Crowd Control: from control theory to applications to road traffic – CroCo

In recent years, the study of collective behavior of a crowd of autonomous agents has drawn a great interest from scientific communities, e.g. in civil engineering (for emergency egress and traffic problems), robotics (coordination of robots), computer science and sociology (social networks), and biology (crowds of animals). In particular, it is well-known that some simple rules of interaction between agents can provide the formation of special patterns, like in formations of bird flocks, lines, etc... This phenomenon is often referred to as self-organization or emerging behavior.

Beside the problem of analyzing the collective behavior of such systems, it is now interesting to understand what changes of behavior can be induced by an external agent (e.g. a policy maker) to the crowd. For example, one can try to enforce the creation of patterns when they are not formed naturally, or break the formation of such patterns. This is the problem of control of crowds.

Two control methods are studied in the project in different frameworks. The first (TASK 1) is the control of crowds with leaders, i.e. by acting on a small and fixed number of agents. The resulting control has the clear advantage of focusing on few agents, then naturally producing sparse (or parsimonious) strategies.

The second method (TASK 2) is the control with local policies, modeled as by perturbation of the crowd dynamics in a small subset of the configuration space, modeling external interventions. This model also produces sparse strategies, in terms of the size of the control domain.

The key application of control of traffic models will be addressed in TASK 3. The goal is both to solve challenging specific problems of traffic control, and to use such application as a benchmark for the theory developed in TASKS 1-2. Indeed, we will study problems of control of traffic both with autonomous vehicles acting as leaders and with time-varying speed limits acting as localized policies.


Problems of control of crowds addressed in the project can be stated in different mathematical frameworks, depending on the model used to describe the crowd dynamics. We address it in three different mathematical frameworks: microscopic models (sizable finite-dimensional dynamical systems), macroscopic models (transport partial differential equations), and multi-scale models (measure evolutions). The control problems will be effectively studied by building bridges between the three mathematical frameworks, by studying in which cases control strategies can be translated from one setting to another.

Several methods for such approach were developed and improved by the project team. In particular:
- mean-field limits, that allow to pass from microscopic to macroscopic models. In the case of controlled systems, such limit must be handled with care, and the P.I. and collaborators have developed specific successful techniques.
- multi-scale models, that mix microscopic and macroscopic approaches. The team solved optimal control problems in the case of leaders-and-followers model, that can be adapted to the measure setting.
- the Wasserstein distance and its generalizations, unifying the three frameworks.
- numerical schemes developed by the P.I. and collaborators for the three frameworks.

The researches developed in this project will be carried out by a team based in Marseille. The project aims to establish a small strong group focused on control of multi-agent systems, connecting approaches from control and applied mathematics. The strength of our team is its interdisciplinarity, that permits to merge different techniques (geometric control, control of partial differential equations, systems theory) and a variety of tools to give a significant contribution in solving challenging problems of crowd control.