2013, 3(3): 389-406. doi: 10.3934/naco.2013.3.389

Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon

1. 

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S.Kovalevskaya str., 16, Ekaterinburg, 620990, Russian Federation, Russian Federation

Received  November 2011 Revised  February 2013 Published  July 2013

The research is focused on an algorithm for constructing solutions of optimal control problems with infinite time horizon arising, for example, in economic growth models.   
       There are several significant difficulties which complicate solution of the problem, such as: (1) stiffness of Hamiltonian systems generated by the Pontryagin maximum principle; (2) non-stability of equilibrium points; (3) lack of initial conditions for adjoint variables.
     The analysis of the Hamiltonian system implemented in this paper for optimal control problems with infinite horizon provides results effective for construction of optimal solutions, namely, (1) if a steady state exists and satisfies regularity conditions then there exists a nonlinear stabilizer for the Hamiltonian system; (2) a nonlinear stabilizer generates the system with excluding adjoint variables whose trajectories (according to qualitative analysis of the corresponding differential equations) approximate solutions of the original Hamiltonian system in a neighborhood of the steady state;(3) trajectories of the stabilized system serve as first approximations and localize search of optimal trajectories.    
    The results of numerical experiments are presented by modeling of an economic growth system with investment in capital and enhancement of the labor efficiency.
Citation: Alexander Tarasyev, Anastasia Usova. Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 389-406. doi: 10.3934/naco.2013.3.389
References:
[1]

S. M. Aseev and A. V. Kryazhimskiy, The Pontryagin maximum principle and optimal economic growth problems, Proceedings of the Steklov Institute of Mathematics, 257 (2007), Pleiades Publishing.

[2]

R. U. Ayres and B. Warr, "The Economic Growth Engine: How Energy and Work Drive Material Prosperity," Edward Elgar Publishing, Cheltenham UK, 2009.

[3]

E. J. Balder, An existance result for optimal economic growth problems, J. Math. Anal. Appl., 95 (1983), 195-213. doi: 10.1016/0022-247X(83)90143-9.

[4]

M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Applied Mathematics and Optimization, 15 (1987), 1-13. doi: 10.1007/BF01442644.

[5]

G. Feichtinger and V. M. Veliov, On a distributed control problem arising in dynamic optimization of a fixed-size population, SIAM J. Optim., 18 (2007), 980-1003. doi: 10.1137/06066148X.

[6]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, "Optimal Control of Nonlinear Processes," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5.

[7]

Ph. Hartman, "Ordinary Differential Equations," J. Wiley and Sons, N. Y., London, Sydney, 1964.

[8]

A. A. Krasovskii, Assessment of the impact of aggregated economic factors on optimal consumption in models of economic growth, IIASA Interim Report IR-06-050, (2006).

[9]

A. N. Krasovskii and N. N. Krasovskii, "Control Under Lack of Information," Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-2568-3.

[10]

N. N. Krasovskii and A. I. Subbotin, "Game-Theoretical Control Problems," Springer, NY, Berlin, 1988. doi: 10.1007/978-1-4612-3716-7.

[11]

A. A. Krasovskii and A. M. Tarasyev, Dynamic optimization of investments in the economic growth models, Automation and Remote Control, 68 (2007), 1765-1777. doi: 10.1134/S0005117907100050.

[12]

A. A. Krasovskii and A. M. Tarasyev, Conjugation of Hamiltonian systems in optimal control problems, Preprints of the 17th World Congress of the International Federation of Automatic Control IFAC, Seoul, Korea, (2008), 7784-7789.

[13]

S. A. Reshmin, A. M. Tarasyev and C. Watanabe, A dynamic model of R & D investment, Journal of Applied Mathematics and Mechanics, 65 (2001), 395-410. doi: 10.1016/S0021-8928(01)00045-4.

[14]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience, New York, 1962.

[15]

W. Sanderson, The SEDIM model: version 0.1, IIASA Interim Report IR-04-041, 42 pages, (2004).

[16]

K. Shell, Applications of Pontryagin's maximum principle to economics, Mathematical Systems Theory and Economics, 1 (1969), 241-292.

[17]

R. M. Solow, "Growth Theory: An Exposition," New York, Oxford University Press, 1970.

[18]

A. M. Tarasyev and A. A. Usova, Construction of a regulator for the Hamiltonian system in a two-sector economic growth model, Proceedings of the Steklov Institute of Mathematics, 271 (2010).

show all references

References:
[1]

S. M. Aseev and A. V. Kryazhimskiy, The Pontryagin maximum principle and optimal economic growth problems, Proceedings of the Steklov Institute of Mathematics, 257 (2007), Pleiades Publishing.

[2]

R. U. Ayres and B. Warr, "The Economic Growth Engine: How Energy and Work Drive Material Prosperity," Edward Elgar Publishing, Cheltenham UK, 2009.

[3]

E. J. Balder, An existance result for optimal economic growth problems, J. Math. Anal. Appl., 95 (1983), 195-213. doi: 10.1016/0022-247X(83)90143-9.

[4]

M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Applied Mathematics and Optimization, 15 (1987), 1-13. doi: 10.1007/BF01442644.

[5]

G. Feichtinger and V. M. Veliov, On a distributed control problem arising in dynamic optimization of a fixed-size population, SIAM J. Optim., 18 (2007), 980-1003. doi: 10.1137/06066148X.

[6]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, "Optimal Control of Nonlinear Processes," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77647-5.

[7]

Ph. Hartman, "Ordinary Differential Equations," J. Wiley and Sons, N. Y., London, Sydney, 1964.

[8]

A. A. Krasovskii, Assessment of the impact of aggregated economic factors on optimal consumption in models of economic growth, IIASA Interim Report IR-06-050, (2006).

[9]

A. N. Krasovskii and N. N. Krasovskii, "Control Under Lack of Information," Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-2568-3.

[10]

N. N. Krasovskii and A. I. Subbotin, "Game-Theoretical Control Problems," Springer, NY, Berlin, 1988. doi: 10.1007/978-1-4612-3716-7.

[11]

A. A. Krasovskii and A. M. Tarasyev, Dynamic optimization of investments in the economic growth models, Automation and Remote Control, 68 (2007), 1765-1777. doi: 10.1134/S0005117907100050.

[12]

A. A. Krasovskii and A. M. Tarasyev, Conjugation of Hamiltonian systems in optimal control problems, Preprints of the 17th World Congress of the International Federation of Automatic Control IFAC, Seoul, Korea, (2008), 7784-7789.

[13]

S. A. Reshmin, A. M. Tarasyev and C. Watanabe, A dynamic model of R & D investment, Journal of Applied Mathematics and Mechanics, 65 (2001), 395-410. doi: 10.1016/S0021-8928(01)00045-4.

[14]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience, New York, 1962.

[15]

W. Sanderson, The SEDIM model: version 0.1, IIASA Interim Report IR-04-041, 42 pages, (2004).

[16]

K. Shell, Applications of Pontryagin's maximum principle to economics, Mathematical Systems Theory and Economics, 1 (1969), 241-292.

[17]

R. M. Solow, "Growth Theory: An Exposition," New York, Oxford University Press, 1970.

[18]

A. M. Tarasyev and A. A. Usova, Construction of a regulator for the Hamiltonian system in a two-sector economic growth model, Proceedings of the Steklov Institute of Mathematics, 271 (2010).

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