Bipartite Consensus for Nonlinear Networked Systems Based on General Lyapunov Functions

This article investigates the bipartite consensus problem for nonlinear networked systems with cooperative-competitive interactions subject to the nonlinear control protocols. Based on general Lyapunov functions instead of quadratic form, a bipartite consensus criterion is established under the connected signed digraph condition. Through the sum-of-squares decomposition for multivariate polynomials, the consensus problem for polynomial systems is transformed into a sum-of-squares programming problem, yielding polynomial Lyapunov functions effectively by our proposed algorithm to realize consensus verification. Moreover, the proposed method is extended to address the bipartite consensus for nonlinear networked systems with the signed digraph having only a directed spanning tree. Finally, the theoretical and algorithmic developments are demonstrated on two numerical examples, where the widely used handcrafted quadratic Lyapunov functions may be nonexistent.