Geodesic
Construction of a regulator for the Hamiltonian system in a two-sector economic growth model
Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 278-298 
A. M. Tarasyev; A. A. Usova. Construction of a regulator for the Hamiltonian system in a two-sector economic growth model. Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 278-298. http://geodesic.mathdoc.fr/item/TRSPY_2010_271_a18/
@article{TRSPY_2010_271_a18,
     author = {A. M. Tarasyev and A. A. Usova},
     title = {Construction of a~regulator for the {Hamiltonian} system in a~two-sector economic growth model},
     journal = {Informatics and Automation},
     pages = {278--298},
     year = {2010},
     volume = {271},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_271_a18/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider an optimal control problem of investment in the capital stock of a country and in the labor efficiency. We start from a model constructed within the classical approaches of economic growth theory and based on three production factors: capital stock, human capital, and useful work. It is assumed that the levels of investment in the capital stock and human capital are endogenous control parameters of the model, while the useful work is an exogenous parameter subject to logistic-type dynamics. The gross domestic product (GDP) of a country is described by a Cobb–Douglas production function. As a utility function, we take the integral consumption index discounted on an infinite time interval. To solve the resulting optimal control problem, we apply dynamic programming methods. We study optimal control regimes and examine the existence of an equilibrium state in each regime. On the boundaries between domains of different control regimes, we check the smoothness and strict concavity of the maximized Hamiltonian. Special focus is placed on a regime of variable control actions. The novelty of the solution proposed consists in constructing a nonlinear stabilizer based on the feedback principle. The properties of the stabilizer allow one to find an approximate solution to the original problem in the neighborhood of an equilibrium state. Solving numerically the stabilized Hamiltonian system, we find the trajectories of the capital of a country and labor efficiency. The solutions obtained allow one to assess the growth rates of the GDP of the country and the level of consumption in the neighborhood of an equilibrium position.

[1] Aseev S.M., Kryazhimskii A.V., Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta, Tr. MIAN, 257, Nauka, M., 2007 | MR

[2] Intriligator M., Matematicheskie metody optimizatsii i ekonomicheskaya teoriya, Airis-press, M., 2002

[3] Krasovskii A.A., Tarasev A.M., “Svoistva gamiltonovykh sistem v printsipe maksimuma Pontryagina dlya zadach ekonomicheskogo rosta”, Tr. MIAN, 262, 2008, 127–145 | MR | Zbl

[4] Arrow K.J., “Applications of control theory to economic growth”, Mathematics of the decision sciences, Part 2, Amer. Math. Soc., Providence (RI), 1968, 85–119 | MR

[5] Ayres R.U., Warr B., The economic growth engine: How energy and work drive material prosperity, Edward Elgar Publ., Cheltenham, 2009

[6] Balder E.J., “An existence result for optimal economic growth problems”, J. Math. Anal. and Appl., 95 (1983), 195–213 | DOI | MR | Zbl

[7] Grossman G.M., Helpman E., Innovation and growth in the global economy, MIT Press, Cambridge (MA), 1993

[8] Hartman P., Ordinary differential equations, J. Wiley and Sons, New York, 1964 | MR | Zbl

[9] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, New York, 1988 | MR | Zbl

[10] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1976 | Zbl

[11] Rockafellar R.T., “Hamilton–Jacobi theory and parametric analysis in fully convex problems of optimal control”, J. Global Optim., 28 (2004), 419–431 | DOI | MR | Zbl

[12] Sanderson W., The SEDIM model: Version 0.1, IIASA Interim Rep. IR-04-041, Laxenburg (Austria), 2004

[13] Shell K., “Applications of Pontryagin's maximum principle to economics”, Mathematical systems theory and economics, v. 1, Springer, Berlin, 1969, 241–292 | DOI | MR

[14] Solow R.M., Growth theory: An exposition, Oxford Univ. Press, New York, 1970