A variational approach to determine the optimal power distribution for cycling in a time trial

J. de Jong; Robbert Fokkink; Geert Jan Olsder; Arend Schwab

doi:10.1016/j.proeng.2016.06.280

A variational approach to determine the optimal power distribution for cycling in a time trial

J. de Jong, Robbert Fokkink^*, Geert Jan Olsder, Arend Schwab

^*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

2 Citations (Scopus)

80 Downloads (Pure)

Abstract

The optimal pacing strategy of a cyclist in an individual time-trial depends on terrain, weather conditions and the cyclists endurance capacity. Previous experimental and theoretical studies have shown that a suboptimal pacing strategy may have a substantial negative effect. In this paper we express the optimal pacing problem as a mathematical optimal control problem which we solve using Pontryagin's maximum principle. Our solution of the pacing problem is partly numerical and partly analytical. It applies to a straight course without bends. It turns out that the optimal pacing problem is a singular control problem. Intricate mathematical arguments are required to prove that the singular control times form a single interval: optimal pacing starts with maximum power and decays through a singular control, which may be degenerate, to minimum power.

Original language	English
Title of host publication	Procedia Engineering
Subtitle of host publication	The Engineering of Sport 11
Editors	F.C.T. van der Helm, A.J. Jansen
Publisher	Elsevier
Pages	907-911
Volume	147
DOIs	https://doi.org/10.1016/j.proeng.2016.06.280
Publication status	Published - 2016
Event	ISEA 2016 - The Engineering of Sport 11 - Delft, Netherlands Duration: 11 Jul 2016 → 14 Jul 2016 http://www.isea2016.com/

Publication series

Name	Procedia Engineering
Volume	147

Conference

Conference	ISEA 2016 - The Engineering of Sport 11
Country/Territory	Netherlands
City	Delft
Period	11/07/16 → 14/07/16
Internet address	http://www.isea2016.com/

Keywords

bicycling
maximum principle
power equation

Access to Document

10.1016/j.proeng.2016.06.280

1-s2.0-S1877705816307275Final published version, 147 KBLicence: CC BY-NC-ND

Cite this

de Jong, J., Fokkink, R., Olsder, G. J., & Schwab, A. (2016). A variational approach to determine the optimal power distribution for cycling in a time trial. In F. C. T. van der Helm, & A. J. Jansen (Eds.), Procedia Engineering: The Engineering of Sport 11 (Vol. 147, pp. 907-911). (Procedia Engineering; Vol. 147). Elsevier. https://doi.org/10.1016/j.proeng.2016.06.280

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title = "A variational approach to determine the optimal power distribution for cycling in a time trial",

abstract = "The optimal pacing strategy of a cyclist in an individual time-trial depends on terrain, weather conditions and the cyclists endurance capacity. Previous experimental and theoretical studies have shown that a suboptimal pacing strategy may have a substantial negative effect. In this paper we express the optimal pacing problem as a mathematical optimal control problem which we solve using Pontryagin's maximum principle. Our solution of the pacing problem is partly numerical and partly analytical. It applies to a straight course without bends. It turns out that the optimal pacing problem is a singular control problem. Intricate mathematical arguments are required to prove that the singular control times form a single interval: optimal pacing starts with maximum power and decays through a singular control, which may be degenerate, to minimum power.",

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editor = "{van der Helm}, F.C.T. and Jansen, {A.J. }",

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note = "ISEA 2016 - The Engineering of Sport 11 ; Conference date: 11-07-2016 Through 14-07-2016",

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de Jong, J, Fokkink, R , Olsder, GJ & Schwab, A 2016, A variational approach to determine the optimal power distribution for cycling in a time trial. in FCT van der Helm & AJ Jansen (eds), Procedia Engineering: The Engineering of Sport 11. vol. 147, Procedia Engineering, vol. 147, Elsevier, pp. 907-911, ISEA 2016 - The Engineering of Sport 11, Delft, Netherlands, 11/07/16. https://doi.org/10.1016/j.proeng.2016.06.280

A variational approach to determine the optimal power distribution for cycling in a time trial. / de Jong, J.; Fokkink, Robbert ; Olsder, Geert Jan et al.
Procedia Engineering: The Engineering of Sport 11. ed. / F.C.T. van der Helm; A.J. Jansen. Vol. 147 Elsevier, 2016. p. 907-911 (Procedia Engineering; Vol. 147).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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AU - Olsder, Geert Jan

AU - Schwab, Arend

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N2 - The optimal pacing strategy of a cyclist in an individual time-trial depends on terrain, weather conditions and the cyclists endurance capacity. Previous experimental and theoretical studies have shown that a suboptimal pacing strategy may have a substantial negative effect. In this paper we express the optimal pacing problem as a mathematical optimal control problem which we solve using Pontryagin's maximum principle. Our solution of the pacing problem is partly numerical and partly analytical. It applies to a straight course without bends. It turns out that the optimal pacing problem is a singular control problem. Intricate mathematical arguments are required to prove that the singular control times form a single interval: optimal pacing starts with maximum power and decays through a singular control, which may be degenerate, to minimum power.

AB - The optimal pacing strategy of a cyclist in an individual time-trial depends on terrain, weather conditions and the cyclists endurance capacity. Previous experimental and theoretical studies have shown that a suboptimal pacing strategy may have a substantial negative effect. In this paper we express the optimal pacing problem as a mathematical optimal control problem which we solve using Pontryagin's maximum principle. Our solution of the pacing problem is partly numerical and partly analytical. It applies to a straight course without bends. It turns out that the optimal pacing problem is a singular control problem. Intricate mathematical arguments are required to prove that the singular control times form a single interval: optimal pacing starts with maximum power and decays through a singular control, which may be degenerate, to minimum power.

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A variational approach to determine the optimal power distribution for cycling in a time trial

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