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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 2(404), Pages 3–64 (Mi umn9467)  

This article is cited in 27 scientific papers (total in 28 papers)

Infinite-horizon optimal control problems in economics

S. M. Aseevab, K. O. Besova, A. V. Kryazhimskiyba

a Steklov Mathematical Institute of the Russian Academy of Sciences
b International Institute for Applied Systems Analysis, Laxenburg, Austria

Abstract: This paper extends optimal control theory to a class of infinite-horizon problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of this sort the initial state is fixed, no constraints are imposed on the behaviour of the admissible trajectories at large times, and the objective functional is given by a discounted improper integral. We develop the method of finite-horizon approximations in a broad context and use it to derive complete versions of the Pontryagin maximum principle for such problems. We provide sufficient conditions for the normality of infinite-horizon optimal control problems and for the validity of the ‘standard’ limit transversality conditions with time going to infinity. As a meaningful example, we consider a new two-sector model of optimal economic growth subject to a random jump in prices.
Bibliography: 53 titles.

Keywords: dynamic optimization, Pontryagin maximum principle, infinite horizon, transversality conditions at infinity, optimal economic growth.

Funding Agency Grant Number
Russian Foundation for Basic Research 09-01-00624-а
Ministry of Education and Science of the Russian Federation НШ-65772.2010.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations


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English version:
Russian Mathematical Surveys, 2012, 67:2, 195–253

Bibliographic databases:

Document Type: Article
UDC: 517.977
MSC: 49K15, 91B62
Received: 18.11.2011

Citation: S. M. Aseev, K. O. Besov, A. V. Kryazhimskiy, “Infinite-horizon optimal control problems in economics”, Uspekhi Mat. Nauk, 67:2(404) (2012), 3–64; Russian Math. Surveys, 67:2 (2012), 195–253

Citation in format AMSBIB
\by S.~M.~Aseev, K.~O.~Besov, A.~V.~Kryazhimskiy
\paper Infinite-horizon optimal control problems in economics
\jour Uspekhi Mat. Nauk
\yr 2012
\vol 67
\issue 2(404)
\pages 3--64
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 2
\pages 195--253

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    This publication is cited in the following articles:
    1. D. V. Khlopin, “O neobkhodimykh kraevykh usloviyakh dlya silno optimalnogo upravleniya v zadachakh na beskonechnom promezhutke”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 1, 49–58  mathnet
    2. Dmitrii V. Khlopin, “Neobkhodimye usloviya ravnovesiya na beskonechnom promezhutke”, MTIP, 5:2 (2013), 105–136  mathnet
    3. D. Khlopin, “Necessity of vanishing shadow price in infinite horizon control problems”, J. Dyn. Control Syst., 19:4 (2013), 519–552  crossref  mathscinet  zmath  isi  elib  scopus
    4. Dmitrii V. Khlopin, “Ob igre Zorgera”, MTIP, 5:3 (2013), 115–119  mathnet
    5. D. V. Khlopin, “O zadachakh upravleniya na beskonechnom promezhutke s monotonnoi dinamikoi”, Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki, 18 (2013), 2727–2729  elib
    6. S. M. Aseev, “On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems”, Proc. Steklov Inst. Math. (Suppl.), 287, suppl. 1 (2014), 11–21  mathnet  crossref  mathscinet  isi  elib  elib
    7. A. J. Zaslavski, “Stability of a turnpike phenomenon for approximate solutions of nonautonomous discrete-time optimal control systems”, Nonlinear Anal., 100 (2014), 1–22  crossref  mathscinet  zmath  isi  elib  scopus
    8. K. O. Besov, “On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function”, Proc. Steklov Inst. Math., 284 (2014), 50–80  mathnet  crossref  crossref  isi
    9. D. V. Khlopin, “Necessity of limiting co-state arcs in Bolza-type infinite horizon problem”, Optimization, 64:11 (2015), 2417–2440  crossref  mathscinet  zmath  isi  elib  scopus
    10. Z. N. Murzabekov, “The Synthesis of the Proportional-Differential Regulators for the Systems with Fixed Ends of Trajectories Under Two-Sided Constraints on Control Values”, Asian Journal of Control, 17:6 (2015), 1–8  crossref  mathscinet  isi  scopus
    11. S. M. Aseev, “Adjoint variables and intertemporal prices in infinite-horizon optimal control problems”, Proc. Steklov Inst. Math., 290:1 (2015), 223–237  mathnet  crossref  crossref  zmath  isi  elib  elib
    12. K. O. Besov, “Problem of optimal endogenous growth with exhaustible resources and possibility of a technological jump”, Proc. Steklov Inst. Math., 291 (2015), 49–60  mathnet  crossref  crossref  isi  elib
    13. A. J. Zaslavski, “Existence and a turnpike property of solutions for a class of nonautonomous optimal control systems with discounting”, J. Nonlinear Convex Anal., 16:6 (2015), 1155–1183  mathscinet  zmath  isi
    14. S. M. Aseev, “On the boundedness of optimal controls in infinite-horizon problems”, Proc. Steklov Inst. Math., 291 (2015), 38–48  mathnet  crossref  crossref  isi  elib
    15. S. M. Aseev, “Existence of an optimal control in infinite-horizon problems with unbounded set of control constraints”, Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 1–10  mathnet  crossref  crossref  mathscinet  isi  elib
    16. A. A. Krasovskii, P. D. Lebedev, A. M. Tarasyev, “Some facts about the Ramsey model”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 123–131  mathnet  crossref  crossref  mathscinet  isi  elib
    17. D. V. Khlopin, “O gamiltoniane v zadachakh upravleniya na beskonechnom promezhutke”, Tr. IMM UrO RAN, 22, no. 4, 2016, 295–310  mathnet  crossref  mathscinet  elib
    18. D. Khlopin, “On Hamiltonian as limiting gradient in infinite horizon problem”, J. Dyn. Control Syst., 23:1 (2017), 71–88  crossref  mathscinet  zmath  isi  scopus
    19. A. A. Krasovskii, P. D. Lebedev, A. M. Tarasyev, “Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints”, Comput. Math. Math. Phys., 57:5 (2017), 770–783  mathnet  crossref  crossref  mathscinet  isi  elib
    20. D. Khlopin, “On boundary conditions at infinity for infinite horizon control problem”, 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V. F. Demyanov) (CNSA), IEEE, 2017, 146–148  isi
    21. K. O. Besov, “On Balder's Existence Theorem for Infinite-Horizon Optimal Control Problems”, Math. Notes, 103:2 (2018), 167–174  mathnet  crossref  crossref  isi  elib
    22. Z. Murzabekov, M. Milosz, K. Tussupova, “The optimal control problem with fixed-end trajectories for a three-sector economic model of a cluster”, Intelligent Information and Database Systems, Aciids 2018, Pt I, Lecture Notes in Artificial Intelligence, 10751, Springer, 2018, 382–391  crossref  isi  scopus
    23. D. V. Khlopin, “O neobkhodimykh predelnykh gradientakh v zadachakh upravleniya na beskonechnom promezhutke”, Vypusk posvyaschen 70-letnemu yubileyu Aleksandra Georgievicha Chentsova, Tr. IMM UrO RAN, 24, no. 1, 2018, 247–256  mathnet  crossref  elib
    24. Avetisyan A.S., Khurshudyan A.Zh., “Exact and Approximate Controllability of Nonlinear Dynamic Systems in Infinite Time: the Green'S Function Approach”, ZAMM-Z. Angew. Math. Mech., 98:11 (2018), 1992–2009  crossref  mathscinet  isi  scopus
    25. Khlopin D.V., “A Maximum Principle For One Infinite Horizon Impulsive Control Problem”, IFAC PAPERSONLINE, 51:32 (2018), 213–218  crossref  isi  scopus
    26. Aseev S., Manzoor T., “Optimal Exploitation of Renewable Resources: Lessons in Sustainability From An Optimal Growth Model of Natural Resource Consumption”, Control Systems and Mathematical Methods in Economics: Essays in Honor of Vladimir M. Veliov, Lecture Notes in Economics and Mathematical Systems, 687, eds. Feichtinger G., Kovacevic R., Tragler G., Springer-Verlag Berlin, 2018, 221–245  crossref  mathscinet  zmath  isi  scopus
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