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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 4, Pages 895–911 (Mi izv1872)  

This article is cited in 5 scientific papers (total in 6 papers)

The frequency theorem for continuous one-parameter semigroups

A. L. Likhtarnikov, V. A. Yakubovich


Abstract: The following is proved under certain, not very restrictive, assumptions. For the existence of a bounded linear operator $H=H^*$ such that the quadratic form $\operatorname{Re}(Ax+bu,Hx)+F(x,u)$ is positive definite on $X\times U$, it is necessary and sufficient that the form $F[(i\omega I-A)^{-1}bu,u]$ $\forall\omega\in R^1$ be positive definite, where $A$ is the infinitesimal generating operator of a strongly continuous semigroup in a Hilbert space $X$, $b$ is a bounded linear operator acting from a Hilbert space $U$ into $X$, and $F(x,u)$ is a quadratic form on $X$. Moreover, there exist bounded linear operators $H_0,h$, and $\varkappa$ such that the representation $\operatorname{Re}(Ax+bu,Hx)+F(x,u)=[\varkappa u-hx]^2$ holds. A similar assertion is proved in the “degenerate” case.
Bibliography: 30 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:4, 849–864

Bibliographic databases:

UDC: 519.9+517.9
MSC: Primary 47D05, 93C15; Secondary 93D15
Received: 09.12.1975

Citation: A. L. Likhtarnikov, V. A. Yakubovich, “The frequency theorem for continuous one-parameter semigroups”, Izv. Akad. Nauk SSSR Ser. Mat., 41:4 (1977), 895–911; Math. USSR-Izv., 11:4 (1977), 849–864

Citation in format AMSBIB
\Bibitem{LikYak77}
\by A.~L.~Likhtarnikov, V.~A.~Yakubovich
\paper The frequency theorem for continuous one-parameter semigroups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 4
\pages 895--911
\mathnet{http://mi.mathnet.ru/izv1872}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=497014}
\zmath{https://zbmath.org/?q=an:0362.93009}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 4
\pages 849--864
\crossref{https://doi.org/10.1070/IM1977v011n04ABEH001748}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. L. Likhtarnikov, “Absolute stability criteria for nonlinear operator equations”, Math. USSR-Izv., 11:5 (1977), 1011–1029  mathnet  crossref  mathscinet  zmath
    2. A. Kh. Gelig, G. A. Leonov, A. L. Fradkov, “Vladimir Andreevich Yakubovich”, Autom. Remote Control, 67:10 (2006), 1530–1546  mathnet  crossref  mathscinet  zmath  elib
    3. S. V. Gusev, A. L. Likhtarnikov, “Kalman-Popov-Yakubovich lemma and the $S$-procedure: A historical essay”, Autom. Remote Control, 67:11 (2006), 1768–1810  mathnet  crossref  mathscinet  zmath  elib  elib
    4. P. V. Pakshin, V. A. Ugrinovskii, “Stochastic problems of absolute stability”, Autom. Remote Control, 67:11 (2006), 1811–1846  mathnet  crossref  mathscinet  zmath  elib  elib
    5. Sergey Popov, Volker Reitmann, “Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions”, DCDS-A, 34:1 (2013), 249  crossref
    6. S. V. Gusev, “Kalman–Popov–Yakubovich lemma for ordered fields”, Autom. Remote Control, 75:1 (2014), 18–33  mathnet  crossref  isi  elib  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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