Pinning control of successive lag synchronization on a dynamical network with noise perturbation

In recent years, the successive lag synchronization (SLS) has been deeply analyzed, however, the control theory of SLS on complex networks with noise perturbation is lacking. To this end, this paper focuses on the pinning control of SLS on a dynamical network with noise perturbation. Both the constant pinning control law and adaptive pinning control law are designed respectively to push the network to achieve the desired SLS. By utilizing the Lyapunov stability theory of stochastic differential equations, several sufficient conditions for the controlled networks to achieve the SLS are obtained. The influences of network structure, noise strength and coupling strength on the realization of SLS are also discussed. We find that the noise can suppress or even destroy the realization of SLS. Meanwhile, we attain the index set of pinned nodes, which can tell us whether a node should be controlled or not. In particular, those nodes with zero in-degree must be controlled to achieve SLS, while those nodes with non-zero in-degree need not to be controlled if the coupling strength is large enough. All theoretical results are verified by numerical simulations.