Necessary and Sufficient Conditions for Finite-Time and Fixed-Time Bipartite Containment Control of Cooperative–Antagonistic Networks

This article addresses the problems of finite-time and fixed-time bipartite containment control for single-integrator cooperative-antagonistic networks with multiple leaders over arbitrary weakly connected signed digraphs. The leaders can be either stationary or evolving dynamically through interacting with other leaders in their neighborhood. Based on the nearest neighbor rule, a unified distributed nonlinear control scheme is proposed. With rigorous Lyapunov-based analysis, it is shown that under the proposed control scheme, all followers can converge toward the convex hull formed by each leader's trajectory and its symmetrical trajectory having the same amplitude but different sign in a finite time or fixed time if and only if the underlying weakly connected signed digraph contains at least one structurally balanced closed strong component. As a byproduct, necessary and sufficient criteria are established for single-integrator cooperative-antagonistic networks to achieve finite-time and fixed-time bipartite consensuses, interval bipartite consensus, and stability. The effectiveness of the presented control scheme is illustrated via simulations.