Preface to the Special Issue Mathematical modelling for complex systems: multi-agents methods

This special issue of CAIM collects eight selected research papers covering a broad number of topics and methods, all of them sharing a common thread. This common treat is given by the approach of the investigation, that consists in treating the different systems under analysis as a complex system, that is, a system composed of several living entities which interact among themselves and with the outer environment, in a nonlinear manner, see [1] [2] [3] and references therein. A broad variety of living systems can be considered as complex, spanning from biological systems to systems whose dynamics receives important inputs from human behaviors, such as for example crowd and traffic dynamics, swarms, opinion formation, political dynamics. In multi-agent settings, the entities or agents not only have to deal with the outer environment, but also with other agents which act, learn, progress and change over time, sharing the experience of the learning process. New ideas and new mathematical methods are needed to understand the main features of the behaviour of such living, and hence, complex systems: as a consequence, the interest of mathematicians in new structures is a fascinating challenging field of research in mathematical sciences. The publication of a special issue on this hot topic field of research on CAIM Journal is definitively appropriate given the journal’s aims and scope, being the Journal of the SIMAI, Italian Society for Industrial and Applied Mathematics. Articles are aimed towards a broad audience with topics covering different mathematical approaches, different mathematical scales and different collective phenomena. More in detail, in the article entitled Selective model-predictive control for flocking systems by Giacomo Albi and Lorenzo Pareschi, [4], the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of a controller, acting in order to enhance consensus. A mean-field limit of the feedback selective constrained dynamics is derived, which can be solved numerically by means of a stochastic algorithm, able to simulate efficiently the selective constrained dynamics. Numerical simulations are reported to show the efficiency of the proposed techniques. The second paper, [5], by Damian A. Knopoff and Germán A. Torres, On an optimal control strategy in a kinetic social dynamics model, deals on an optimal control problem applied to a socio-economic kinetic model for wealth distribution in a society. The modeling approach is based on the tools of the so called kinetic theory of active particles. The main idea is to provide a ruler with tools that could let him make strategic decisions in terms of wealth distribution policies, given that when he rises to power he has a vision about what kind of society he expects to get in the future. The third article, High-order variational time integrators for particle dynamics by Edie Miglio, Nicola Parolini, Mattia Penati and Roberto Porcù, [6], presentes a new framework for the definition and analysis of Galerkin variational integrators of arbitrary order. The classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to numerical test cases, and in particular to the simulation of brownout phenomena occurring in helicopter take-off and landing.