Research On Flocking Behavior Of Cucker-Smale Models With Leader

Flocking behavior refers to the orderly flock motion formed by a large number of particles through interaction.This behavior is a specific description of the behavior of biological groups by researchers in the fields of biology,control,and mathematics.Recently,the study of flock behavior has attracted the attention of many scholars at home and abroad.Most of the flock systems adopt asymptotic convergence,and the convergence time tends to infinity.Therefore,the convergence speed of the particle system has become an important indicator to measure the performance of the system.The Cucker-Smale system constructs a self-propelled particle system by identifying birds and fish schools to achieve flocking.That is to say,the method of adding control items to optimize the flock motion of the particle system.However,particles will inevitably be disturbed by the outside world in the process of information transmission within a limited time.In order to study the stability of the real system,this thesis first introduces the basic concepts of flocks,ordinary differential equations and other related knowledge,and then introduces the main research content of this thesis.Details as follows:Firstly,to determine the flocking behavior of the Cucker-Smale model with a leader in a limited time,a Cucker-Smale mathematical mechanism with two nonlinear control terms is established.Based on the finite-time stability theory,matrix theory,algebraic graph theory and Lyapunov function are used to prove the rationality of the sufficient conditions for the flocking of particles in the theorem.Numerical experiments verify the correctness of the obtained results.Secondly,aiming at the flocking behavior of the random Cucker-Smale model with a leader in a finite time,the method of adding noise terms and nonlinear control terms to the deterministic model is adopted,which overcomes the difficulty of requiring a long time for particles to flock.Using the finite-time stability theory of stochastic differential equations,comparison theorem,Lyapunov function method and algebraic graph theory,etc.,the rationality of the sufficient conditions for the flocking of particles in the given theorem is proved.Finally,a numerical simulation of the particle system is carried out to verify the correctness and reliability of the theoretical results.