Boltzmann type opinion consensus
26
Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Boltzmann type control of opinion consensus Mattia Zanella Department of Mathematics and Computer Science, University of Ferrara, Italy Joint research with: G. Albi (Munich, Germany) L. Pareschi (Ferrara, Italy) XXXIX Summer School on Mathematical Physics Ravello, September 15-27, 2014 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 1 / 26
-
Upload
mattia-zanella -
Category
Education
-
view
118 -
download
0
description
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.
Transcript of Boltzmann type opinion consensus
- 1. Boltzmann typecontrol of opinionconsensusMattia ZanellaThe BoltzmannEquationComplexityreductionSHBEMMConstrainedself-organizedsystemsOpinion controlthrough leadersTheBoltzmann-typeoptimal controlFokker-PlanckModelingNumerical resultsTest 1Test 2aTest 2bConclusionsBibliographyBoltzmann type control of opinion consensusMattia ZanellaDepartment of Mathematics and Computer Science,University of Ferrara, ItalyJoint research with:G. Albi (Munich, Germany) L. Pareschi (Ferrara, Italy)XXXIX Summer School on Mathematical PhysicsRavello, September 15-27, 2014Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 1 / 26
2. Boltzmann typecontrol of opinionconsensusMattia ZanellaThe BoltzmannEquationComplexityreductionSHBEMMConstrainedself-organizedsystemsOpinion controlthrough leadersTheBoltzmann-typeoptimal controlFokker-PlanckModelingNumerical resultsTest 1Test 2aTest 2bConclusionsBibliographySketch of the presentation1 The Boltzmann EquationComplexity reductionSHBEMM2 Constrained self-organized systemsOpinion control through leadersThe Boltzmann-type optimal control3 Fokker-Planck Modeling4 Numerical resultsTest 1Test 2aTest 2b5 Conclusions6 BibliographyMattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 2 / 26 3. Boltzmann typecontrol of opinionconsensusMattia ZanellaThe BoltzmannEquationComplexityreductionSHBEMMConstrainedself-organizedsystemsOpinion controlthrough leadersTheBoltzmann-typeoptimal controlFokker-PlanckModelingNumerical resultsTest 1Test 2aTest 2bConclusionsBibliographyThe Boltzmann EquationLet us consider DR3 open, limited and regular, and we considerx 2 D and v 2 R3 Then for t 2 [0; T] the Boltzmann Equation is@f(x; v; t)@t+ vrxf(x; v; t) = Q(f; f)f(x; v; 0) = f0(x; v);where we interpret f : DR3[0;+1) ! R+ as a probabilitydensity function and where we de 4. ned the collision operatorQ(f; f)(x; v; t) =ZR3S2[f(x; v; t)f(x;w; t)
