Skip to main content

Existence and Structure of Solutions of Autonomous Discrete Time Optimal Control Problems

  • Conference paper
Recent Advances in Optimization

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 563))

Summary

In this paper we consider autonomous discrete time optimal control problems. We discuss the reduction to finite cost and the representation formula, the existence of optimal solutions on infinite horizon and their structure, and the structure of optimal solutions on finite intervals.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B.D.O. Anderson and J.B. Moore. Linear Optimal Control. Prentice-Hall, Englewood Cliffs, NJ, 1971.

    Google Scholar 

  2. H. Atsumi. Neoclassical growth and the efficient program of capital accumulation. Review of Econ. Studies, 32: 127–136, 1965.

    Article  Google Scholar 

  3. S. Aubry and P.Y. Le Daeron. The discrete Renkel-Kontorova model and its extensions. Physics D, 8: 381–422, 1983.

    Article  ADS  Google Scholar 

  4. J. Blot and P. Cartigny. Optimality in infinite-horizon variational problems under sign conditions. J. Optim. Theory Appl., 106: 411–419, 2000.

    Article  MathSciNet  Google Scholar 

  5. J. Blot and P. Michel. The value-function of an infinite-horizon linear quadratic problem. Appl. Math. Lett., 16: 71–78, 2003.

    Article  MathSciNet  Google Scholar 

  6. W.A. Brock. On existence of weakly maximal programs in a multi-sector economy. Review of Econ. Studies, 37: 275–280, 1970.

    Article  MATH  Google Scholar 

  7. B.D. Coleman, M. Marcus and V.J. Mizel. On the thermodynamics of periodic phases. Arch. Rational Mech. Anal., 117: 321–347, 1992.

    Article  MathSciNet  ADS  Google Scholar 

  8. F.S. De Blasi and J. Myjak. Sur la porosité des contractions sans point fixe. C. R. Acad. Sci. Paris, 308: 51–54, 1989.

    Google Scholar 

  9. F. S. De Blasi, J. Myjak and P. L. Papini. Porous sets in best approximation theory. J. London Math. Soc., 44: 135–142, 1991.

    Article  MathSciNet  Google Scholar 

  10. Z. Dzalilov, A.F. Ivanov and A.M. Rubinov. Difference inclusions with delay of economic growth. Dynam. Systems Appl., 10: 283–293, 2001.

    MathSciNet  Google Scholar 

  11. Z. Dzalilov, A.M. Rubinov and P.E. Kloeden. Lyapunov sequences and a turnpike theorem without convexity. Set-Valued Analysis, 6: 277–302, 1998.

    Article  MathSciNet  Google Scholar 

  12. D. Gale. On optimal development in a multisector economy. Rev. of Econ. Studies, 34: 1–19, 1967.

    Article  MathSciNet  Google Scholar 

  13. M. Ali Khan and T. Mitra. On choice of technique in the Robinson-Solow-Srinivasan model. Int. J. Econ. Theory, accepted.

    Google Scholar 

  14. A. Leizarowitz. Infinite horizon autonomous systems with unbounded cost. Appl. Math. Optim., 13: 19–43, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Leizarowitz and V.J. Mizel. One dimensional infinite horizon variational problems arising in continuum mechanics. Arch. Rational Mech. Anal., 106: 161–194, 1989.

    Article  MathSciNet  ADS  Google Scholar 

  16. V.L. Makarov, M.J. Levin and A.M. Rubinov. Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. North-Holland, Amsterdam, 1995.

    Google Scholar 

  17. V.L. Makarov and A.M. Rubinov. Mathematical Theory of Economic Dynamics and Equilibria. Nauka, Moscow, 1973, English trans. Springer-Verlag, New York, 1977.

    Google Scholar 

  18. M.A. Mamedov and S. Pehlivan. Statistical convergence of optimal paths. Math. Japon., 52: 51–55, 2000.

    MathSciNet  Google Scholar 

  19. M.A. Mamedov and S. Pehlivan. Statistical cluster points and turnpike theorem in nonconvex problems. J. Math. Anal. Appl., 256: 686–693, 2001.

    Article  MathSciNet  Google Scholar 

  20. M. Marcus and A.J. Zaslavski. The structure of extremals of a class of second order variational problems. Ann. Inst. H. Poincaré, Anal. non linéaire, 16: 593–629, 1999.

    Article  MathSciNet  Google Scholar 

  21. L.W. McKenzie. Turnpike theory. Econometrics 44: 841–866, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Radner. Path of economic growth that are optimal with regard only to final states; a turnpike theorem. Rev. Econom. Stud., 28: 1961, 98–104, 1961.

    Article  Google Scholar 

  23. A.M. Rubinov. Superlinear Multivalued Mappings and their Applications in Economic Mathematical Problems. Nauka, Leningrad, 1980.

    Google Scholar 

  24. A.M. Rubinov. Economic dynamics. J. Soviet Math., 26: 1975–2012, 1984.

    Article  MATH  Google Scholar 

  25. P.A. Samuelson. A catenary turnpike theorem involving consumption and the golden rule. Amer. Econ. Review, 55: 486–496, 1965.

    Google Scholar 

  26. C.C. von Weizsacker. Existence of optimal programs of accumulation for an infinite horizon. Rev. Econ. Studies, 32: 85–104, 1965.

    Article  Google Scholar 

  27. A.J. Zaslavski. Ground states in Frenkel-Kontorova model. Math. USSR Izvestiya, 29: 323–354, 1987.

    Article  Google Scholar 

  28. A.J. Zaslavski. Optimal programs on infinite horizon 1. SIAM J. Control Optim., 33: 1643–1660, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  29. A.J. Zaslavski. Optimal programs on infinite horizon 2. SIAM J. Control Optim., 33: 1661–1686, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  30. A.J. Zaslavski. Dynamic properties of optimal solutions of variational problems. Nonlinear Analysis, 27: 1996, 895–932, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  31. A.J. Zaslavski. Structure of extremals for one-dimensional variational problems arising in continuum mechanics. J. Math. Anal. Appl., 198: 893–921, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  32. A.J. Zaslavski. Existence and structure of optimal solutions of variational problems. Proc. Special Session on Optimization and Nonlinear Analysis, Joint AMSIMU Conference, Jerusalem, May 1995, Contemporary Mathematics, 204: 247–278, 1997.

    MATH  MathSciNet  Google Scholar 

  33. A.J. Zaslavski. Turnpike theorem for convex infinite dimensional discrete-time control systems. J. Convex Analysis, 5: 237–248, 1998.

    MATH  MathSciNet  Google Scholar 

  34. A.J. Zaslavski. Turnpike property for infinite dimensional convex discrete-time control systems in a Banach space. Int. J. Pure and Applied Math., 7: 295–309, 2003.

    MATH  MathSciNet  Google Scholar 

  35. A.J. Zaslavski. Turnpike theorem for a class of discrete time optimal control problems. Proc. 2001 of the 7th Int. Conf. on Nonlinear Functional Analysis and Applications. Fixed Point Theory and Applications, Nova Science Publishers, Inc., New York, 5: 175–182, 2003.

    Google Scholar 

  36. A.J. Zaslavski. Minimal solutions for discrete-time control systems in metric spaces. Numerical Func. Anal. Optim., 24: 637–651, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  37. A.J. Zaslavski and A. Leizarowitz. Optimal solutions of linear control systems with nonperiodic integrands. Math. Oper. Res., 22: 726–746, 1997.

    Article  MathSciNet  Google Scholar 

  38. A.J. Zaslavski and A. Leizarowiitz. Optimal solutions of linear periodic control systems with convex integrands. Appl. Math. Optim., 37: 127–150, 1998.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zaslavski, A.J. (2006). Existence and Structure of Solutions of Autonomous Discrete Time Optimal Control Problems. In: Seeger, A. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28258-0_16

Download citation

Publish with us

Policies and ethics