Abstract
We focus on the cluster synchronization problem for a kind of general networks with nondelayed and delayed coupling. Based on the pinning control scheme, a small fraction of the nodes in each cluster are pinned for successful control, and the states of the whole dynamical networks can be globally forced to the objective cluster states. Sufficient conditions are derived to guarantee the realization of the cluster synchronization pattern for all initial values by means of the Lyapunov stability theorem and linear matrix inequalities (LMIs). By using the adaptive update law, relative smaller control gains are obtained, and hence the control cost can be substantially lower. Numerical simulations are also exploited to demonstrate the effectiveness and validity of the main result.
Highlights
In the past few decades, synchronization and control problems are being widely studied [1,2,3,4,5,6,7,8,9]
This paper considered a class of complex dynamical networks with time nondelayed and delayed coupling
Instead of controlling all nodes, pinning adaptive control was used to pin a small fraction of the nodes in each cluster for successfully control, and the states of the whole dynamical networks can be globally exponentially forced to the objective cluster states
Summary
In the past few decades, synchronization and control problems are being widely studied [1,2,3,4,5,6,7,8,9]. In many circumstances, this simplification does not match satisfactorily the peculiarities of real networks: there exists the information communication of nodes at time t or t − τ and at time t, t − τ This general complex dynamical network model with time nondelayed and delayed coupling was investigated in [14, 20,21,22,23] and so forth. In [21], Xu and Su studied the general complex dynamical networks for synchronization and pinning control, they obtained less conservative criteria for both continuous-time and discrete-time complex dynamical networks with time delayed and nondelayed coupling by linearizing the system, and their results were established locally. The dimension of these vectors and matrices will be cleared in the context
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