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Uspekhi Mat. Nauk, 1972, Volume 27, Issue 3(165), Pages 21–77 (Mi umn5057)  

This article is cited in 21 scientific papers (total in 21 papers)

Convexity of values of vector integrals, theorems on measurable choice and variational problems

V. I. Arkin, V. L. Levin


Abstract: We give an account of applications of measurable many-valued mappings and theorems on convexity of finite-dimensional vector integrals to several variational problems. Theorems on convexity are carried over to vector integrals with values in function spaces, and with the help of these we obtain a aximum principle as a ecessary and sufficient extremum condition and an existence theorem for a on-linear variational problem with operator constraints of integral equality type, similar to Monge's problem on mass displacement.

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English version:
Russian Mathematical Surveys, 1972, 27:3, 21–85

Bibliographic databases:

UDC: 517.4+519.3+519.9
MSC: 28B05, 46G10, 26B25, 30C80, 47J20
Received: 17.01.1972

Citation: V. I. Arkin, V. L. Levin, “Convexity of values of vector integrals, theorems on measurable choice and variational problems”, Uspekhi Mat. Nauk, 27:3(165) (1972), 21–77; Russian Math. Surveys, 27:3 (1972), 21–85

Citation in format AMSBIB
\Bibitem{ArkLev72}
\by V.~I.~Arkin, V.~L.~Levin
\paper Convexity of values of vector integrals, theorems on measurable choice and variational problems
\jour Uspekhi Mat. Nauk
\yr 1972
\vol 27
\issue 3(165)
\pages 21--77
\mathnet{http://mi.mathnet.ru/umn5057}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=405097}
\zmath{https://zbmath.org/?q=an:0254.49006}
\transl
\jour Russian Math. Surveys
\yr 1972
\vol 27
\issue 3
\pages 21--85
\crossref{https://doi.org/10.1070/RM1972v027n03ABEH001378}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. L. Levin, “Convex integral functionals and the theory of lifting”, Russian Math. Surveys, 30:2 (1975), 119–184  mathnet  crossref  mathscinet  zmath
    2. A. G. Chentsov, “On a game problem of converging at a given instant of time”, Math. USSR-Sb., 28:3 (1976), 353–376  mathnet  crossref  mathscinet  zmath  isi
    3. V. S. Klimov, “Imbedding theorems and geometric inequalities”, Math. USSR-Izv., 10:3 (1976), 615–638  mathnet  crossref  mathscinet  zmath
    4. M. A. Krasnoselskii, A. V. Pokrovskii, “O razryvnom operatore superpozitsii”, UMN, 32:1(193) (1977), 169–170  mathnet  mathscinet  zmath
    5. Phan Văn Chu'o'ng, “On a smooth selection theorem and its application to multivalued integral equations”, Math. USSR-Sb., 34:4 (1978), 547–559  mathnet  crossref  mathscinet  zmath
    6. Phan Van Chu'o'ng, “Vector versions of a density theorem and applications to problems of control theory”, Journal of Mathematical Analysis and Applications, 95:2 (1983), 379  crossref
    7. Phan Vâan Chû??ng, “Some Results on Density of Extreme Selections for Measurable Multifunctions”, Math Nachr, 126:1 (1986), 311  crossref  mathscinet  zmath  isi
    8. R. J. Chitashvili, M. G. Mania, “Optimal locally absolutely continuous change of measure. finite set of decisions. part ii:optimization problems”, Stochastics, 21:3 (1987), 187  crossref
    9. R. J. Chitashvili, M. G. Mania, “Optimal locally absolutely continuous change of measure. finite set of decisions. part i”, Stochastics, 21:2 (1987), 131  crossref
    10. Pham T Nhu, “Suboptimal linear feedback for a class of stochastic discrete-time systems”, Journal of Mathematical Analysis and Applications, 150:2 (1990), 307  crossref
    11. I. L. Averbakh, “An iterative method of solving two-stage discrete stochastic programming problems with additively separable variables”, U.S.S.R. Comput. Math. Math. Phys., 31:6 (1991), 21–27  mathnet  mathscinet  zmath  isi
    12. G. E. Ivanov, “Optimal guaranteed control of linear systems under disturbances”, Math. Notes, 60:2 (1996), 147–152  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. Fabián Flores-Bazán, Stefania Perrotta, “Nonconvex Variational Problems Related to a Hyperbolic Equation”, SIAM J Control Optim, 37:6 (1999), 1751  crossref  mathscinet  zmath  isi
    14. E Giner, “On pareto minima of vector-valued integral functionals”, Optimization, 48:1 (2000), 107  crossref
    15. Fabián Flores-Bazán, Jean-Pierre Raymond, “A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders”, Journal of Mathematical Analysis and Applications, 261:2 (2001), 448  crossref
    16. V. S. Klimov, “Symmetrization of functions from Sobolev spaces”, Russian Math. (Iz. VUZ), 46:11 (2002), 41–47  mathnet  mathscinet  zmath  elib
    17. A. V. Akparova, A. A. Shananin, “Model proizvodstva v usloviyakh nesovershennoi kreditnoi sistemy i nestabilnoi realizatsii produktsii”, Matem. modelirovanie, 17:9 (2005), 60–76  mathnet  mathscinet  zmath
    18. N. Yu. Lukoyanov, “On optimality conditions for the guaranteed result in control problems for time-delay systems”, Proc. Steklov Inst. Math. (Suppl.), 268, suppl. 1 (2010), S175–S187  mathnet  crossref  isi  elib
    19. N. P. Osmolovskii, “Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints”, J Math Sci, 2012  crossref
    20. F. S. Stonyakin, “Anti-compacts and their applications to analogs of Lyapunov and Lebesgue theorems in Frechét spaces”, Journal of Mathematical Sciences, 218:4 (2016), 526–548  mathnet  crossref
    21. F. S. Stonyakin, “Sequential analogues of the Lyapunov and Krein–Milman theorems in Fréchet spaces”, Journal of Mathematical Sciences, 225:2 (2017), 322–344  mathnet  crossref
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