Geodesic
On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 15-24 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a class of infinite horizon optimal control problems that appear in studies on economic growth processes, the properties of the adjoint variable in the relations of the Pontryagin maximum principle defined by a formula similar to the Cauchy formula for the solutions to linear differential systems are studied. It is shown that, under a dominating discount condition, the adjoint variable defined in this way satisfies both the core relations of the maximum principle (the adjoint system and the maximum condition) in the normal form and the additional stationarity condition for the Hamiltonian. In addition, a new economic interpretation of the adjoint variable based on this formula is considered.
Keywords: optimal economic growth problems, infinite horizon, Pontryagin’s maximum principle, stationarity condition for the Hamiltonian.
Mots-clés : adjoint variable 
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S. M. Aseev. On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 15-24. http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a1/

[1] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1979, 432 pp. | MR

[2] 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

[3] Aseev S. M., Kryazhimskii A. V., “Printsip maksimuma Pontryagina dlya zadachi optimalnogo upravleniya s funktsionalom, zadannym nesobstvennym integralom”, Dokl. AN, 394:5 (2004), 583–585 | MR | Zbl

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

[5] Aseev S. M., Kryazhimskii A. V., “Ob odnom klasse zadach optimalnogo upravleniya, voznikayuschikh v matematicheskoi ekonomike”, Tr. MIAN, 262, 2008, 16–31 | MR | Zbl

[6] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 391 pp. | MR

[7] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970, 720 pp. | MR

[8] Acemoglu D., Introduction to modern economic growth, Princeton Univ. Press, Princeton, N.J., 2008, 552 pp.

[9] Aseev S. M., Kryazhimskiy A. V., “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons”, SIAM J. Control Optim., 43:3 (2004), 1094–1119 | DOI | MR | Zbl

[10] Aseev S. M., Veliov V. M., “Maximum principle for infinite-horizon optimal control problems with dominating discount”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19:1–2 (2012), 43–63 | MR | Zbl

[11] Aseev S. M., Veliov V. M., Needle variations in infinite-horizon optimal control, Research Report 2012-04, ORCOS Institute of Mathematical Methods in Economics, Vienna University of Technology, Vienna, 2012, 19 pp. http://orcos.tuwien.ac.at/fileadmin/t/orcos/Research_Reports/2012-04_Ase-VV_rev.pdf | MR

[12] 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

[13] Barro R. J., Sala-i-Martin X., Economic growth, McGraw Hill, N.Y., 1995, 654 pp.

[14] Bellman R., Dynamic programming, Princeton Univ. Press, Princeton, 1957, 342 pp. | MR | Zbl

[15] Benveniste L. M., Scheinkman J. A., “Duality theory for dynamic optimization models of economics: the continuous time case”, J. Econ. Theory, 27 (1982), 1–19 | DOI | MR | Zbl

[16] Carlson D. A., Haurie A. B., Leizarowitz A., Infinite horizon optimal control. Determenistic and Stochastic Systems, Springer-Verlag, Berlin, 1991, 332 pp. | Zbl

[17] Dorfman R., “An economic interpretation of optimal control theory”, Am. Econ. Rev., 59 (1969), 817–831

[18] Halkin H., “Necessary conditions for optimal control problems with infinite horizons”, Econometrica, 42 (1974), 267–272 | DOI | MR | Zbl

[19] Michel P., “On the transversality conditions in infinite horizon optimal problems”, Econometrica, 50:4 (1982), 975–985 | DOI | MR | Zbl

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

[21] Seierstad A., “Necessary conditions for nonsmooth, infinite-horizon optimal control problems”, J. Optim. Theory Appl., 103:1 (1999), 201–229 | DOI | MR | Zbl