Final Report Summary - HDSPCONTR (High-Dimensional Sparse Optimal Control) Our research work focused on a novel mathematical field, i.e. the sparse control and the automatic learning of multi-agent dynamical systems and their mean-field approximations, modeling both social and economical scenarios. The main questions addressed in our work are:1. Can we learn automatically – by means of suitable algorithms – the governing equations of the dynamics of a group socially or economically interacting by observing it for long time enough?2. Once we have a model of the dynamics can we sparsely control it? The sparse controllability means the possibility of influencing the dynamics of an entire group by steering the behavior of very few key elements of the group. Is sparse controllability depending on the number of agents?A rigorous mathematical classification and numerical simulation of such learnable and sparsely controllable dynamical systems can have a significant impact on several real life scenarios in policy making for social interactions, networks, financial market, bank systems etc.Among other results of practical relevance achieved in the project, we proved that for models of opinion formation, sparse control, i.e. controlling in appropriate way the opinion of few key agents of the systems, leads very fast to opinion consensus. This model could explain, for instance, how, in absence of political opposition, a totalitarian regime can conquer quite easily and fast the population obedience.We also obtain results, which predict that in cases of emergency it is possible induce the safe and orderly evacuation of pedestrians from a building by controlling the movement of a few informed agents. One not yet published result investigated during the project explains how, under extremely idealised conditions, one may be able to predict the realisation of an electoral campaign: assume to stylise political parties as points in the domain on the political opinions. Moreover, assume that the opinion of each elector is instantaneously influenced by campaign strategies of the parties, which are Nash equilibria. We proved then that the electors will migrate in time towards the barycentre of the political opinions (and the party closer to the barycentre is destined to win). These are just a few relevant examples of how the mathematics explored in this project could be applied to explain, predict, and control multi-agent models describing socio-economic systems.
