Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

Abstract

This chapter reviews several approaches to study convergence of networks of nonlinear dynamical systems based on the use of contraction theory. Rather than studying the properties of the collective asymptotic solution of interest, the strategy focuses on finding sufficient conditions for any pair of trajectories of two agents in the network to converge towards each other. The key tool is the study, in an appropriate metric, of the matrix measure of the agents' or network Jacobian. The effectiveness of the proposed approach is illustrated via a set of representative examples. IntroductionThe problem of steering the collective behaviour of a network of dynamical agents towards a desired common target solution is a fundamental problem in network science and control theory [1][2][3]. A typical problem is that of achieving consensus or synchronization in a network of linear or nonlinear systems. Here, the key challenge, once the coupling function among the nodes has been selected, is to prove convergence of all nodes towards the desired common asymptotic behaviour. This convergence problem is usually solved locally by means of the Master Stability Function Approach (MSF) [4] or globally via Lyapunov stability theory [5].Global convergence is extremely useful particularly when the agents are subject to high level of noise or perturbations as it is often the case in applications; think for example of the recent application of synchronization to biological systems [6].

Qualitative Control Strategies for Synchronization of Bistable Gene Regulatory Networks

“…Inspired by works in the smooth case (see, for instance [26], [12]), we conjecture that under Condition (A), for every α ≥ α s , the solution x α (t) of Equation ( 16) satisfies x α (t) → Φ 10 or Φ 01 when t → +∞.…”

Non-Euclidean Contractivity of Recurrent Neural Networks

“…It is known that the corresponding matrix measure is then µ p,R (A) = µ p (RAR −1 ). For diagonally weighted 1 , ∞ , and 2 norms, it is known [1], [10] that…”

Distributed and time-varying primal-dual dynamics via contraction analysis

“…An introduction and survey on contraction theory can be found in [2]. Different variants of contraction theory exists, and, in particular, partial contraction [34,14], used for the study of convergence to linear subspaces, has proven to be useful in the synchronization analysis of diffusively-coupled network systems [34,12,3]; however, its study in applications related to distributed algorithms is still missing, and our paper provides such contribution.…”

“…Corollary 4.3 (Contraction analysis of the augmented primal-dual dynamics). Consider the constrained optimization problem (3), its standing assumptions, and its associated augmented primaldual dynamics (14) with ρ > 0.…”