A Fokker–Planck Feedback Control-Constrained Approach for Modelling Crowd Motion

ABSTRACT

A Fokker–Planck control approach to model crowd motion is investigated. This strategy is formulated as a bilinear optimal control-constrained problem governed by the Fokker–Planck equation modeling the evolution of the probability density function of the stochastic motion of the crowd. Theoretical results on existence and regularity of controls are provided. For computational purposes, the resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, L² stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments demonstrate the efficiency of the proposed control framework.

KEYWORDS:

• Crowd motion
• Fokker-Planck equation
• numerical analysis
• optimal control theory
• stochastic process

Acknowledgements

We thank Gabriele Ciaramella and the referee for many useful comments.

Funding

This work was supported in part by the European Union under Grant Agreement NO. 304617 Marie Curie Research Training Network “Multi-ITN STRIKE – Novel Methods in Computational Finance”. S. Roy was also supported by the DAAD Passage to India Program and the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in TIFR/ICTS, Bangalore.