Abstract
In this Letter, we investigate the exponential synchronization problem for an array of N linearly coupled complex networks with Markovian jump and mixed time-delays. The complex network consists of m modes and the network switches from one mode to another according to a Markovian chain with known transition probability. The mixed time-delays are composed of discrete and distributed delays, both of which are mode-dependent. The nonlinearities imbedded with the complex networks are assumed to satisfy the sector condition that is more general than the commonly used Lipschitz condition. By making use of the Kronecker product and the stochastic analysis tool, we propose a novel Lyapunov–Krasovskii functional suitable for handling distributed delays and then show that the addressed synchronization problem is solvable if a set of linear matrix inequalities (LMIs) are feasible. Therefore, a unified LMI approach is developed to establish sufficient conditions for the coupled complex network to be globally exponentially synchronized in the mean square. Note that the LMIs can be easily solved by using the Matlab LMI toolbox and no tuning of parameters is required. A simulation example is provided to demonstrate the usefulness of the main results obtained.
Highlights
The last decade has witnessed rapidly growing research interests on the dynamics analysis of complex networks since the pioneering work of Watts and Strogatz [1]
Synchronization has proven to be one of the most important controlling activities to excite the collective behavior of complex dynamical networks, and has received increasing research attention in, for example, the large-scale and complex networks of chaotic oscillators [2,3,4,5], the coupled systems exhibiting spatio-temporal chaos and autowaves [6,7], and the array of coupled neural networks [8,9,10]
By utilizing a novel Lyapunov–Krasovskii functional and the Kronecker product, we show that the addressed synchronization problem is solvable if a set of linear matrix inequalities (LMIs) are feasible
Summary
The last decade has witnessed rapidly growing research interests on the dynamics analysis of complex networks since the pioneering work of Watts and Strogatz [1]. Synchronization problems for various networks with discrete and/or distributed time-delays have extensively studied, see e.g. In this Letter, we deal with the synchronization problem for an array of coupled complex networks with simultaneous presence of both the discrete and distributed time-delays. The main novelty of this Letter can be summarized as follows: (1) a new class of complex networks is proposed that contain Markovian jumping parameters; (2) both the discrete and distributed time-delays are considered that are dependent on the jumping mode; (3) rather than the commonly used Lipschitz-type function, a more general sector-like nonlinear function is employed; and (4) a new Lyapunov–Krasovskii functional is exploited to cater the mode-dependent distributed delays. The arguments of a function will be omitted in the analysis when no confusion can arise
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