Group System Of Collective Behavior And Stability

Swarming behavior is ubiquitous in nature and society. It is of both theoretical and practical importance to investigate mechanism of coordination and cooperation emerging in swarms. Firstly, we considered a swarm of M individuals in an n-dimensional Euclidean space, and established a general swarm model with time delays under disturbances for the quadratic attractant/repellant profiles. It is proved that the swarm members will converge and form a cohesive cluster around the center in a finite time under certain conditions in the presence of communication delays and disturbances. For multi-modal Quadratic attractant/repellant profiles and multi-modal Gaussian attractant/repellant profiles, all the swarm members will converge to more favorable areas in the presence of noise disturbances.Secondly, we considered the consensus problem of the multiple agents control for a discrete-time case under fixed information topology. We considered a swarm of N+1 individuals in an Euclidean space, constructed a second-order discrete model, designed the control laws based on the neighbor to solve the consensus problem of multi-agent systems. We proved that each follower can track the active leader. And we studied the effect of disturbances on the system.Finally, we studied the complex network with time delays pinning synchronization. We considered a swarm of N individuals in an Euclidean space, constructed pinning synchronization model, through adaptive parameters control group, so as to achieve a small individual control of the entire group effect. We proved that the system can achieve synchronization under the local control law. During the study, we applied the graph theory and Lyapunov function related theory. Numerical simulations illustrate the theoretical results.