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The turnpike result for approximate solutions of nonautonomous variational problems

Part of: Existence theories Spaces with richer structures

Published online by Cambridge University Press:  09 April 2009

Alexander J. Zaslavski
Alexander J. Zaslavski
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel, e-mail: ajzasl@techunix.technion.ac.il
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Abstract

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In this work we study the structure of approximate solutions of variational problems with continuous integrands f: [0, ∞) × Rn × Rn → R1 which belong to a complete metric space of functions. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.

Keywords

Complete metric spacegood functionintegrandturnpike property.

MSC classification

Secondary: 49J99: None of the above, but in this section 54E52: Baire category, Baire spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 105 - 130
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Berkovitz, L. D., ‘Lower semicontinuity of integral functionals’, Trans. Amer. Math. Soc. 192 (1974), 51–57.CrossRefGoogle Scholar
[2]Dzalilov, Z., Ivanov, A. F. and Rubinov, A. M., ‘Difference inclusions with delay of economic growth’, Dynam. Systems Appl. 10 (2001), 283–293.Google Scholar
[3]Dzalilov, Z., Rubinov, A. M. and Kloeden, P. E., ‘Lyapunov sequences and a turnpike theorem without convexity’, Set-Valued Analysis 6 (1998), 277–302.CrossRefGoogle Scholar
[4]Gale, D., ‘On optimal development in a multisector economy’, Rev. Econom. Stud. 34 (1967), 1–19.CrossRefGoogle Scholar
[5]Makarov, V. L., Levin, M. J. and Rubinov, A. M., Mathematical economic theory: pure and mixed types of economic mechanisms (North-Holland, Amsterdam, 1995).Google Scholar
[6]Makarov, V. L. and Rubinov, A. M., Mathematical theory of economic dynamics and equilibria (Nauka, Moscow, 1973); English translation (Springer, New York, 1977).Google Scholar
[7]Mamedov, M. A. and Pehlivan, S., ‘Statistical convergence of optimal paths’, Math. Japon. 52 (2000), 51–55.Google Scholar
[8]Mamedov, M. A. and Pehlivan, S., ‘Statistical cluster points and turnpike theorem in nonconvex problems’, J. Math. Anal. Appl. 256 (2001), 686–693.CrossRefGoogle Scholar
[9]McKenzie, L. W., ‘Turnpike theory’, Econometrica 44 (1976), 841–866.CrossRefGoogle Scholar
[10]Radner, R., ‘Path of economic growth that are optimal with regard only to final states; a turnpike theorem’, Rev. Econom. Stud. 28 (1961), 98–104.CrossRefGoogle Scholar
[11]Rubinov, A. M., ‘Economic dynamics’, J. Soviet Math. 26 (1984), 1975–2012.CrossRefGoogle Scholar
[12]Samuelson, P. A., ‘A catenary turnpike theorem involving consumption and the golden rule’, American Economic Review 55 (1965), 486–496.Google Scholar
[13]Zaslavski, A. J., ‘Existence and uniform boundedness of approximate solutions of variational problems without convexity assumptions’, Dynam. Systems Appl. 13 (2004), 161–178.Google Scholar
[14]Zaslavski, A. J., ‘The structure of approximate solutions of variational problems without convexity’, J. Math. Anal. Appl. 296 (2004), 578–593.CrossRefGoogle Scholar
[15]Zaslavski, A. J., ‘The turnpike property for approximate solutions of variational problems without convexity’, Nonlinear Analysis 58 (2004), 547–569.CrossRefGoogle Scholar
[16]Zaslavski, A. J., ‘Existence and uniform boundedness of optimal solutions of variational problems’, Abstract and Appl. Analysis 3 (1998), 265–292.CrossRefGoogle Scholar
[17]Zaslavski, A. J., ‘The turnpike property for extremals of nonautonomous variational problems with vector-valued functions’, Nonlinear Analysis 42 (2000), 1465–1498.CrossRefGoogle Scholar
[18]Zaslavski, A. J. and Leizarowitz, A., ‘Optimal solutions of linear control systems with nonperiodic integrands’, Math. Oper. Res. 22 (1997), 726–746.CrossRefGoogle Scholar
[19]Zaslavski, A. J. and Leizarowitz, A., ‘Optimal solutions of linear periodic control systems with convex integrands’, Appl. Math. Optim. 37 (1998), 127–150.CrossRefGoogle Scholar