Boltzmann type control of opinion consensus through leaders

TL;DR

This paper introduces a Boltzmann type control method for guiding opinion consensus in social groups, utilizing leaders' strategies embedded in microscopic interactions, with theoretical analysis and explicit stationary solutions.

Contribution

It develops a novel Boltzmann type control framework for opinion dynamics, integrating leader strategies into microscopic interactions and deriving explicit stationary solutions.

Findings

The Boltzmann control approach effectively guides opinions towards consensus.

Explicit stationary solutions are derived from the Fokker-Planck asymptotic limits.

The method demonstrates strategic influence of leaders in opinion formation.

Abstract

The study of formations and dynamics of opinions leading to the so called opinion consensus is one of the most important areas in mathematical modeling of social sciences. Following the Boltzmann type control recently introduced in [G. Albi, M. Herty, L. Pareschi arXiv:1401.7798], we consider a group of opinion leaders which modify their strategy accordingly to an objective functional with the aim to achieve opinion consensus. The main feature of the Boltzmann type control is that, thanks to an instantaneous binary control formulation, it permits to embed the minimization of the cost functional into the microscopic leaders interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived which allow to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann type control approach and the capability of the leaders control to strategically lead the followers opinion.