In a variety of problems in engineering and applied science, the goal is to design or control a network of dynamical agents so as to achieve some desired asymptotic behaviour. Examples include consensus and rendez-vous problems in robotics, synchronization of generator angles in power grids or coordination of oscillations in bacterial populations. A pressing challenge in all of these problems is to derive appropriate analytical tools to prove convergence towards the target behaviour. Such tools are not only invaluable to guarantee the desired performance, but can also provide important guidelines for the design of decentralized control strategies to steer the collective behaviour of the network of interest in a desired manner. During this talk, a methodology for analysis and design of convergence in networks will be presented which is based on the use of a classical, yet not fully exploited, tool for convergence analysis: contraction theory. As opposed to classical methods for stability analysis, the idea is to look at convergence between trajectories of a system of interest rather that at their asymptotic convergence towards some solution of interest. After introducing the problem, a methodology will be derived based on the use of matrix measures induced by non-Euclidean norms that will be exploited to design strategies to control the collective behaviour of networks of dynamical agents. Representative examples will be used to illustrate the theoretical results.
