Journal of Optimization Theory and Applications

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

Optimal Control and the Fibonacci Sequence

  • Thomas von BraschEmail author
  • Johan Byström
  • Lars Petter Lystad


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.


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



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.


  1. 1.
    McReynolds, S.R.: A successive sweep method for solving optimal programming problems. Ph.D. Thesis, Harvard University (1966) Google Scholar
  2. 2.
    Lystad, L.P.: Bruk av reguleringstekniske metoder for analyse og utvikling av økonomiske modeller (the use of control theory for analysis and development of economic models). Ph.D. Thesis, NTH, Institutt for sosialøkonomi, p. 228, Meddelse nr. 28 (1975) Google Scholar
  3. 3.
    Magill, M.J.P.: A local analysis of n-sector capital accumulation under uncertainty. J. Econ. Theory 15(1), 211–219 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Magill, M.J.P.: Some new results on the local stability of the process of capital accumulation. J. Econ. Theory 15(1), 174–210 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Judd, K.L.: Numerical Methods in Economics. MIT Press, Cambridge (1998) zbMATHGoogle Scholar
  6. 6.
    Levine, P., Pearlman, J., Pierse, R.: Linear-quadratic approximation, external habit and targeting rules. J. Econ. Dyn. Control 32(10), 3315–3349 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Benigno, P., Woodford, M.: Linear-quadratic approximation of optimal policy problems. Discussion paper 0809-01, Department of Economics, Columbia University (2008) Google Scholar
  8. 8.
    Benavoli, A., Chisci, L., Farina, A.: Fibonacci sequence, golden section, Kalman filter and optimal control. Signal Process. 89(8), 1483–1488 (2009) zbMATHCrossRefGoogle Scholar
  9. 9.
    Capponi, A., Farina, A., Pilotto, C.: Expressing stochastic filters via number sequences. Signal Process. 90(7), 2124–2132 (2010) zbMATHCrossRefGoogle Scholar
  10. 10.
    Donoghue, J.: State estimation and control of the Fibonacci system. Signal Process. 91(5), 1190–1193 (2011) zbMATHCrossRefGoogle Scholar
  11. 11.
    Byström, J., Lystad, L.P., Nyman, P.-O.: Using generalized Fibonacci sequences for solving the one-dimensional LQR problem and its discrete-time Riccati equation. Model. Identif. Control 31(1), 1–18 (2010) CrossRefGoogle Scholar
  12. 12.
    Ljungqvist, L., Sargent, T.J.: Recursive Macroeconomic Theory, 2nd edn. MIT Press, Cambridge (2004) Google Scholar
  13. 13.
    Castellanos, D.: Rapidly converging expansions with Fibonacci coefficients. Fibonacci Q. 24, 70–82 (1986) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Brock, W.A., Mirman, L.J.: Optimal economic growth and uncertainty: the discounted case. J. Econ. Theory 4(3), 479–513 (1972) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lewis, F.L., Vrabie, D., Syrmos, V.L.: Optimal Control, 3rd edn. Wiley, New York (2012) zbMATHCrossRefGoogle Scholar
  16. 16.
    Von Brasch, T., Byström, J., Lystad, L.P.: Optimal control and the Fibonacci sequence. Discussion paper 674. Statistics Norway (2012) Google Scholar
  17. 17.
    Sydsæter, K., Hammond, P., Seierstad, A., Strøm, A.: Further Mathematics for Economic Analysis. Financial Times/Prentice Hall, London/New York (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Thomas von Brasch
    • 1
    • 2
    Email author
  • 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

Personalised recommendations