Abstract

We study the structure of approximate solutions of an autonomous nonconcave discrete-time control system with a compact metric space of states. This control system is described by a bounded upper semicontinuous objective function which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. We show that the turnpike property of approximate solutions on finite intervals follows from the asymptotic turnpike property of approximate solutions of the corresponding infinite horizon optimal control problem. We also show that this asymptotic turnpike property for the corresponding infinite horizon optimal control problem holds for most objective functions in the sense of Baire category.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexander J. Zaslavski

Dept. of Mathematics, Technion, Israel Institute of Technology, 32000 Haifa, Israel

ajzasl@tx.technion.ac.il

Keywords

  • Compact metric space
  • generic property
  • good program
  • turnpike property

Mathematics Subject Classification

49J99

References

  1. B. D. O. Anderson, J. B. Moore: Linear Optimal Control, Prentice-Hall, Englewood Cliffs (1971).Google Scholar
  2. S. M. Aseev, V. M. Veliov: Maximum principle for infinite-horizon optimal control problems with dominating discount, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19 (2012) 43–63.Google Scholar
  3. J. P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley Interscience, New York (1984).Google Scholar
  4. S. Aubry, P. Y. Le Daeron: The discrete Frenkel-Kontorova model and its extensions I, Physica D 8 (1983) 381–422.https://doi.org/10.1016/0167-2789(83)90233-6Google Scholar
  5. J. Baumeister, A. Leitão, G. N. Silva: On the value function for nonautonomous optimal control problems with infinite horizon, Syst. Control Lett. 56 (2007) 188–196.https://doi.org/10.1016/j.sysconle.2006.08.011Google Scholar
  6. J. Blot: Infinite-horizon Pontryagin principles without invertibility, J. Nonlinear Convex Anal. 10 (2009) 177–189.Google Scholar
  7. J. Blot, P. Cartigny: Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl. 106 (2000) 411–419.https://doi.org/10.1023/A:1004611816252Google Scholar
  8. P. Cartigny, A. Rapaport: Nonturnpike optimal solutions and their approximations in infinite horizon, J. Optim. Theory Appl. 134 (2007) 1–14.https://doi.org/10.1007/s10957-007-9206-0Google Scholar
  9. I. V. Evstigneev, S. D. Flam: Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators, Set-Valued Anal. 6 (1998) 61–81.https://doi.org/10.1023/A:1008606332037Google Scholar
  10. D. Gale: On optimal development in a multi-sector economy, Rev. Econ. Stud. 34 (1967) 1–18.https://doi.org/10.2307/2296567Google Scholar
  11. A. Leizarowitz: Infinite horizon autonomous systems with unbounded cost, Appl. Math. Optim. 13 (1985) 19–43.https://doi.org/10.1007/BF01442197Google Scholar
  12. A. Leizarowitz: Tracking nonperiodic trajectories with the overtaking criterion, Appl. Math. Optim. 14 (1986) 155–171.https://doi.org/10.1007/BF01442233Google Scholar
  13. A. Leizarowitz, V. J. Mizel: One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Ration. Mech. Anal. 106 (1989) 161–194.https://doi.org/10.1007/BF00251430Google Scholar
  14. V. Lykina, S. Pickenhain, M. Wagner: Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl. 340 (2008) 498–510.https://doi.org/10.1016/j.jmaa.2007.08.008Google Scholar
  15. V. L. Makarov, A. M. Rubinov: Mathematical Theory of Economic Dynamics and Equilibria, Springer, New York (1977).https://doi.org/10.1007/978-1-4612-9886-1Google Scholar
  16. A. B. Malinowska, N. Martins, D. F. M. Torres: Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett. 5 (2011) 41–53.https://doi.org/10.1007/s11590-010-0189-7Google Scholar
  17. M. Marcus, A. J. Zaslavski: The structure of extremals of a class of second order variational problems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999) 593–629.https://doi.org/10.1016/s0294-1449(99)80029-8Google Scholar
  18. L. W. McKenzie: Turnpike theory, Econometrica 44 (1976) 841–866.https://doi.org/10.2307/1911532Google Scholar
  19. B. Mordukhovich: Minimax design for a class of distributed parameter systems, Automat. Remote Control 50 (1990) 1333–1340.Google Scholar
  20. B. Mordukhovich, I. Shvartsman: Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in: Optimal Control, Stabilization and Nonsmooth Analysis (Baton Rouge, 2003), M. de Queiroxz et al. (ed.), Lecture Notes Control Inform. Sci. 301, Springer, Berlin (2004) 121–132.https://doi.org/10.1007/978-3-540-39983-4_8Google Scholar
  21. J. Moser: Minimal solutions of variational problems on a torus, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 229–272.https://doi.org/10.1016/s0294-1449(16)30387-0Google Scholar
  22. Z. Nitecki: Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, MIT Press, Cambridge (1971).Google Scholar
  23. E. Ocana Anaya, P. Cartigny, P. Loisel: Singular infinite horizon calculus of variations. Applications to fisheries management, J. Nonlinear Convex Anal. 10 (2009) 157–176.Google Scholar
  24. S. Pickenhain, V. Lykina, M. Wagner: On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet. 37 (2008) 451–468.Google Scholar
  25. P. A. Samuelson: A catenary turnpike theorem involving consumption and the golden rule, Amer. Econ. Rev. 55 (1965) 486–496.Google Scholar
  26. A. J. Zaslavski: Ground states in Frenkel-Kontorova model, Math. USSR, Izv. 29 (1987) 323–354.https://doi.org/10.1070/IM1987v029n02ABEH000972Google Scholar
  27. A. J. Zaslavski: Optimal programs on infinite horizon 1,2, SIAM J. Control Optim. 33 (1995) 1643–1686.https://doi.org/10.1137/S0363012993257271Google Scholar
  28. A. J. Zaslavski: Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York (2006).Google Scholar
  29. A. J. Zaslavski: Turnpike results for a discrete-time optimal control system arising in economic dynamics, Nonlinear Anal. 67 (2007) 2024–2049.https://doi.org/10.1016/j.na.2006.08.029Google Scholar
  30. A. J. Zaslavski: Two turnpike results for a discrete-time optimal control system, Nonlinear Anal. 71 (2009) 902–909.https://doi.org/10.1016/j.na.2008.12.053Google Scholar
  31. A. J. Zaslavski: Optimization on Metric and Normed Spaces, Springer, New York (2010).https://doi.org/10.1007/978-0-387-88621-3Google Scholar
  32. A. J. Zaslavski, A. Leizarowitz: Optimal solutions of linear control systems with nonperiodic integrands, Math. Oper. Res. 22 (1997) 726–746.https://doi.org/10.1287/moor.22.3.726Google Scholar

Suggested citation

A. J. Zaslavski. “Turnpike Properties of Approximate Solutions of Nonconcave Discrete-Time Optimal Control Problems.” Journal of Convex Analysis 21 (2014), No. 3, 681–701. https://doi.org/10.68381/jca21036

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.