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.

Recent Advances in Opinion Modeling: Control and Social Influence

“…In general this leads to a complicated nonlinear system [7], however in the simplified situation of P and R symmetric and S ≡ 1 we obtain the following closed system of differential equations for the mean opinions…”

“…A similar analysis can be performed for the second moment ϕ(w) = w 2 , showing the decay of the energy for particular choices of the interaction potential (cf. [99,6,7] for further details on the proprieties of the moment of (1.4)).…”

“…To derive the system of kinetic equation we introduce a density distribution of followers fF (w, t) and leaders fL(v, t) depending on the opinion variables w, v ∈ I and time t ≥ 0, see [7,57]. It is assumed that the densities of the followers and the leaders satisfy is…”

“…Note that, since the optimal control problem acts only over the leader dynamics we can approximate its solution by a model predictive control approximation as in Section 2.1. Next we can build the corresponding constrained binary Boltzmann dynamic following [7].…”

“…Two relevant situations will be explored, first a distributed control, which models the action of an external force acting as a policy maker, like the effects of the media [6], next an indirect internal control, where we assume that the control corresponds to the strategies of opinion leaders, aiming to influence the consensus of the whole population [7]. In order to characterize the kinetic structure of the optimal control of consensus dynamics, we will start to derive it as a feedback control from a general optimal control problem for the corresponding microscopic model, and thereafter we will connect it to the binary dynamics.…”