Maximum Principle for an Optimal Control Problem with an Asymptotic Endpoint Constraint

S. M. Aseev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 2, pp. 35-48

Résumé

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.

Détail

Keywords: optimal control, infinite horizon, Pontryagin maximum principle, asymptotic endpoint constraint, growth models, sustainable development.

Bibliographie

[1] Acemoglu D., Introduction to modern economic growth, Princeton Univ. Press, Princeton, 2008, 1008 pp.

[2] Alekseev V.M., Tikhomirov V.M., Fomin S.V., Optimalnoe upravlenie, Nauka. Glav. red. fiz.-mat. lit., Moskva, 1979, 430 pp. | MR

[3] Aseev S.M., Besov K.O., Kryazhimskii A.V., “Zadachi optimalnogo upravleniya na beskonechnom intervale vremeni v ekonomike”, Uspekhi mat. nauk, 67:2 (2012), 3–64 | DOI | MR | Zbl

[4] Aseev S.M, Besov K.O., Kaniovskii S.Yu., “Optimizatsiya ekonomicheskogo rosta v modeli Dasgupty - Khila - Solou - Stiglitsa pri nepostoyannoi otdache ot rasshireniya masshtabov proizvodstva”, Tr. MIAN, 304 (2019), 83–122 | DOI | Zbl

[5] Aseev S.M., Kryazhimskii A.V., “Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta”, Tr. MIAN, 257 (2007), 3–271

[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, Lect. Notes Econ. Math. Syst., 687, eds. G. Feichtinger, R. Kovacevic, G. Tragler, Springer, Cham, 2018, 221–245 | DOI | MR | Zbl

[7] Aseev S.M., Veliov V.M., “Maximum principle for infinite-horizon optimal control problems with dominating discount”, Dynamics of Continuous, Discrete and Impulsive Systems. Ser. B: Applications Algorithms, 19:1–2 (2012), 43–63 | MR | Zbl

[8] Aseev S.M., Veliov V.M., “Needle variations in infinite-horizon optimal control”, Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics, 619, eds. G. Wolansky, A.J. Zaslavski, Amer. Math. Soc., Providence, 2014, 1–17 | DOI | MR | Zbl

[9] Aseev S.M., Veliov V.M., “Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:3 (2014), 41–57 | MR

[10] Aseev S.M., Velov V.M., “Drugoi vzglyad na printsip maksimuma dlya zadach optimalnogo upravleniya s beskonechnym gorizontom v ekonomike”, Uspekhi mat. nauk, 74:6 (2019), 3–54 | DOI | MR | Zbl

[11] Aubin J.-P., Clarke F.H., “Shadow prices and duality for a class of optimal control problems”, SIAM J. Control Optim., 17 (1979), 567–586 | DOI | MR | Zbl

[12] Barro R.J., Sala-i-Martin X., Economic growth, McGraw Hill, N Y, 1995, 539 pp.

[13] Benchekroun H., Withhagen C., “The optimal depletion of exhaustible resources: A complete characterization”, Resource and Energy Economics, 33:3 (2011), 612–636 | DOI

[14] Besov K.O., “O neobkhodimykh usloviyakh optimalnosti dlya zadach ekonomicheskogo rosta s beskonechnym gorizontom i lokalno neogranichennoi funktsiei mgnovennoi poleznosti”, Tr. MIAN, 284 (2014), 56–88 | DOI | MR | Zbl

[15] Brodskii Yu.I., “Neobkhodimye usloviya slabogo ekstremuma dlya zadach optimalnogo upravleniya na beskonechnom intervale vremeni”, Mat. sb., 105 (147):3 (1978), 371–388 | MR

[16] Cesari L., Optimization - Theory and applications. Problems with ordinary differential equations, Springer-Verlag, Berlin; Heidelberg; N Y, 1983, 542 pp. | MR | Zbl

[17] Dasgupta P., Heal G.M., “The optimal depletion of exhaustible resources”, The Review of Economic Studies (Symposium on the Economics of Exhaustible Resources), 41:5 (1974), 3–28 | DOI

[18] Filippov A.F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka. Glav. red. fiz.-mat. lit., Moskva, 1985, 224 pp.

[19] Hartman P., Ordinary differential equations, J. Wiley Sons, N Y; London; Sydney, 1964, 612 pp. | MR | Zbl

[20] Kolmogorov A.N., Fomin S.V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka. Glav. red. fiz.-mat. lit., Moskva, 1976, 543 pp. | MR

[21] Kuratovskii K., Topologiya, v. 1, Mir, Moskva, 1966, 606 pp.

[22] Meadows D.H., Meadows D.L., Randers J., and Behrens III W.W., The limits to growth: A report for the Club of Rome's project on the predicament of mankind, Universe Books, N Y, 1972, 205 pp.

[23] Pezzey J., Sustainable development concepts: an economic analysis, World Bank environment paper, no. 2, The World Bank, Washington, 1992 | DOI

[24] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, Moskva, 1961, 400 pp. | MR

[25] Ramsey F.P., “A mathematical theory of saving”, Econ. J., 38 (1928), 543–559 | DOI

[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 (2015), 100–130 | DOI

[27] Solow R.M., “Intergenerational equity and exhaustible resources”, The Review of Economic Studies (Symposium on the Economics of Exhaustible Resources), 41 (1974), 29–45 | DOI

[28] Stiglitz J., “Growth with exhaustible natural resources: Efficient and optimal growth paths”, The Review of Economic Studies (Symposium on the Economics of Exhaustible Resources), 41 (1974), 123–137 | DOI | MR

[29] Tauchnitz N., Pontryagin's maximum principle for infinite horizon optimal control problems with bounded processes and with state constraints, 2020, 27 pp., arXiv: 2007.09692 | MR

[30] Valente S., “Sustainable development, renewable resources and technological progress”, Environmental and Resource Economics, 30:1 (2005), 115–125 | DOI

[31] Valente S., “Optimal growth, genuine savings and long-run dynamics”, Scottish Journal of Political Economy, 55:2 (2008), 210–226 | DOI