Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

Kinetic description of optimal control problems and applications to opinion consensus

Pages: 1407 – 1429



Giacomo Albi (Department of Mathematics and Computer Science, University of Ferrara, Italy)

Michael Herty (Department of Mathematics, RWTH Aachen University, Aachen, Germany)

Lorenzo Pareschi (Department of Mathematics and Computer Science, University of Ferrara, Italy)


In this paper, an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model, we analyze a microscopic model of opinion formation under constraints. For this problem, a Boltzmann–type equation based on a model predictive control formulation is introduced and discussed. In particular, the receding horizon strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker–Planck asymptotic limit is also derived and explicit expressions of stationary solutions are given. Several numerical results showing the robustness of the present approach are finally reported.


Boltzmann equation, optimal control, consensus modeling, model predictive control, collective behavior, mean-field limit, simulation methods

2010 Mathematics Subject Classification

49-xx, 65C05, 76P05, 91C20

Published 13 May 2015