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Optimal Control and the Fibonacci Sequence

Author

Listed:
  • Thomas Brasch

    (Statistics Norway
    NUPI)

  • Johan Byström

    (Luleå University of Technology)

  • Lars Petter Lystad

    (Narvik University College)

Abstract

We bridge mathematical number theory with optimal control and show that a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady-state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.

Suggested Citation

  • Thomas Brasch & Johan Byström & Lars Petter Lystad, 2012. "Optimal Control and the Fibonacci Sequence," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 857-878, September.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:3:d:10.1007_s10957-012-0061-2
    DOI: 10.1007/s10957-012-0061-2
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    References listed on IDEAS

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    1. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    2. Thomas von Brasch & Johan Byström & Lars Petter Lystad, 2012. "Optimal control and the Fibonacci sequence," Discussion Papers 674, Statistics Norway, Research Department.
    3. William A. Brock & Leonard J. Mirman, 2001. "Optimal Economic Growth And Uncertainty: The Discounted Case," Chapters, in: W. D. Dechert (ed.), Growth Theory, Nonlinear Dynamics and Economic Modelling, chapter 1, pages 3-37, Edward Elgar Publishing.
    4. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
    5. Magill, Michael J. P., 1977. "Some new results on the local stability of the process of capital accumulation," Journal of Economic Theory, Elsevier, vol. 15(1), pages 174-210, June.
    6. Levine, Paul & Pearlman, Joseph & Pierse, Richard, 2008. "Linear-quadratic approximation, external habit and targeting rules," Journal of Economic Dynamics and Control, Elsevier, vol. 32(10), pages 3315-3349, October.
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