What is the Optimal Way to Lie? From Microscopic to Kinetic Descriptions of Consensus Control

We establish an approach for consensus control of opinion dynamics by introducing a liar to the classical system. The liar's aim is to steer the population toward consensus at their goal opinion by showing 'apparent opinions', or 'lies', to members of the population. We analyse this as an optimal control problem for how best to lie to a population in order to guarantee the consensus that the liar desires. We consider a range of regularisations, each motivated by some social convention, such as the liar wanting to present an opinion close to their goal opinion. For each regularisation, we demonstrate the effect of instantaneous controls. Furthermore, we introduce a Boltzmann-type description for the corresponding kinetic system and present analysis and numerical results for the resulting Boltzmann and Fokker-Planck equations.