Optimal consensus and opinion dynamics

Abstract: In the following thesis we study the influence of the communication graph on the behavior of multi-agent systems. Specifically we investigate two issues the first is concerned with the existence of consensus control for linear dynamics, the second is a study of the behavior of a nonlinear dynamical related to opinion dynamics. For the finite optimal consensus problem of multi-agent system, we formulate the problem as an optimization problem on a Hilbert space to model the graph neighborhood constraints. Then we show that completeness of the graph is a necessary and sufficient condition of the existence of a finite time linear control that guarantees consensus in finite time. As a extension of this result we show that the optimal control we get is also optimal among the larger class of non linear control, and that it can be implemented as optimal for connected but not complete graphs if we replace the neighborhood restriction by a feedback control using the information of all edges of the graph. The second part is a study of a modified version of continuous opinion dynamic model by introduced Hegselmann an Krause. To modify the model we introduce stubborn agents; agent whose opinion do not change over time, specifically we introduce two types of agents: one that can influence the whole distribution at ones and we call it of positive influence and the other with a bounded influence and we call it of non negative influence. For each type introduced we study the topological properties of the distribution and the clustering phenomena observed but also the statistical properties and we do so in the presence of one or two stubborn agent. We finally end this part by two possible applications of the use of stubborn agents for reaching consensus or tracking trajectories.