Finite flocking time of the nonlinear Cucker--Smale model with Rayleigh friction type using the discrete $p$-Laplacian

The study of collective behavior in multi-agent systems has attracted the attention of many researchers due to its wide range of applications. Among them, the Cucker-Smale model was developed to study the phenomenon of flocking, and various types of extended models have been actively proposed and studied in recent decades. In this study, we address open questions of the Cucker--Smale model with norm-type Rayleigh friction: {\bf (i)} The positivity of the communication weight, {\bf (ii)} The convergence of the norm of the velocities of agents, {\bf (iii)} The direction of the velocities of agents. For problems (i) and (ii), we present the nonlinear Cucker--Smale model with norm-type Rayleigh friction, where the nonlinear Cucker--Smale model is generalized to a nonlinear model by applying a discrete $p$-Laplacian operator. For this model, we present conditions that guarantee that the norm for velocities of agents converges to 0 or a positive value, and we also show that the regular communication weight satisfies the conditions given in this study. In particular, we present a condition for the initial configuration to obtain that the norm of agent velocities converges to only some positive value. By contrast, problem (iii) is not solved by the norm-type nonlinear model. Thus, we propose a nonlinear Cucker--Smale model with a vector-type Rayleigh friction for problem (iii). In parallel to the first model, we show that the direction of the agents' velocities can be controlled by parameters in the nonlinear Cucker--Smale model with the vector-type Rayleigh friction.Comment: 26 page