Giacomo Albi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella                                                                         

Active Particles, Volulme 1. Advances in Theory, Models, and Applications, pp. 49-98, 2017. (Preprint arXiv)

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.

Michael Herty, Mattia Zanella

Discrete and Continuous Dynamical Systems – Series A, 37(4): 2023-2043, 2017. (Preprint arXiv)

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

G. Albi, L. Pareschi, M. Zanella 

System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology, vol 494, 2016. (Preprint arXiv)

In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.