Optimally Controlling Nutrition and Propulsion Force in a Long Distance Running Race

TL;DR

This paper formulates a long-distance running race as an optimal control problem, optimizing nutrition intake and propulsion force to minimize race time, and confirms that steady pacing is most effective.

Contribution

It introduces a novel differential equation model incorporating nutrition and energy compartments for optimizing long-distance running strategies.

Findings

Steady pacing without energy depletion is optimal.

Model confirms traditional race strategies.

Provides a framework for personalized running optimization.

Abstract

Runners competing in races are looking to optimize their performance. In this paper, a runner's performance in a race, such as a marathon, is formulated as an optimal control problem where the controls are: the nutrition intake throughout the race and the propulsion force of the runner. As nutrition is an integral part of successfully running long distance races, it needs to be included in models of running strategies. We formulate a system of ordinary differential equations to represent the velocity, fat energy, glycogen energy, and nutrition for a runner competing in a long-distance race. The energy compartments represent the energy sources available in the runner's body. We allocate the energy source from which the runner draws, based on how fast the runner is moving. The food consumed during the race is a source term for the nutrition differential equation. With our model, we are investigating strategies to manage the nutrition and propulsion force in order to minimize the running time in a fixed distance race. This requires the solution of a nontrivial singular control problem. Our results confirm the belief that the most effective way to run a race is to run approximately the same pace the entire race without letting one's energies hit zero.