title: Boltzmann-type control of opinion consensus through leaders.
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 modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders' interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived, which allow one 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.
authors.affiliation: Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, Ferrara 44121, Italy.
authors.forename: G
authors.initials: G
authors.lastname: Albi
authors.affiliation: Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, Ferrara 44121, Italy lorenzo.pareschi@unife.it.
authors.forename: L
authors.initials: L
authors.lastname: Pareschi
authors.affiliation: Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, Ferrara 44121, Italy.
authors.forename: M
authors.initials: M
authors.lastname: Zanella
keywords: Boltzmann equation; collective behaviour; consensus dynamic; multi-agent systems; opinion leader; optimal control