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Journal of Optimization Theory and Applications

, Volume 154, Issue 3, pp 857–878 | Cite as

Optimal Control and the Fibonacci Sequence

  • Thomas von Brasch
  • Johan Byström
  • Lars Petter Lystad
Article

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.

Keywords

Brock–Mirman model Fibonacci sequence Golden ratio Mathematical number theory Optimal control 

Notes

Acknowledgements

Thanks are due to three anonymous reviewers and to Ådne Cappelen, John Dagsvik, Pål Boug, and Anders Rygh Swensen for useful comments. The usual disclaimer applies.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Thomas von Brasch
    • 1
    • 2
  • Johan Byström
    • 3
  • Lars Petter Lystad
    • 4
  1. 1.Research Department, Unit for MacroeconomicsStatistics NorwayOsloNorway
  2. 2.Department of International EconomicsNUPIOsloNorway
  3. 3.Department of Engineering Sciences and MathematicsLuleå University of TechnologyLuleåSweden
  4. 4.Department of TechnologyNarvik University CollegeNarvikNorway

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