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 Izv. Vyssh. Uchebn. Zaved. Mat., 2016, Number 2, Pages 75–86 (Mi ivm9084)

On the structure of a solution set of controlled initial-boundary value problems

A. V. Chernovab

a Chair of Applied Mathematics, Nizhni Novgorod State Technical University, 24 Minin str., Nizhni Novgorod, 603950 Russia
b Chair of Mathematical Physics and Optimal Control, Nizhni Novgorod State University, 23 Gagarin ave., Nizhni Novgorod, 603950 Russia

Abstract: For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As concerns boundedness and precompactness, we mean certain conditions of the majorant type without Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. In the capañity of examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation and verification the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.

Keywords: reachability tube, convexity conditions, total preservation of solvability, functional-operator equation of the Hammerstein type, nonlinear distributed system, parabolic equation, elliptic equation.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 172702.B.49.21.0003

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2016, 60:2, 62–71

Bibliographic databases:

UDC: 517.988

Citation: A. V. Chernov, “On the structure of a solution set of controlled initial-boundary value problems”, Izv. Vyssh. Uchebn. Zaved. Mat., 2016, no. 2, 75–86; Russian Math. (Iz. VUZ), 60:2 (2016), 62–71

Citation in format AMSBIB
\Bibitem{Che16} \by A.~V.~Chernov \paper On the structure of a~solution set of controlled initial-boundary value problems \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2016 \issue 2 \pages 75--86 \mathnet{http://mi.mathnet.ru/ivm9084} \transl \jour Russian Math. (Iz. VUZ) \yr 2016 \vol 60 \issue 2 \pages 62--71 \crossref{https://doi.org/10.3103/S1066369X16020109} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000409281900010} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84961377367} 

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• http://mi.mathnet.ru/eng/ivm/y2016/i2/p75

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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. Sumin, “Volterra functional-operator equations in the theory of optimal control of distributed systems”, IFAC-PapersOnLine, 51:32 (2018), 759–764
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