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The study of collective behavior in multi-agent systems has attracted theattention of many researchers due to its wide range of applications. Amongthem, the Cucker-Smale model was developed to study the phenomenon of flocking,and various types of extended models have been actively proposed and studied inrecent decades.  In this study, we address open questions of the Cucker--Smale model withnorm-type Rayleigh friction: {\bf (i)} The positivity of the communicationweight, {\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 Rayleighfriction, where the nonlinear Cucker--Smale model is generalized to a nonlinearmodel by applying a discrete $p$-Laplacian operator. For this model, we presentconditions that guarantee that the norm for velocities of agents converges to 0or a positive value, and we also show that the regular communication weightsatisfies the conditions given in this study. In particular, we present acondition for the initial configuration to obtain that the norm of agentvelocities 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 Rayleighfriction for problem (iii). In parallel to the first model, we show that thedirection of the agents' velocities can be controlled by parameters in thenonlinear Cucker--Smale model with the vector-type Rayleigh friction.

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