# 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.

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