  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.
