Pattern formation of Cucker–Smale system with nonlinear velocity couplings

In this paper, we propose a novel Cucker–Smale system with nonlinear velocity couplings and a targeted driving force. Firstly, by imposing assumptions on the initial state, the system can achieve flocking behavior. Secondly, the collision avoidance results under various velocity coupling degrees are deduced by applying the triangle inequality. Thirdly, by applying the Barb$\breve{a}$lat’s lemma, all agents eventually reach the prescribed line-shaped formation with a targeted driving force. In particular, for $1/2<\alpha <1$ and $1<\beta <3/2$, the finite-time and fixed-time line-shaped formation can be successfully achieved without the symbolic function, and an upper bound on the settling time is obtained. Finally, the results of the theoretical analysis are verified by numerical simulations, and the final formation’s trajectory equation is derived by using the least squares method.