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This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

This paper investigates the matrix projective synchronization (MPS) problem for time-varying disturbed complex dynamical networks (CDNs) with uncertain nonlinear structures and different dimensional nodes. In order to well describe the practical networks, the unavoidable unknown nonlinear structure and the inevitable uncertain disturbance of each node are considered into our network model, which is different from the previous works. Besides, it is worth pointing out that the outer coupling configuration matrix, which represents the coupling strength and the topological structure, is not restricted by the dissipatively coupled condition in this paper. The definition of MPS is also introduced for the networks with different dimensional nodes. Moreover, several MPS schemes are respectively put forward for our network model according to the norm bound of the uncertain nonlinear structures and disturbances being unknown or not. Finally, two proper examples associated with numerical simulations are given to verify the effectiveness of our theoretical results.