The Cucker–Smale Model With Nonlinear Velocity Couplings and Power-Law Attractive Potentials

When the communication weight has a short range, the distance between agents could tend to infinity for some initial conditions, which leads to the nonexistence of well-studied flocking behavior in the Cucker–Smale model with (or without) nonlinear velocity couplings. To control the distance, it is natural to consider this model with pairwise attractive potentials. Therefore, this note investigates the Cucker–Smale model with nonlinear velocity couplings and attractive potentials. When the attractive potential is any power-law function, it is proved that this model not only exhibits flocking but also achieves consensus regardless of initial conditions. More importantly, by constructing two delicate Lyapunov functions, the precise convergence rates (including polynomial convergence, exponential convergence, finite time consensus, and consensus independent of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> ) are derived for different kinds of nonlinear velocity couplings and power-law potentials.