Topics on the Average Consensus Problems

Driven by fundamental technological achievements like the digital revolution in communication and computing and the miniaturization of electronic components, a growing interest in the field of networked systems has appeared, in recent years, in different scientific areas like physics, communication and control engineering, economy, and lately, mathematics. Besides differences among the various approaches, the basic model consists of a certain number of agents, namely, systems living in a common environment and which communicate among each other according to some pre-specified communication pattern. Such a pattern may be fixed or varying according to their physical position, to their internal state and can possibly be affected by noises. The evolution law of each system typically depends on the information obtained through its communication links and is, in general, modeled by a difference or differential equation. For this type of models, the interest is, in general, in studying time evolution, asymptotic behaviors, and in formulating and solving related control problem. In particular, a fundamental issue is understanding the emerging of a group behavior from the individual dynamics and the communication pattern. While many models have already been proposed, the mechanisms which determine how the various individual actions get reflect in the group behavior are, in general, hard to understand and, up to now, few results are known. In this thesis we will focus on the so-called consensus problem, where the group of agents has to reach an agreement on key pieces of information or on a common decision (represented by scalar or vector values) that enable them to cooperate in a coordinate fashion. We will consider a standard algorithm proposed in literature to solve this problem. We will provide some theoretical developments by • characterizing the speed of convergence of this algorithm for particular communication pattern exhibiting symmetries, • investigating the realistic and practical situation in which the systems can communicate each other only through digital channels and hence can exchange only quantized information, • proposing the application of this algorithm to a problem of distributed estimation.