Research On Asymptotic Flocking Behavior Of The Hybrid Cucker-Smale Model

In recent years,the Cucker-Smale model has received a lot of attention and re-search from researchers,and one of the most studied questions is about the flocking of this model.Most of the available research results are about continuous-time dynamic model and discrete-time dynamic model.However,in the real situation,natural and artificial individuals can achieve flocking by working together.So,in this paper,we focus on the asymptotic flocking problem for the hybrid Cucker-Smale model,which has not been studied before,and the results can be summarized as follows:(1)Asymptotic behavior of a hybrid Cucker-Smale model with symmetric inter-action information.The hybrid Cucker-Smale model is proposed based on the existing Cucker-Smale model and the hybrid multi-agent systems.Then,the definition of its asymptotic flocking is given based on this hybrid model.based on which we give the definition of its asymptotic flocking.The super-linear inequality for the derivative of the velocity is obtained by the derivative of the velocity variance and the associated auxiliary lemma.From this inequality we can construct sufficient conditions to make the hybrid model asymptotically flocking,and lead to our main theorem.Finally,we verify the validity of the theorem by using some simulation examples.(2)Asymptotic behavior study of the hybrid Cucker-Smale model with normalized communication weight.Still the hybrid Cucker-Smale model is considered,but here we use the normalized communication weight function.This approach overcomes the shortcomings of the normalized pre-factor_N~1and better reflects the interaction effects among the agents,but the setup breaks the symmetry of the model.Therefore,in order to be able to conclude that the hybrid model is asymptotically flocking,we need to establish a super-linear inequality on the velocity by means of a categorical discussion and the L∞method.Then,a sufficient condition for the asymptotic flocking of the model is obtained from this inequality.Finally,the validity of the results is verified by two examples.