Optimal leader-following consensus of fractional opinion formation models

Abstract

This paper deals with a control strategy enforcing consensus in a fractional opinion formation model with leadership, where the interaction rates between followers and the influence rate of the leader are functions of deviations of opinions between agents. The fractional-order derivative determines the impact of the memory during the opinion evolution. The problem of leader-following consensus control is cast in the framework of nonlinear optimal control theory. We study a finite horizon optimal control problem, in which deviations of opinions between agents and with respect to the leader are penalized along with the control that is applied only to the leader. The existence conditions for optimal consensus control are proved and necessary optimality conditions for the considered problem are derived. The results of the paper are illustrated by some examples.