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Exponential synchronization of complex networks with Markovian jump and mixed delays. (English) Zbl 1220.90040
Summary: 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-Krasovskiĭ 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.
MSC:
 90B15 Network models, stochastic (optimization) 34F05 ODE with randomness 34K20 Stability theory of functional-differential equations 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 93E15 Stochastic stability
Software:
LMI toolbox; Matlab