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Izv. Vyssh. Uchebn. Zaved. Mat., 2009, Number 1, Pages 3–43 (Mi ivm1252)  

This article is cited in 24 scientific papers (total in 24 papers)

Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum

A. V. Arguchintseva, V. A. Dykhtab, V. A. Srochkoa

a Irkutsk State University
b Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton–Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.

Keywords: maximum principle, Hamilton–Jacobi inequalities, nonlocal computational methods, variational maximum principle.

Full text: PDF file (423 kB)
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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:1, 1–35

Bibliographic databases:

UDC: 517.9
Received: 21.05.2008

Citation: A. V. Arguchintsev, V. A. Dykhta, V. A. Srochko, “Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 1, 3–43; Russian Math. (Iz. VUZ), 53:1 (2009), 1–35

Citation in format AMSBIB
\Bibitem{ArgDykSro09}
\by A.~V.~Arguchintsev, V.~A.~Dykhta, V.~A.~Srochko
\paper Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2009
\issue 1
\pages 3--43
\mathnet{http://mi.mathnet.ru/ivm1252}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2530588}
\zmath{https://zbmath.org/?q=an:1183.49003}
\elib{https://elibrary.ru/item.asp?id=11642260}
\transl
\jour Russian Math. (Iz. VUZ)
\yr 2009
\vol 53
\issue 1
\pages 1--35
\crossref{https://doi.org/10.3103/S1066369X09010010}


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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. Sorokin S.P., “Monotonnye resheniya neravenstv Gamiltona–Yakobi v optimalnom upravlenii”, Vestn. Tambovskogo un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 14:4 (2009), 800–802
    2. V. A. Dykhta, “Analiz dostatochnykh uslovii optimalnosti s mnozhestvom funktsii Lyapunova”, Tr. IMM UrO RAN, 16, no. 5, 2010, 66–75  mathnet  elib
    3. O. N. Samsonyuk, “Sostavnye funktsii tipa Lyapunova v zadachakh upravleniya impulsnymi dinamicheskimi sistemami”, Tr. IMM UrO RAN, 16, no. 5, 2010, 170–178  mathnet  elib
    4. V. A. Dykhta, O. N. Samsonyuk, “Hamilton–Jacobi inequalities in control problems for impulsive dynamical systems”, Proc. Steklov Inst. Math., 271 (2010), 86–102  mathnet  crossref  mathscinet  isi  elib  elib
    5. V. A. Srochko, S. N. Ushakova, “Improvement of extreme controls and the steepest ascent method in the norm maximization problem on the reachable set”, Comput. Math. Math. Phys., 50:5 (2010), 805–815  mathnet  crossref  adsnasa  isi
    6. S. P. Sorokin, “Dostatochnye usloviya optimalnosti v forme printsipa maksimuma Pontryagina dlya zadach upravleniya gibridnymi sistemami”, Sib. zhurn. industr. matem., 14:1 (2011), 102–113  mathnet  mathscinet
    7. Dykhta V., Samsonyuk O., “Some applications of Hamilton–Jacobi inequalities for classical and impulsive optimal control problems”, Eur. J. Control, 17:1 (2011), 55–69  crossref  mathscinet  zmath  isi  elib  scopus
    8. V. A. Dykhta, S. P. Sorokin, “Positional solutions of Hamilton–Jacobi equations in control problems for discrete-continuous systems”, Autom. Remote Control, 72:6 (2011), 1184–1198  mathnet  crossref  mathscinet  zmath  isi
    9. V. A. Dykhta, S. P. Sorokin, “Hamilton–Jacobi inequalities and the optimality conditions in the problems of control with common end constraints”, Autom. Remote Control, 72:9 (2011), 1808–1821  mathnet  crossref  mathscinet  zmath  isi
    10. V. A. Dykhta, O. N. Samsonyuk, “The canonical theory of the impulse process optimality”, Journal of Mathematical Sciences, 199:6 (2014), 646–653  mathnet  crossref  mathscinet
    11. Dykhta V.A., Sorokin S.P., “O realizatsii nestandartnoi dvoistvennosti v zadachakh optimalnogo upravleniya”, Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki, 16:4 (2011), 1071–1073  mathscinet  elib
    12. Dykhta V.A., Sorokin S.P., Yakovenko G.N., “Upravlyaemye sistemy: usloviya ekstremalnosti, optimalnosti i identifikatsiya algebraicheskoi struktury”, Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 3:3 (2011), 122–131  elib
    13. V. A. Srochko, V. G. Antonik, N. S. Rozinova, “Metody bilineinykh approksimatsii dlya resheniya zadach optimalnogo upravleniya”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 4:3 (2011), 146–157  mathnet
    14. V. A. Srochko, “On solving the optimization problem for chemotherapy process in terms of the maximum principle”, Russian Math. (Iz. VUZ), 56:7 (2012), 55–59  mathnet  crossref  mathscinet
    15. V. A. Srochko, “Ekstremalnye rezhimy upravleniya v zadache optimizatsii protsessa terapii”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 2012, no. 3, 113–119  mathnet
    16. V. A. Srochko, E. V. Aksenyushkina, “Lineino-kvadratichnaya zadacha optimalnogo upravleniya: obosnovanie i skhodimost nelokalnykh metodov resheniya”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 6:1 (2013), 89–100  mathnet
    17. A. V. Chernov, “On the convergence of the conditional gradient method as applied to the optimization of an elliptic equation”, Comput. Math. Math. Phys., 55:2 (2015), 212–226  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    18. V. M. Aleksandrov, “Optimal control of linear systems with interval constraints”, Comput. Math. Math. Phys., 55:5 (2015), 749–765  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. E. V. Aksenyushkina, V. A. Srochko, “Sufficient optimality conditions for a class of nonconvex control problems”, Comput. Math. Math. Phys., 55:10 (2015), 1642–1652  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    20. V. A. Srochko, V. G. Antonik, “Optimality conditions for extremal controls in bilinear and quadratic problems”, Russian Math. (Iz. VUZ), 60:5 (2016), 75–80  mathnet  crossref  isi
    21. V. G. Antonik, V. A. Srochko, “Optimality conditions of the maximum principle type in bilinear control problems”, Comput. Math. Math. Phys., 56:12 (2016), 2023–2034  mathnet  crossref  crossref  isi  elib
    22. O. V. Morzhin, “Nelokalnoe uluchshenie upravlenii v nelineinykh diskretnykh sistemakh”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 19 (2017), 150–163  mathnet  crossref
    23. V. M. Aleksandrov, “Optimal resource consumption control with interval restrictions”, J. Appl. Industr. Math., 12:2 (2018), 201–212  mathnet  crossref  crossref  elib
    24. V. A. Srochko, E. V. Aksenyushkina, “Parametrizatsiya nekotorykh zadach upravleniya lineinymi sistemami”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 30 (2019), 83–98  mathnet  crossref
  • Известия высших учебных заведений. Математика Russian Mathematics (Izvestiya VUZ. Matematika)
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