A CUCKER-SMALE Model With Noise And Variable Time-delays

The Cucker-Smale(C-S)model and its extensions,as well as the related mathematical problem have received more and more attentions in recent years.One of the most important extensions is to consider time delay,in which the model is described by a stochastic delay differential equation.This thesis concentrates on C-S model with time delay and mainly considers almost surely stability of the C-S model with variable delay and multi-dimensional white noise by two methods.The first method is from [41],but Mao's methods cannot be used directly since the C-S model cannot be expressed by the common stochastic delay differential equation.In this thesis,we make some changes for this C-S model by the following steps.First,every component minus their average and the new vector is obtained.Second,when this new vector is multiplied by the Laplacian matrix from the left then a common stochastic differential equation is obtained.By using Mao's methods,this thesis obtains the following results:(1)when the time delay is very small,every component will converge to the average speed;(2)The upper bound of the time delay is obtained.The second method is from [30].By applying a theorem in the report,we conclude that the solutions of the C-S model with fixed time delay,are mean square consensus and almost surely consensus.