Optimal leader-following consensus of fractional opinion formation models

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.publishe

The Phillip island penguin parade (A mathematical treatment)

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters", such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade

The flux measure of influence in engineering networks

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-115).The objective of this project is to characterize the influence of individual nodes in complex networks. The flux metric developed here achieves this goal by considering the difference between the weighted outdegree and indegree of a node. This technique differentiates among nodes that traditional centrality measures treat as identical units. The behavior and proper interpretation of the flux metric are demonstrated on a variety of weighted and directed networks. Simulations of fluid flow, opinion sharing, epidemic dynamics, and resource allocation reveal the practical capabilities of the flux metric. An engineering design challenge may also be framed as a network analysis problem so that the the flux metric contributes to understanding the relationships among the system's subcomponents and objectives. A case study that investigates the design of autonomous underwater vehicles (AUVs) for use in the offshore oil and gas industry demonstrates these insights. In all of the applications explored here, the flux metric consistently emerges as a reliable indicator of the influence of a node.by Kyle Michael Schwing.S.M