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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 3(411), Pages 5–38 (Mi umn9525)  

This article is cited in 12 scientific papers (total in 12 papers)

Lagrange's principle in extremum problems with constraints

E. R. Avakova, G. G. Magaril-Il'yaevbc, V. M. Tikhomirovc

a Institute of Control Sciences of the Russian Academy of Sciences
b Institute for Information Transmission Problems of the Russian Academy of Sciences
c Moscow State University

Abstract: In this paper a general result concerning Lagrange's principle for so-called smoothly approximately convex problems is proved which encompasses necessary extremum conditions for mathematical and convex programming, the calculus of variations, Lyapunov problems, and optimal control problems with phase constraints. The problem of local controllability for a dynamical system with phase constraints is also considered. In an appendix, results are presented that relate to the development of a ‘Lagrangian approach’ to problems where regularity is absent and classical approaches are meaningless.
Bibliography: 33 titles.

Keywords: extremum problem, optimal control, phase constraints, mix, controllability, abnormality.

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-00188


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English version:
Russian Mathematical Surveys, 2013, 68:3, 401–433

Bibliographic databases:

UDC: 517.977
MSC: Primary 49J40; Secondary 49M05
Received: 11.10.2012

Citation: E. R. Avakov, G. G. Magaril-Il'yaev, V. M. Tikhomirov, “Lagrange's principle in extremum problems with constraints”, Uspekhi Mat. Nauk, 68:3(411) (2013), 5–38; Russian Math. Surveys, 68:3 (2013), 401–433

Citation in format AMSBIB
\by E.~R.~Avakov, G.~G.~Magaril-Il'yaev, V.~M.~Tikhomirov
\paper Lagrange's principle in extremum problems with constraints
\jour Uspekhi Mat. Nauk
\yr 2013
\vol 68
\issue 3(411)
\pages 5--38
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 3
\pages 401--433

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    This publication is cited in the following articles:
    1. E. R. Avakov, G. G. Magaril-Il'yaev, “Mix of controls and the Pontryagin maximum principle”, J. Math. Sci., 217:6 (2016), 672–682  mathnet  crossref  mathscinet
    2. A. D. Ioffe, “On necessary conditions for a minimum”, J. Math. Sci., 217:6 (2016), 751–772  mathnet  crossref  mathscinet
    3. N. Tauchnitz, “The Pontryagin Maximum Principle for Nonlinear Optimal Control Problems with Infinite Horizon”, J. Optim. Theory Appl., 167:1 (2015), 27–48  crossref  mathscinet  zmath  isi  elib  scopus
    4. I. V. Orlov, S. I. Smirnova, “Invertibility of multivalued sublinear operators”, Eurasian Math. J., 6:4 (2015), 44–58  mathnet
    5. J. M. Borwein, Q. J. Zhu, “A Variational Approach to Lagrange Multipliers”, J. Optim. Theory Appl., 2016  crossref  mathscinet  scopus
    6. Avakov E.R., Magaril-Il'yaev G.G., “Pontryagin maximum principle, relaxation, and controllability”, Dokl. Math., 93:2 (2016), 193–196  crossref  mathscinet  zmath  isi  elib  scopus
    7. Ioffe A.D., “Metric Regularity-a Survey Part II. Applications”, J. Aust. Math. Soc., 101:3 (2016), 376–417  crossref  mathscinet  zmath  isi  elib  scopus
    8. Tikhomirov V., “Survey of the Theory of Extremal Problems”, Advances in Mathematical Economics, Vol 20, Advances in Mathematical Economics, 20, eds. Kusuoka S., Maruyama T., Springer-Verlag Singapore Pte Ltd, 2016, 131–150  crossref  mathscinet  zmath  isi
    9. E. R. Avakov, G. G. Magaril-Il'yaev, “Relaxation and controllability in optimal control problems”, Sb. Math., 208:5 (2017), 585–619  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. E. R. Avakov, G. G. Magaril-Il'yaev, “Controllability and second-order necessary conditions for optimality”, Sb. Math., 210:1 (2019), 1–23  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. E. R. Avakov, G. G. Magaril-Il'yaev, “Local infimum and a family of maximum principles in optimal control”, Sb. Math., 211:6 (2020), 750–785  mathnet  crossref  crossref  mathscinet  isi  elib
    12. Idczak D., Walczak S., “An Extremum Principle For Smooth Problems”, Games, 11:4 (2020), 56  crossref  mathscinet  zmath  isi
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