# Optimal Control and the Fibonacci Sequence

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## 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.

## References

- 1.McReynolds, S.R.: A successive sweep method for solving optimal programming problems. Ph.D. Thesis, Harvard University (1966) Google Scholar
- 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.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.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.Judd, K.L.: Numerical Methods in Economics. MIT Press, Cambridge (1998) zbMATHGoogle Scholar
- 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.Benigno, P., Woodford, M.: Linear-quadratic approximation of optimal policy problems. Discussion paper 0809-01, Department of Economics, Columbia University (2008) Google Scholar
- 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.Capponi, A., Farina, A., Pilotto, C.: Expressing stochastic filters via number sequences. Signal Process.
**90**(7), 2124–2132 (2010) zbMATHCrossRefGoogle Scholar - 10.Donoghue, J.: State estimation and control of the Fibonacci system. Signal Process.
**91**(5), 1190–1193 (2011) zbMATHCrossRefGoogle Scholar - 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.Ljungqvist, L., Sargent, T.J.: Recursive Macroeconomic Theory, 2nd edn. MIT Press, Cambridge (2004) Google Scholar
- 13.Castellanos, D.: Rapidly converging expansions with Fibonacci coefficients. Fibonacci Q.
**24**, 70–82 (1986) MathSciNetzbMATHGoogle Scholar - 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.Lewis, F.L., Vrabie, D., Syrmos, V.L.: Optimal Control, 3rd edn. Wiley, New York (2012) zbMATHCrossRefGoogle Scholar
- 16.Von Brasch, T., Byström, J., Lystad, L.P.: Optimal control and the Fibonacci sequence. Discussion paper 674. Statistics Norway (2012) Google Scholar
- 17.Sydsæter, K., Hammond, P., Seierstad, A., Strøm, A.: Further Mathematics for Economic Analysis. Financial Times/Prentice Hall, London/New York (2008) Google Scholar