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Turnpike theorem for nonautonomous infinite dimensional discrete-time control systems. (English) Zbl 0968.49004

Summary: We study the structure of “approximate” solutions for a nonautonomous infinite-dimensional discrete-time control system determined by a sequence of continuous functions \(v_i: X\times X\to R^1\), \(i= 0,\pm 1,\pm 2,\dots\), where \(X\) is a complete metric space. We show that for a generic sequence of functions \(\{v_i\}^\infty_{i=-\infty}\) there exists a sequence \(\{y_i\}^\infty_{i=-\infty}\subset X\) (the “turnpike”) such that the following properties hold: (i) \(\{y_i\}^{k_2}_{i= k_1}\) is an optimal solution for any finite interval \([k_1,k_2]\); (ii) given \(\varepsilon> 0\), each “approximate” solution on an interval \([k_1,k_2]\) with sufficiently large \(k_2-k_1\) is within \(\varepsilon\) of the “turnpike” for all \(i\in \{L+ k_1,\dots, k_2- L\}\), where \(L\) is a constant which depends only on \(\varepsilon\).

MSC:

49J10 Existence theories for free problems in two or more independent variables
49J27 Existence theories for problems in abstract spaces
49K40 Sensitivity, stability, well-posedness
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[1] DOI: 10.1016/0167-2789(83)90233-6 · Zbl 1237.37059 · doi:10.1016/0167-2789(83)90233-6
[2] DOI: 10.1007/BF01442197 · Zbl 0591.93039 · doi:10.1007/BF01442197
[3] DOI: 10.1007/BF01448190 · Zbl 0687.49015 · doi:10.1007/BF01448190
[4] DOI: 10.1007/BF00251430 · Zbl 0672.73010 · doi:10.1007/BF00251430
[5] Makarov V, Mathematical theory of economic dynamics and equilibria (1973)
[6] DOI: 10.2307/1911532 · Zbl 0356.90006 · doi:10.2307/1911532
[7] Moser J, Ann. Inst. H. Poincare, Anal. Non Lineare 3 pp 229– (1986)
[8] DOI: 10.2307/2295707 · doi:10.2307/2295707
[9] Rubinov A.M, Superlinear multivalued mappings and their applications in economic mathematical problems (1980)
[10] DOI: 10.1007/BF01084444 · Zbl 0544.90016 · doi:10.1007/BF01084444
[11] Samuelson P.A, American Economic Review 55 pp 486– (1965)
[12] DOI: 10.1070/IM1987v029n02ABEH000972 · Zbl 0646.58040 · doi:10.1070/IM1987v029n02ABEH000972
[13] DOI: 10.1137/S0363012993257271 · Zbl 0847.49022 · doi:10.1137/S0363012993257271
[14] DOI: 10.1006/jmaa.1996.0120 · Zbl 0881.49001 · doi:10.1006/jmaa.1996.0120
[15] Zaslavski, A.J. 1997. Existence and Structure of Optimal Solutions of Variational Problems. Proceedings of the Special Session on Optimization and Nonlinear Analysis. Joint AMS-IMU Conference. May1997, Jerusalem. Vol. 204, pp.247–278. Contemporary Mathematics · Zbl 0868.49001
[16] Zaslavski A.J, Convex Analysis 5 pp 237– (1998)
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