||Many researchers considered for cycling time-trials the speed minimizing time for
fixed energy expenditure. The purpose of this contribution is to recall and extend a
fundamental result overlooked in the literature. In 1994, Maronski proved that for
moderate slope profiles, the minimum-time pacing strategy for cycling has three
phases: (1) a short initial maximum-power phase, (2) a long constant-speed central
phase, (3) and a short final maximal-power phase (Maronski, 1994). We call the
constant-speed phase the Maronski effect.
It seems counter-intuitive that the optimal speed is constant during climbing and
descent. The Maronski effect can be explained by the dual problem to minimize
energy expenditure for fixed minimum time. Then any variation of the constant
speed in the central phase (2) causes an increase in energy due to the quadratic
dependence of aerial resistance on the speed.
Here we point out that the Maronski effect is not limited to the simple model used.
For example, consider replacing the fixed-work constraint by the 3-parameter criti-
cal power model (Morton, 1996), introducing critical power, anaerobic capacity, and
maximum power as a function of remaining anaerobic resources as parameters.
We found that in this case the optimal solution, although quantitatively different, is
qualitatively equivalent to that of Maronski.
Concluding, we have confirmed the Maronski effect for an extended optimal cycling
control problem that includes a physiological component. It remains open to further
generalize the optimization problem to more realistic physiological models for which
we expect the Maronski effect to be overcome. For a first step in that direction, see
Finally, we point out that the Maronski effect implies that the second phase of the
solution is a singular arc. Such arcs bear principal difficulties in numerical computa-
tion because precisely on those arcs higher-order optimality conditions are required
to uniquely define the solution. For computing solutions with several singular arcs
we report progress following Riedinger (2013) by incorporating higher-order opti-
mality conditions using complementarity constraints and GPOPS (Rao, 2010).