## The asymptotic consensus problem on convex metric spaces

**Title**: The asymptotic consensus problem on convex metric spaces

**Authors**:

Baras, John, S.

**Conference**: 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys'10) pp. 287- 292

**Date**: September 13 - September 14, 2010

We consider the consensus problem of a group of dynamic agents whose communication network is modeled by a directed time-varying graph. In this paper we generalize the asymptotic consensus problem to convex metric spaces. A convex metric space is a metric space endowed with a convex structure. Using this convex structure we define convex sets and in particular the convex hull of a (finite) set. Under minimal connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the convex hull generated by the agentâ€™s current state and the states of its neighbors, then asymptotic agreement is achieved. In addition, we give bounds on the distance between the consensus point and the initial values of the agents.