Optimization of Running Strategies According to the Physiological Parameters for a Two-Runners Model
In order to describe the velocity and the anaerobic energy of two runners competing against each other for middle-distance races, we present a mathematical model relying on an optimal control problem for a system of ordinary differential equations. The model is based on energy conservation and on Newton’s second law: resistive forces, propulsive forces and variations in the maximal oxygen uptake are taken into account. The interaction between the runners provides a minimum for staying 1 m behind one’s competitor. We perform numerical simulations and show how a runner can win a race against someone stronger by taking advantage of staying behind, or how they can improve their personal record by running behind someone else. Our simulations show when it is the best time to overtake, depending on the difference between the athletes. Finally, we compare our numerical results with real data from the men’s 1500 m finals of different competitions.
KeywordsOptimization Running strategies Mathematics of sport Optimal control Middle-distance races
I would like to thank Frédéric Bonnans for his very helpful comments and A. Aftalion for suggesting this interesting topic of research and for her many remarks. A first version of this work, of which she is co-author, can be found on arXiv (http://arxiv.org/abs/1508.00523v1).
- Bonnans J-F, Giorgi D, Grelard V, Maindrault S, Martinon P (2014) BOCOP—a toolbox for optimal control problems. http://bocop.org
- YouTube Channel: Beijing 2008 Athletics Gymnastics Aquatics (2008) Athletics—men’s 1500M—Beijing 2008 summer olympic games (video file). https://www.youtube.com/watch?v=0HcGVbDLhI8
- YouTube Channel: Athletics (2014) Men’s 1500m final IAAF diamond league Rome 2014 (video file). https://www.youtube.com/watch?v=CTVUGLapmPY
- YouTube Channel: Sport Singapore (2015) Athletics men’s 1500m final (day 6) | 28th SEA games Singapore 2015 (video file). https://www.youtube.com/watch?v=pfeVSzDnv-I