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.
Highlights
In many real-world systems, we observe that a number of individuals achieve a common objective without any central decision maker
Necessary optimality conditions for the optimal control problem governed by a fractional opinion formation model with leadership are derived
We have proved the existence conditions for a globally optimal solution and the necessary optimality conditions for the optimal control problem P
Summary
In many real-world systems, we observe that a number of individuals (animal or human) achieve a common objective without any central decision maker. Individuals ( called agents) only through local interactions and computations reach an agreement upon a common state, referred to as consensus Examples of such collective behavior include flocking of birds, swarming of bees and schooling of fish [1,2]. It turned out that fractional operators are excellent tools for modeling the behavior of human beings, especially psychological processes that depend on the experience in the past [22,23] This brings us to consider opinion formation models in the framework of fractional calculus. Necessary optimality conditions for the optimal control problem governed by a fractional opinion formation model with leadership are derived. Those conditions ensure optimal leader-following consensus of considered fractional opinion formation model.
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