Asymptotic analysis of the linear formation model with an undirected connected topology

This paper introduces linear formation into an undirected connected Cucker–Smale model. Firstly, the projection system corresponding to the original system is established, and the mutual control relationship between the displacement difference between the original system and the projection system is given. Secondly, the boundedness of the projection system’s displacement is derived by using graph theory and matrix theory, and the convergence of the system speed is given by using Lyapunov stability theory. The research results indicate that under certain conditions, for any given direction, the multi-agent system can asymptotically converge to a flock and form a straight line in the presented direction. Moreover, the velocity of the agent converges to the average of the initial velocity. At last, the validity of the results is verified by numerical simulations.