Successive Lag Synchronization On Complex Dynamical Networks Via Intermittent Control

The meaning of achieving successive lag synchronization is that for a network whose nodes are numbered in advance,the first node reaches a certain state,and then the second node also reaches this state after a certain time interval,up to the last node.That is highly similar to the phenomenon of vehicles passing through tunnel in an orderly manner and aircraft queues passing through certain locations during parade.In the early research,the control methods of successive lag synchronization were mostly continuous control.As we know,continuous control takes more time than intermittent control,and time means cost,so successive lag synchronization on dynamical networks under continuous control may be lavish.In order to reduce control cost,this paper uses intermittent control strategy to realize successive lag synchronization.Moreover,it is noted that successive lag synchronization is a generalized form of complete synchronization.To realize complete synchronization on complex dynamical networks,this paper proposes an intermittently sampled-data pinning control strategy based on the intermittent control strategy.The main results are as follows:Successive lag synchronization on nonlinear dynamical networks by using aperiodically intermittent control is investigated.Different from previous works about the successive lag synchronization,the proposed controllers could be discontinuous and aperiodic.Aperiodically intermittent controllers are proposed to realize successive lag synchronization on the dynamical networks with and without communication delay,and the coupling functions are designed to guarantee the invariance of synchronization manifold.Furthermore,several sufficient conditions are obtained by applying the Lyapunov function method to make successive lag synchronization achieve global stability.Finally,a Loop-shaped network example and a Chain-shaped network example are provided to verify correctness of theoretical results.Due to complete synchronization is a typical synchronization pattern,this paper studies the problem of intermittent control strategy to achieve complete synchronization.As we know,the well-studied intermittent control,sampled-data control and pinning control are three effective and economical control methods for synchronization on complex dynamical networks.However,how to integrate them into a control method to get simpler and more applicable synchronization conditions is still a challenging topic.In view of this,in this paper,an intermittently sampled-data pinning control is firstly developed for achieving the synchronization on complex dynamical networks.The intermittently sampled-data pinning control strategy is applied to solve simultaneously three difficulties:(i)the process of receiving feedback signal by controllers may be discontinuous;(ii)the controllers are difficult to keep working continuously in practical applications;(iii)it is usually hard to control all nodes in a dynamical network as the network size is very large.Secondly,some sufficient conditions for realizing synchronization on dynamical networks are obtained,and then time delay is taken into account when the feedback control signal is received untimely.Compared with previous results,these conditions seem simpler and more applicable as they only relate to matrix's spectral radius.Finally,two numerical examples are presented to demonstrate the reasonableness and effectiveness of the control method.