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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 5, Pages 1064–1083 (Mi izv1880)  

This article is cited in 3 scientific papers (total in 3 papers)

Absolute stability criteria for nonlinear operator equations

A. L. Likhtarnikov


Abstract: Conditions are obtained for the stability in the large of solutions of nonlinear equations of the form
\begin{equation} \frac{dx}{dt}=Ax+bu+f,\qquad u=\varphi(y,t),\quad y=Cx. \end{equation}
Here $A$ is the infinitesimal generator of a semigroup of class $C_0$, the maps $b\colon U\to X$ and $C\colon X\to Y$ are bounded linear operators, and $U,X$ and $Y$ are (generally different) Hilbert spaces. The equations (1) describe a wide class of distributed parameter control systems. The results obtained have the following features:
a) The stability conditions pertain not to an individual system but to classes of systems; the stability holds uniformly in a certain sense for all systems of a particular class (“absolute stability in a given class of nonlinearities”).
b) For some classes of nonlinearities, the conditions are not only sufficient but necessary.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:5, 1011–1029

Bibliographic databases:

UDC: 517.9
MSC: 34G05, 34H05, 34K20
Received: 26.03.1976

Citation: A. L. Likhtarnikov, “Absolute stability criteria for nonlinear operator equations”, Izv. Akad. Nauk SSSR Ser. Mat., 41:5 (1977), 1064–1083; Math. USSR-Izv., 11:5 (1977), 1011–1029

Citation in format AMSBIB
\Bibitem{Lik77}
\by A.~L.~Likhtarnikov
\paper Absolute stability criteria for nonlinear operator equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 5
\pages 1064--1083
\mathnet{http://mi.mathnet.ru/izv1880}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=508655}
\zmath{https://zbmath.org/?q=an:0364.93027|0391.93024}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 5
\pages 1011--1029
\crossref{https://doi.org/10.1070/IM1977v011n05ABEH001756}


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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. V. I. Gorbachuk, A. V. Knyazyuk, “Boundary values of solutions of operator-differential equations”, Russian Math. Surveys, 44:3 (1989), 67–111  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. 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
    3. S. Popov, F. Raitmann, S. Skopinov, “Ogranichennost i ustoichivost na konechnykh intervalakh dlya mnogoznachnykh dvazhdy nelineinykh evolyutsionnykh sistem, porozhdennykh zadachei mikrovolnovogo nagreva”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 148–163  mathnet  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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