Convergence Problems on Second-Order Signed Networks: A Lyapunov-Based Analysis Approach

This article focuses on exploring a class of Lyapunov analysis approaches to deal with the convergence problems on second-order signed networks (SOSNs) under arbitrary strongly connected directed topologies. A new class of Laplacian potentials is first proposed for SOSNs by exploiting the properties of Laplacian matrices. Then the relation between convergence behaviors of all agents and those of Laplacian potentials can be disclosed, which makes it possible to solve the convergence issues of SOSNs from the viewpoint of the Lyapunov stability theory even though the weight-balanced condition is not satisfied. Furthermore, the proposed Laplacian potential can be leveraged to deal with the convergence problems of distributed averaging for SOSNs. It is shown that the signed-average consensus objective is reached under the structurally balanced signed digraphs by designing a distributed control protocol according to the proposed Laplacian potential. Additionally, simulation examples are given to demonstrate the validity of our potential-based distributed control results.