Graph Laplacian Potential and Lyapunov Functions for Multi-Agent Systems

Abstract

In this chapter we show that for networked multi-agent systems, there is an energy-like function, called the graph Laplacian potential, that depends on the communication graph topology. The Laplacian potential captures the notion of a virtual potential energy stored in the graph. We shall study the Laplacian potential for both undirected graphs and directed graphs. The Laplacian potential is further used here to construct Lyapunov functions that are suitable for the analysis of cooperative control systems on graphs. These Lyapunov functions depend on the graph topology, and based on them a Lyapunov analysis technique is introduced for cooperative multi-agent systems on graphs. Control protocols coming from such Lyapunov functions are distributed in form, depending only on information about the agent and its neighbors.