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Maximum principle for an optimal control problem with an asymptotic endpoint constraint. (English. Russian original) Zbl 07466401

Proc. Steklov Inst. Math. 315, Suppl. 1, S42-S54 (2021); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 27, No. 2, 35-48 (2021).
Summary: Under conditions characterizing the dominance of the discounting factor, a complete version of the Pontryagin maximum principle for an optimal control problem with infinite time horizon and a special asymptotic endpoint constraint is developed. Problems of this type arise in mathematical economics in the studies of growth models.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
91B62 Economic growth models

Software:

DYNAMO
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References:

[1] Acemoglu, D., Introduction to Modern Economic Growth (2008), Princeton: Princeton Univ. Press, Princeton · Zbl 1158.91001
[2] Alekseev, VM; Tikhomirov, VM; Fomin, SV, Optimal Control (1979), Moscow: Nauka, Moscow · Zbl 0516.49002
[3] Aseev, SM; Besov, KO; Kryazhimskii, AV, Infinite-horizon optimal control problems in economics, Russ. Math. Surv., 67, 2, 195-253 (2012) · Zbl 1248.49023
[4] Aseev, SM; Besov, KO; Kaniovskii, SYu, Optimal policies in the Dasgupta-Heal-Solow-Stiglitz model under nonconstant returns to scale, Proc. Steklov Inst. Math., 304, 74-109 (2019) · Zbl 1422.91487
[5] Aseev, SM; Kryazhimskii, AV, The Pontryagin maximum principle and optimal economic growth problems, Proc. Steklov Inst. Math., 257, 1-255 (2007) · Zbl 1215.49001
[6] 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, 221-245 (2018), Cham: Springer, Cham · Zbl 1416.91310
[7] Aseev, SM; Veliov, VM, Maximum principle for infinite-horizon optimal control problems with dominating discount, Dyn. Contin. Discrete Impuls. Syst., 19, 1-2, 43-63 (2012) · Zbl 1266.49003
[8] Aseev, SM; Veliov, VM, Needle variations in infinite-horizon optimal control, Variational and Optimal Control Problems on Unbounded Domains, 1-18 (2014), Providence: Amer. Math. Soc., Providence · Zbl 1329.49005
[9] Aseev, SM; Veliov, VM, Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions, Proc. Steklov Inst. Math., 291, Suppl. 1, S22-S39 (2015) · Zbl 1336.49024
[10] Aseev, SM; Veliov, VM, Another view of the maximum principle for infinite-horizon optimal control problems in economics, Russ. Math. Surv., 74, 6, 963-1011 (2019) · Zbl 1480.49022
[11] Aubin, JP; Clarke, FH, Shadow prices and duality for a class of optimal control problems, SIAM J. Control Optim., 17, 567-586 (1979) · Zbl 0439.49018
[12] Barro, RJ; Sala-i-Martin, X., Economic Growth (1995), New York: McGraw Hill, New York · Zbl 0825.90067
[13] Benchekroun, H.; Withhagen, C., The optimal depletion of exhaustible resources: A complete characterization, Resour. Energy Econ., 33, 3, 612-636 (2011)
[14] Besov, KO, On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function, Proc. Steklov Inst. Math., 284, 50-80 (2014) · Zbl 1310.91101
[15] Brodskii, YuI, Necessary conditions for a weak extremum in optimal control problems on an infinite time interval, Math. USSR-Sb., 34, 3, 327-343 (1978) · Zbl 0421.49018
[16] Cesari, L., Optimization — Theory and Applications: Problems with Ordinary Differential Equations (1983), New York: Springer, New York · Zbl 0506.49001
[17] Dasgupta, P.; Heal, GM, The optimal depletion of exhaustible resources, Rev. Econ. Stud., 41, 5, 3-28 (1974) · Zbl 0304.90018
[18] Filippov, AF, Differential Equations with Discontinuous Righthand Sides (1985), Moscow: Nauka, Moscow
[19] Hartman, P., Ordinary Differential Equations (1964), New York: Wiley, New York · Zbl 0125.32102
[20] Kolmogorov, AN; Fomin, SV, Elements of the Theory of Functions and Functional Analysis (1976), Moscow: Nauka, Moscow
[21] Kuratowski, K., Topology (1966), New York: Academic, New York · Zbl 0158.40901
[22] Meadows, DH; Meadows, DL; Randers, J.; Behrens III, WW, The Limits to Growth: A Report for the Club of Rome’s Project on the Predicament of Mankind (1972), New York: Universe Books, New York
[23] Pezzey, J., Sustainable Development Concepts: An Economic Analysis (1992), Washington: The World Bank, Washington
[24] Pontryagin, LS; Boltyanskii, VG; Gamkrelidze, RV; Mishchenko, EF, The Mathematical Theory of Optimal Processes (1961), Moscow: Fizmatgiz, Moscow · Zbl 0102.31901
[25] Ramsey, FP, A mathematical theory of saving, Econ. J., 38, 152, 543-559 (1928)
[26] Seierstad, A., A maximum principle for smooth infinite horizon optimal control problems with state constraints and with terminal constraints at infinity, Open J. Optim., 4, 100-130 (2015)
[27] Solow, RM, Intergenerational equity and exhaustible resources, Rev. Econ. Stud., 41, 5, 29-45 (1974) · Zbl 0304.90019
[28] Stiglitz, J., Growth with exhaustible natural resources: Efficient and optimal growth paths, Rev. Econ. Stud., 41, 5, 123-137 (1974) · Zbl 0353.90010
[29] N. Tauchnitz, Pontryagin’s maximum principle for infinite horizon optimal control problems with bounded processes and with state constraints (2020). https://arxiv.org/pdf/2007.09692.pdf
[30] Valente, S., Sustainable development, renewable resources and technological progress, Environ. Resour. Econ., 30, 1, 115-125 (2005)
[31] Valente, S., Optimal growth, genuine savings and long-run dynamics, Scott. J. Polit. Econ., 55, 2, 210-226 (2008)
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