Volume 9 (2018): Issue 2 (December 2018)
Special Issue on Mathematical modelling for complex systems: multi-agents methods. Guest Editor: Elena De Angelis

In this paper 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. Two types of selective controls have been presented: an homogeneous control filtered by a selective function and a distributed control active only on a selective set. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we derive the mean-field limit of the feedback selective constrained dynamics, which eventually will be solved numerically by means of a stochastic algorithm, able to simulate effciently the selective constrained dynamics. Finally, several numerical simulations are reported to show the effciency of the proposed techniques.

Kinetic models have so far been used to model wealth distribution in a society. In particular, within the framework of the kinetic theory for active particles, some important models have been developed and proposed. They involve nonlinear interactions among individuals that are modeled according to game theoretical tools by introducing parameters governing the temporal dynamics of the system. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile.

The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This 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 the simulation of brownout phenomena occurring in helicopter take-off and landing.

The political replacement effect is an interesting socio-political hypothesis introduced by Acemoglu and Robinson and statistically tested. It may determine, under some conditions, the phenomenon of innovation blocking, possibly leading to economic backwardness in a society. In a previous paper, we have introduced a kinetic model with stochastic evolutive game-type interactions, analyzing the relationship between the level of political competition in a society and the degree of economic liberalization. In the present paper we model we model the possibility of having a sort of phase transition occurring in the system when the phenomenon of blocking of the introduction of technological innovation, intended in a broad sense, appears. Crossing a critical point, the rules of interactions change by means of slightly different transition probabilities nevertheless determining very significant differences in the resulting long-term solutions.

In this paper, we propose a numerical approach to solve a kinetic model of chemotaxis phenomena. This scheme is shown to be uniformly stable with respect to the small parameter, consistent with the uid-di usion limit (Keller-Segel model). Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model that couples a kinetic equation with macroscopic ones. This method is validated by various test cases and compared to other standard methods.

This paper deals with the modeling of interactions between the immune system and cancer cells, in the framework of the mathematical kinetic theory for active particles. The work deepens a previous paper of Belloquid et al. that assumes spatial homogeneity and discrete values of the activity of cancer and immune cells. A number of simulations are made with the aim to investigate how the state of the various cell populations evolves in time depending on the choice of the free parameters.

This paper aims at bridging Mathematical Modelling and Mathematics Education by studying the opinion dynamics of students who work in small groups during mathematics classrooms. In particular, we propose a model which hinges upon the pioneering work of Hegselmann and Krause on opinion dynamics and integrates recent results of interactionist research in Mathematical Education. More precisely, the proposed model incorporates the following features: 1) the feelings of each student towards the classmates (building upon the so-called \I can" -\you can" framework); 2) the different levels of preparation of the students; 3) the presence of the teacher, who may or may not intervene to drive the students towards the correct solution of the problem. Several numerical experiments are presented to assess the capability of the model in reproducing typical realistic scenarios.

In the present paper we propose two mathematical models describing, respectively at the microscopic level and at the mesoscopic level, a system of interacting tumor cells and cells of the immune system. The microscopic model is in terms of a Markov chain defined by the generator, the mesoscopic model is developed in the framework of the kinetic theory of active particles. The main result is to prove the transition from the microscopic to mesoscopic level of description.