# The optimal power distribution for cycling a time trial

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## Abstract

In this thesis a problem of determining the optimal pacing strategy to minimize travel time is considered. The problem is restricted to a straight race track with constant slope and rolling resistance, and no headwind. It is expressed as an optimal control problem that can be solved using Pontryagin's Maximum Principle. The control variable is the cyclist's power, which is modelled according to a hyperbolic power-time relationship, where a maximum power level is assumed. The Hamiltonian is linear with respect to this control variable. The minimum time problem is redefined as a maximum excursion problem, which is related to Goddard’s problem of a rocket's ascent through the atmosphere. It turns out that the optimal pacing problem is a singular control problem. Such problems are difficult to solve, both numerically and analytically, and they only occur sporadically in control theory. It is proven that the singular control only accurs during a single interval; optimal pacing starts with maximum power and decays through a singular control to minimum power. The singular arc may be degenerate; a bang-bang control might be optimal, depending on the length of the race track and the amount of available energy. The solution of the pacing problem has been obtained partly numerical and partly analytical. It applies to a straight course without bends, but it can be extended to an arbitrary course by dividing it into straight segments between bends and optimize over all distributions of energy over the segments.