the results of S. N. Popova are used to receive sufficient conditions for global reducibility to systems of scalar type and for global control over Lyapunov exponents and moreover, for global Lyapunov reducibility in the case of periodic $A(\cdot)$ and $B(\cdot)$.

Keywords: linear control system, uniform complete controllability, system in the Hessenberg form, global control over asymptotic invariants.

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Document Type: Article
UDC: 517.977.1+517.926
MSC: 93B05, 93C05
Received: 15.05.2015

Citation: V. A. Zaitsev, “Uniform complete controllability and global control over asymptotic invariants of linear systems in the Hessenberg form”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:3 (2015), 318–337

Citation in format AMSBIB
\Bibitem{Zai15}
\by V.~A.~Zaitsev
\paper Uniform complete controllability and global control over asymptotic invariants of linear systems in the Hessenberg form
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2015
\vol 25
\issue 3
\pages 318--337
\mathnet{http://mi.mathnet.ru/vuu487}
\elib{http://elibrary.ru/item.asp?id=24237239}


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    This publication is cited in the following articles:
    1. A. A. Kozlov, “O dostatochnom uslovii globalnoi skalyarizuemosti lineinykh upravlyaemykh sistem s lokalno integriruemymi koeffitsientami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:2 (2016), 221–230  mathnet  crossref  mathscinet  elib
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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